HANDLING COMPLEXITY VIA STATISTICAL METHODS A ...

HANDLING COMPLEXITY VIA STATISTICAL METHODS

A Dissertation

Submitted to the Faculty

Purdue University

Evidence S. Matangi

In Partial Fulfillment of the

Requirements for the Degree

Doctor of Philosophy

December 2019

Purdue University

West Lafayette, Indiana

THE PURDUE UNIVERSITY GRADUATE SCHOOL

STATEMENT OF DISSERTATION APPROVAL

Prof. George P. McCabe

Department of Statistics

Prof. Nilupa Gunaratna

Department of Nutrition Science

Prof. Alexander Gluhovsky

Department of Statistics

Prof. Hao Zhang, Head

Department of Statistics

Approved by:

Prof. Jun Xie

Head of the School Graduate Program

iii

I dedicate this thesis to my sweetheart, wife, friend and confidant Jessey,

and wonderful three (TAM) Arlene, Ardele, and Anele

for their invaluable love and support.

ACKNOWLEDGMENTS

Firstly, I would like to express my sincere gratitude to my co-advisors Prof. George

P. McCabe and Prof. Nilupa Gunaratna for the continuous support of my Ph.D study,

for their patience, motivation, persistence and immense knowledge. Their guidance

helped me in all the time of research and writing of this thesis. I could not have

imagined having better advisors and mentors for my Ph.D study.

Besides my co-advisors, I would like to thank the rest of my thesis committee:

Prof. Alexander Gluhovsky, and Prof. Hao Zhang, for their insightful comments and

encouragement, but also for their incise questions which helped me widen my research

concepts.

My sincere thanks also goes to the Food Agriculture and Natural Resources Policy

Analysis Network (FANRPAN), especially their ATONU team who provided me an

opportunity to work with their data for my thesis. Without their precious support it

would not be possible to conduct part of this research.

Hats off!!! To my Statistics department cohort for your unwavering support and

understanding of me and my family and we journeyed through the academic terrain

as Boilermakers. Forever Boilermakers, Go too far!!!

I am also grateful to my sponsors, Fulbright who provided me with such an amaz-

ing opportunity to study in USA, Statistical Consulting Services, who not only spon-

sored but also equipped me for statistical collaboration and consulting work, and

the Gunaratna lab, Purdue summer grant, and the department of statistics. The

sponsorship meant a lot to my family, and nation Zimbabwe.

Last but not the least, I would like to thank my family: my wife Jessey, and

amazing kiddos, Arlene, Ardele and Anele, Maphango family, Jon Smith, Chi Alpha

Christian family, and River City Church for supporting me spiritually throughout my

Boiler life journey.

TABLE OF CONTENTS

Page

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Chapter Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Introduction and Background . . . . . . . . . . . . . . . . . . . . . . . 11.3 The Rationale for the Study . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Contributions of the Study . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Research Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.6 Methodology and Main Findings . . . . . . . . . . . . . . . . . . . . . . 91.7 Structure of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 STATISTICAL CONSIDERATIONS FOR HIERARCHICALLY IMPLE-MENTED BUNDLED INTERVENTIONS . . . . . . . . . . . . . . . . . . . 122.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Introduction and background . . . . . . . . . . . . . . . . . . . . . . . 122.3 Statistical considerations and recommendations . . . . . . . . . . . . . 16

2.3.1 Bundling innovation . . . . . . . . . . . . . . . . . . . . . . . . 172.3.2 Heterogeneous implementation . . . . . . . . . . . . . . . . . . . 182.3.3 Hierarchical/vertical implementation . . . . . . . . . . . . . . . 202.3.4 Varying context . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 PROCESS-DRIVEN METRICS AND PROCESS EVALUATION OF BUN-DLED INTERVENTIONS: THE AGRICULTURE TONUTRITION (ATONU)TRIAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2.1 ATONU intervention . . . . . . . . . . . . . . . . . . . . . . . . 283.2.2 Study area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2.3 Implementation dynamics for ATONU intervention . . . . . . . 293.2.4 Participation metrics . . . . . . . . . . . . . . . . . . . . . . . . 313.2.5 Variance decomposition and Mediation analysis . . . . . . . . . 34

Page3.2.6 Determinants of change in female dietary diversity scores for

ATONU bundled intervention . . . . . . . . . . . . . . . . . . . 393.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3.1 Variation decomposition for process-driven participation metrics 403.3.2 Determinants of participation andWRA dietary diversity scores

for ATONU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4 SIMULATION STUDY OF TIME SERIES MODELS GENERATED BYUNDERLYING DYNAMICS . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2 Motivating Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.3.1 Modern Statistical inference . . . . . . . . . . . . . . . . . . . . 554.3.2 Dynamical systems theory and nonlinear time series analysis . . 564.3.3 Atmospheric systems and statistical inference . . . . . . . . . . 574.3.4 Subsampling Confidence intervals . . . . . . . . . . . . . . . . . 614.3.5 The challenge of short record length for atmosphere data . . . . 644.3.6 Time series modeling challenge for atmospheric data . . . . . . 644.3.7 Related Works . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.4 G-Models and subsampling confidence interval for atmosphere data . . 704.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.1 Handling complexity through Statistics . . . . . . . . . . . . . . . . . . 805.2 Statistical input for bundled interventions implementation and evaluation805.3 G-models and inference on atmospheric data . . . . . . . . . . . . . . . 845.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.5 Future research on bundled interventions . . . . . . . . . . . . . . . . . 865.6 Future research on subsampling and G-models . . . . . . . . . . . . . . 87

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

vii

LIST OF TABLES

Table Page

3.1 Same coverage, good retention scenario. . . . . . . . . . . . . . . . . . . . 31

3.2 Same coverage, poor retention scenario . . . . . . . . . . . . . . . . . . . . 32

3.3 Same household coverage, different gender composition scenarios . . . . . . 33

3.4 Variance decomposition for the compliance and BICR metrics for ATONU 41

3.5 Determinants of compliance for ATONU bundled intervention . . . . . . . 42

3.6 Determinants of BICR for ATONU bundled intervention . . . . . . . . . . 43

3.7 Determinants of men’s participation for ATONU bundled intervention . . . 44

3.8 Determinants of women’s participation for ATONU bundled intervention . 45

3.9 Determinants of joint participation for ATONU bundled intervention . . . 46

3.10 Determinants of WRA end of the intervention 24-hour recall dietary di-versity score for ATONU bundled intervention . . . . . . . . . . . . . . . . 47

3.11 Determinants of WRA end of the intervention 7-days recall dietary diver-sity score for ATONU bundled intervention . . . . . . . . . . . . . . . . . . 49

4.1 Subsampling confidence intervals . . . . . . . . . . . . . . . . . . . . . . . 78

viii

LIST OF FIGURES

Figure Page

2.1 Hierarchy structure for ATONU implementation . . . . . . . . . . . . . . . 16

3.1 ATONU study regions in Ethiopia (circled) . . . . . . . . . . . . . . . . . 29

3.2 Implementation dynamics of the bundled components for ATONU in Ethiopiaand Tanzania . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 Mechanism of impact for hierarchical structured bundled ATONU intervention35

3.4 Error bars for the compliance and BICR metrics for ATONU intervention . 40

4.1 Record of 20-Hz vertical velocity measurements over Lake Michigan. Fig-ure from [73] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2 Actual coverage probabilities of 90% subsampling CIs with β = 0.42 (inred) and β = 0.5 (in black) using Model A for the skewness of nonlineartime series. Figure is adjusted and adopted from that in [106] . . . . . . . 69

4.3 Actual coverage probabilities of 90% subsampling CIs with β = 0.65 usingModel B for the skewness of nonlinear time series . . . . . . . . . . . . . . 75

4.4 Actual coverage probabilities of 95% subsampling CIs with β = 0.61 usingModel B for the skewness of nonlinear time series . . . . . . . . . . . . . . 75

4.5 Actual coverage probabilities of 99% subsampling CIs with β = 0.57 usingModel B for the skewness of nonlinear time series . . . . . . . . . . . . . . 76

4.6 Actual coverage probabilities of 90% subsampling CIs with β = 0.74 usingModel C for the skewness of nonlinear time series . . . . . . . . . . . . . . 76

4.7 Actual coverage probabilities of 95% subsampling CIs with β = 0.71 usingModel C for the skewness of nonlinear time series . . . . . . . . . . . . . . 77

4.8 Actual coverage probabilities of 99% subsampling CIs with β = 0.67 usingModel C for the skewness of nonlinear time series . . . . . . . . . . . . . . 77

ABBREVIATIONS

AR Auto-regressive

ATONU Agriculture to Nutrition

CI Confidence interval

DGM Data generating mechanism

FANRPAN Food Agriculture Natural Resources Policy Analysis Network

LMIC Low and middle income countries

NEAR Newer exponential auto-regressive

NGO Non governmental organization

PAR product auto-regressive

RCT Randomized controlled trial

WASH Water and sanitation hygiene

WRA Women of reproductive age

ABSTRACT

Matangi, Evidence S. Ph.D., Purdue University, December 2019. Handling Complex-ity via Statistical Methods. Major Professors: McCabe G.P. Professor, Nilupa S.Gunaratna, Assistant Professor.

Phenomena investigated from complex systems are characteristically dynamic,

multi-dimensional, and nonlinear. Their traits can be captured through data gen-

erating mechanisms (DGM) that explain the interactions among the systems’ com-

ponents. Measurement is fundamental to advance science, and complexity requires

deviation from linear thinking to handle it. Simplifying the measurement of complex

and heterogeneity of data in statistical methodology can compromise their accuracy.

In particular, conventional statistical methods make assumptions on the DGM that

are rarely met in real world, which can make inference inaccurate. We posit that

causal inference for complex systems phenomena requires at least the incorporation

of subject-matter knowledge and use of dynamic metrics in statistical methods to

improve on its accuracy.

This thesis consists of two separate topics on handling data and data generating

mechanisms complexities: the evaluation of bundled nutrition interventions and mod-

eling atmospheric data.

Firstly, when a public health problem requires multiple ways to address its con-

tributing factors, bundling of the approaches can be cost-effective. Scaling up bundled

interventions geographically requires a hierarchical structure in implementation, with

central coordination and supervision of multiple sites and staff delivering a bundled

intervention. The experimental design to evaluate such an intervention becomes com-

plex to accommodate the multiple intervention components and hierarchical imple-

mentation structure. The components of a bundled intervention may impact targeted

outcomes additively or synergistically. However, noncompliance and protocol devia-

tion can impede this potential impact, and introduce data complexities. We identify

several statistical considerations and recommendations for the implementation and

evaluation of bundled interventions.

The simple aggregate metrics used in clustering randomized controlled trials do

not utilize all available information, and findings are prone to the ecological fallacy

problem, in which inference at the aggregate level may not hold at the disaggregate

level. Further, implementation heterogeneity impedes statistical power and conse-

quently the accuracy of the inference from conventional comparison with a control

arm. The intention-to-treat analysis can be inadequate for bundled interventions. We

developed novel process-driven, disaggregated participation metrics to examine the

mechanisms of impact of the Agriculture to Nutrition (ATONU) bundled intervention

(ClinicalTrials.gov Identifier: NCT03152227). Logistic and beta-logistic hierarchical

models were used to characterize these metrics, and generalized mixed models were

employed to identify determinants of the study outcome, dietary diversity for women

of reproductive age. Mediation analysis was applied to explore the underlying deter-

minants by which the intervention affects the outcome through the process metrics.

The determinants of greater participation should be the targets to improve imple-

mentation of future bundled interventions.

Secondly, observed atmospheric records are often prohibitively short with only

one record typically available for study. Classical nonlinear time series models ap-

plied to explain the nonlinear DGM exhibit some statistical properties of the phenom-

ena being investigated, but have nothing to do with their physical properties. The

data’s complex dependent structure invalidates inference from classical time series

models involving strong statistical assumptions rarely met in real atmospheric and

climate data. The subsampling method may yield valid statistical inference. Atmo-

spheric records, however, are typically too short to satisfy asymptotic conditions for

the method’s validity, which necessitates enhancements of subsampling with the use

of approximating models (those sharing statistical properties with the series under

study).

xii

Gyrostat models (G-models) are physically sound low-order models generated from

the governing equations for atmospheric dynamics thus retaining some of their funda-

mental statistical and physical properties. We have demonstrated statistic that using

G-models as approximating models in place of traditional time series models results

in more precise subsampling confidence intervals with improved coverage probabili-

ties. Future works will explore other types of G-models as approximating models for

inference on atmospheric data. We will adopt this idea for inference on phenomena

for AstroStatistics and pharmaco*kinetics.

1. INTRODUCTION

1.1 Chapter Overview

This chapter introduces the research problem and outlines the background and

rationale for the present study. It subsequently describes the research questions and

provides a chapter by chapter overview of the thesis.

1.2 Introduction and Background

Complexity is an attribute of a system under investigation, and not necessarily a

trait of the mechanism through which an investigation is conducted [1]. It is defined

through the dynamical interactions of the processes underlying or generated through

a system. It is distinguished on the metaphor used to define the system under inves-

tigation as either a machine or an organism. The former advocates for linear thinking

that is associated with simplicity, predictability, and that knowledge of the whole ma-

chine can be learnt from what is gathered from its parts. The latter view of a system

as an organism accommodates for the interconnection of its parts, nonlinearity, and

unpredictability in its dynamics. The basis of most statistical methods has been the

machine-view of systems with assumptions that are being observed to be rarely met

in real world systems. On the other hand, complexity in scientific research questions

an be attributed to technological advancements and the emergence of new scientific

research fields contributing to the complexity in their associated data. To make ac-

curate inference for complex systems it is important to consider how measurement

is conducted, and how are the statistical models generated and under what assump-

tions for the data generating mechanisms. In order for Statistics to contribute to the

scientific goals and challenges exemplified by the 2030 Agenda for Sustainable Devel-

opment and the global warming, there is a need to account for the context-dependent

public health interventions, and the contributions of the underlying dynamics on at-

mospheric phenomena.

Addressing estimation and reliable inference problems is integral for the appli-

cation of Statistics to other fields of study. The objective is to improve on solving

real world problems through the contributions of statistical methods. The central

mandate of statistical inference is the separation of signal from noise in data [2], as

we seek to relate data with hypotheses. We endeavor to ensure that statistical signif-

icance complement subject-matter significance, and gain traction in their appeal to

subject-matter audience. The complexity of the questions that scientists are seeking

to solve, and the varying dynamics in the generation of their data, point to the need

for advanced statistical inference methods [3]. Under such situations, inferential prob-

lems can be handled through considerations on how Statistics handle measurement

and contextual factors, and the statistical assumptions postulated on the underlying

data generating mechanisms (DGM) for the systems or organisms under study.

Systems and organisms consist of multiple and often interconnected components

[4, 5], and they are characteristically dynamic, unpredictable, and multidimensional.

They typically generate complex and heterogeneous data, whose reality can be lost in

the simplification by statistical models. Climate, societies, and ecology are complex

systems, and assuming that they work like machines leads to misleading estimation

and inference [5]. Complexity theory moves from complex to simple, based on the

interchange amongst a systems’ components [6]. In order to understand phenomena

in complex systems, complexity theory requires that we comprehend how things are

connected, configured and constrained by systematic perturbations. The nature of

causality in complex systems is non-linear (small change can have big effects) which

introduces disproportionality in causal statements between machines and systems [5].

Emerging scientific fields, such as implementation science, translation science, com-

plexity science, and systems science are a reservoir of theory that seek to handle such

challenges. Investigations of complexity challenges traditional scientific approaches

that uphold linear causal statements [7].

The reasoning behind most statistical model building is data-driven, which may

fail to incorporate subject-matter expertise thereby limiting inference. The role of

higher order statistical moments for climate and atmosphere phenomena emanates

from the acknowledgement that their data are non-normal [8]. Heterogeneity in the

generation of atmosphere data that is attributed to the underlying dynamics is a typi-

cal cause of skewness. Modern statistical methods such as bootstrap, and subsampling

have taken a lead in making inference on complex data based on the empirical dis-

tribution function of the observed data. These methods are alternatives to statistical

inference which often hinge on the assumption of parametric model underlying the

observed data or where parametric inference requires complicated formulas for the

calculation of standard errors. We envisage that there is a need for subject matter

knowledge (data-centric approach) to be employed in the approximating distribution

to ensure retention of data attributes from the complex DGM, getting away from

the often rigid assumption on the DGM, for comprehensive and contextual relevant

inference to be obtained.

The need to address the emerging and underlying determinants of public health

problems has led to the development of bundled interventions, as an implementation

innovation. These are multi-faceted interventions whose components work simultane-

ously to promote positive outcomes. The multi-pronged dimensions of public health

challenges such as nutrition as exemplified by the double burden of malnutrition on

obesity and under-nutrition, necessitate complex intertwining of strategies and ap-

proaches to handle them. Bundled interventions have been labeled "high-impact in-

vestments" but are often offset by the low quality of implementation in low-resource

settings [9]. Culturally acceptable health promoting programs (behavioral change

communication) together with the harnessing of agriculture can help alleviate mal-

nutrition [10].

There is a need for interventions to illuminate the processes and mechanisms

leading to the outcome, thereby providing useful information for their adoption for

different populations and context [11]. The potential heterogeneity in populations

of low and middle income countries’ (LMICs) communities can put a strain on the

reliability and relevance of bundled interventions inference, since the expected inter-

cluster differences may be huge leading to possible confounding associations. The cul-

tural/gender norms within the wider low-resource communities such as gendered rela-

tionships/patriarchy can deter the implementation of counter-cultural components of

bundled interventions such as women empowerment, and may have a domino effect on

the whole intervention participation, and consequently adoption. Non-consideration

of masculine issues in development initiatives can challenge women’s participation in

patriarchal societies [12], which can gloss over the distinction between implementation

effectiveness and intervention effectiveness resulting in non-adoption of potentially ef-

fective practices to curb public health issues.

In complex interventions, contextual dynamics impact on data quality and qual-

ity; and the hierarchical structure is a potential source of variation and bias which

can influence the decisions on effectiveness evaluations. Implementation effectiveness

precludes intervention effectiveness, and is immensely influenced by context. Adjust-

ing for clustering and covariates offer a great advantage in the evaluation of complex

interventions. The interactions between hierarchy and intervention components can

contribute to the process dynamics in the implementation of bundled interventions

which can be a helpful source for the explanation of the variation in the outcome

of interest. The ability to capture the traits of the process-driven metrics allows

for the understanding of the interplay of context, delivery and reception of interven-

tions. These will serve to inform implementation quality and attribution of change

in outcome of interest to the intervention, objectively. Such metrics can facilitate

actionable courses to be undertaken for implementation improvement, which helps

make the causal pathways become more clear.

Under a hierarchy structure and contextual dynamics, observational data are

prone to the effects of immeasurable confounding variables, limiting the relevance

of inference made. Process data can capture some of the confounding effects through

metrics that are tied to the process dynamics, which are often not easily capture

through conventional data collection methods. The role of technology in data col-

lection allows for the capture of such intricate and yet vital data, as exemplified by

the open data kit (ODK). This is a useful tool especially for resource-constrained

environments that ensures privacy, and high participation rates, and also helps curb

the prevalent challenge of social desirability bias. Statistical considerations on the

implementation complexities can improve the understanding of the process dynamics

of interventions to ensure sound recommendations on practice based on research find-

ings. This allows for the adoption, sustainability, and scaling of interventions within

the contexts of their study.

1.3 The Rationale for the Study

Complexity in systems cannot be explained objectively through linear thinking,

when it is evident that such systems are inherently nonlinear. Creative approaches

to the statistical inference are required to handle data arising from complex systems.

Accommodating for this reality in our investigations aids our quest to address es-

timation and inference problems in statistical applications. Such adjustments puts

traditional and conventional metrics, statistical methods and data generating mech-

anisms (DGM) assumptions on the spotlight, and calls for data-centric approaches

that combine expertise knowledge and data for objective inference. The data revolu-

tion and the emergence of new scientific fields allows for more avenues for statistical

applications requiring that we be confident of our tools on their relevance to such

challenges. The endeavor to lead with Statistics entails that there is a need for statis-

ticians to be pro-active and not necessarily reactive to the myriad of issues at the

centre of scientific exploits. Developments in statistical sciences should strive to meet

and address the needs of the ever-exploding world of science.

We seek to clarify the role and importance of Statistics methods and subject-

matter theory in the evaluation and analysis of nonlinear systems whose underlying

dynamics contribute to both the complexity and variability in data. Statistical signif-

icance should contribute to substantive or subject-matter significance for meeting the

actual needs of the users. Measurement variation at cluster (aggregate) and individ-

ual (dis-aggregate) levels pose a difficult in causal statements for cluster randomized

trials of complex interventions. This coupled with the fact that components of bun-

dled interventions are often key facilitators to the expected positive changes, their

combined effort makes it no mean endeavor ascertaining the causal pathway in a

bundled intervention. We assert that causality can be attributed to the dynamics

introduced from each of the levels of administration of the intervention leading to the

outcome of interest. Practical and statistical considerations should be embedded in

the implementation and evaluation design of bundled interventions, especially under

resource-constrained environments.

The focus on first and second moments have ensured that statistical models as-

sume on higher moments to validate inference on the former, which can be a source of

missing information for the science being investigated as such higher moments could

be containing the crucial information for their understanding. Given that atmosphere

data is non-normal, inference on higher order moments, starting with skewness will

present useful information on endeavors to understand them, The empirical data-

driven distributions approximating the underlying DGM for the original data for

subsampling method estimation and inference are simple and exhibit some of the

statistical properties of the data. They however, have nothing to do with the subject

matter properties of the original data which impedes on the relevance of inference

that is obtained from them.

1.4 Contributions of the Study

This study will contribute to the current literature in the following ways. The

proliferation and acknowledged relevance of bundled interventions in handling public

health problems requires a statistical address on their implementation and evaluation

design. This is particularly so for low resource settings where their postulated iter-

ative and integrated design is flouted due to complexities attributed to the bundles’

interactions with context within the hierarchy structure of their implementation. We

highlighted the statistical issues that point to the implementation quality for bun-

dled interventions and their consequence on effectiveness assessment and offered rec-

ommendations for handling them. Unlike traditional study designs that answer to

specified problems singly, bundled interventions answer to a host of problems, which

creates complexities in streamlining the implementation dynamics to adequately as-

sess their effectiveness on the particular problems being investigated. The interplay

amongst the intervention components contribute to their additive, synergistic, and

antagonistic effects on the outcome of interest.These effects should be acknowledged

in the theory of change to ensure the attribution of the change in outcome to the

intervention, which is pivotal for their adoption, and sustainability. We developed

and applied process-driven participation metrics that capture the implementation

dynamics that are missed by the traditional simple and aggregate metric for inter-

vention evaluations. We proposed a different set of statistical methodology for varia-

tion decomposition and identification of the determinants of the participation levels

for bundled interventions. Different strategies and decisions were recommended for

addressing the variation structures for the participation metrics to enhance the mech-

anism of impact for bundled interventions. Further assessment was conducted on how

the proposed process-driven metrics enhanced the link between the intervention and

the outcomes while accounting for the effect of contextual factors on them and the

outcome.

The assumptions and necessary conditions for each problem assessment should

be handled both uniquely and objectively within the confines of both the evaluation

and implementation design with recognition of contextual influence. An application

of these statistical consideration in the analysis of a bundled intervention will serve

to highlight the importance of process data in handling them for low resource set-

tings and giving credence to the process-outcome links envisaged. The hierarchical

structure of bundled interventions is mainly for the purpose of applying an interven-

tion on a wide spectrum of area and population settings. It can also emanate from

the multi-disciplinary of the research team members and the multi-sectoral nature of

the intervention components, including nutrition, agriculture, water and sanitation

hygiene (WASH), that often work simultaneously. The hierarchical influence on the

process dynamics, in particular on process variation attribution and how it relates

to the variations in the outcomes of interest allows for process improvement through

addressing how these impact implementation quality.

A data-centric approach to atmosphere data handling enhances the foray of sta-

tistical analysis and modeling in the geosciences. We seek to show how time series

models derived from the governing equations of the underlying dynamics of the atmo-

sphere can be used in statistical inference on atmospheric data. We seek to widen the

applicability of subsampling methods in handling data with a dependent structure

through a relaxation on the assumption on their underlying data generating mech-

anism (DGM). This is essential in ensuring the reliability of the inference made as

they retain both the physics and statistical properties of the original data. The flexi-

bility of such models to incorporate more mechanisms akin to the explanation of the

underlying dynamics, offers a leeway for their further expansion to ensure that the

DGM captures the reality of the original data.

The possibility of adopting such models opens a door for statistical modeling of

data in domains where mathematical modeling has mostly been used, which include

but not limited to pharmaco*kinetics, disease modeling, and the linking of astrostatis-

tics data to its underlying theory.

1.5 Research Questions

This research seeks to address the following research questions emanating from two

studies undertaken concurrently on bundled nutrition intervention and atmospheric

data handling.

(i) What are the statistical issues that need to be taken into consideration for the

successful implementation and evaluation of bundled interventions?

(ii) Does controlling for clustering together with process-driven participation met-

rics improve causality statements for bundled interventions?

(iii) Can data-centric approximating models for the underlying atmospheric dynam-

ics facilitate reliable inference on atmospheric data?

1.6 Methodology and Main Findings

The use of process data which captures dis-aggregate data, and reveals the sources

of variation in its hierarchy structure, in the linear mixed modeling of bundled inter-

ventions data allows for process improvement. This highlights the areas that need to

be improved on for implementation quality, and ascertain the effectiveness assessment

of such interventions on addressing the problems consortium under investigation.

The implementation of the Agriculture to Nutrition (ATONU) nutrition sensitive

agriculture bundled intervention in Ethiopia and Tanzania was characteristically het-

erogeneous. This had an impact on the intervention’s implementation quality and

effective assessment, and vital statistical considerations have to be adjusted for to

handle these aspects. Process-driven participation metrics for ATONU intervention

on the dietary diversity index for women of reproductive age (WRA) in Ethiopia

showed that significant variation in them was attributed to both intrahousehold and

inter-household variation within the unit of randomization. In conventional cluster-

ing randomized controlled trial (cRCT) studies such information is not revealed as

metrics are often aggregated at cluster level for the assessment of population level

change. Both ecological fallacy and aggregation bias (loss of detail due to aggrega-

tion) can be attributed to the challenges that so often surrounds the adoption of

misaligned effective interventions that fail to be translated to practice and policy for

public health issues.

Statistical procedures seek to reach a decision on postulated hypotheses, and to

do so they rely on the assumptions of the statistical models. In our aim to make

inference on atmosphere data, we employed G-models for subsampling confidence

interval construction, and obtained narrower intervals. These are physically sound

models that are derived from the underlying governing equations for atmospheric

dynamics [13]. AR(1) models have been frequently used to model climate data be-

cause of their ability to handle correlated time series [14]. G-models’ advantage over

AR(1)-based nonlinear models is in their ability to capture both the physics and

the statistical properties of the atmospheric data. The accuracy of such confidence

intervals hinge on the determination of the subsample size, otherwise considered as

the block size b, which helps in ensuring that the actual coverage is in sync with

the target coverage for appropriate interpretation of the findings. The block sizes we

obtained was comparable to those used in previous works done for subsampling con-

fidence intervals for atmosphere data. The subsampling confidence intervals obtained

with G-models as approximations of the underlying dynamics were narrower than all

previously computed ones.

1.7 Structure of Thesis

Here is an overview of the chapters in this thesis; chapter one focuses on the in-

troduction, rationale, motivation, the research problems being investigated, and the

major findings made. Chapter two focuses on highlighting the statistical consider-

ations in the implementation and evaluation design for bundled interventions and

possible solutions to address them. Chapter three offers an application of process-

driven metrics in the evaluation of a bundled intervention, showcasing some solutions

on handling statistical considerations on heterogeneity in implementation. Chapter

four gives an overview on investigating inferential relevance based on Monte Carlo

(MC) simulations for atmospheric data through time series models generated from

their underlying dynamics. The study seeks to utilize these models for subsampling

confidence interval for parameters of non-normal atmospheric data, as they allow for

the incorporation of the physics defining the data. Lastly, chapter five offers conclu-

sions drawn from the studies, recommendations, and future research suggestions.

2. STATISTICAL CONSIDERATIONS FOR HIERARCHICALLY

IMPLEMENTED BUNDLED INTERVENTIONS

2.1 Abstract

Although the randomized controlled trial (RCT) is considered the gold standard

for assessing interventions, many nutrition studies use experimental designs with more

complex structures. We examine one class of such designs, hierarchically-implemented

bundled nutrition interventions, with particular focus on the unique statistical issues

associated with these studies. Hierarchically-implemented studies involve several lev-

els, such as the individual, the household, the village, and the region, that must be

carefully taken into account in the planning and execution of the study. A bundled

intervention includes a collection of interventions, with separate but often comple-

mentary objectives, that can be implemented at different levels of the hierarchy. Sta-

tistical considerations for bundling and hierarchical implementation are described,

and recommendations are proposed which include the development of process-driven

participation metrics, power and sample size optimization, context and spillover mea-

surement, and the use of analytical methods that take into account both clustering

and covariates.

2.2 Introduction and background

Nutrition interventions often address problems with connected underlying causes

such as the double burden of malnutrition. They require a sound evidence base for

adoption for the at-risk-populations. Such interventions need to be implemented in

the context of sound Theory of Change (ToC); which are often complex and consist

of multiple pathways to the target nutrition outcomes. Communal public health is-

sues are often multidimensional and cannot be addressed through single interventions.

The bundling of interventions is an innovative design, which is defined as multiple

interventions combined to address public health problems. The bundled components

can be instructional sessions, reminder messages, and activities. They can contribute

additively or synergistically to the target nutritional outcomes. The effectiveness for

bundled interventions hinges on the accounting for the complexity involved in imple-

menting their components.

When bundled interventions are scaled to target geographically dispersed popu-

lations, their implementation becomes hierarchical structured. The dissemination of

their components requires consolidated support systems through hierarchical struc-

tures to achieve the desired public health impact [15]. Decision-making, mobilization

initiatives, and interpersonal communication during the implementation process can

influence participation dynamics.

The analysis of complex social interventions as single entities without comprehen-

sive integration of the components is challenging [11]. There is a need to address

the possible consequences of interactions among bundled components and with the

hierarchy levels. A complex systems approach to such interventions, viewed as events

in systems, emphasizes the role of context [16]. The careful examination of the im-

plementation process can help to assess how the target effects are attained [15]. The

ToC should provide a framework for describing the pathway on how and why a de-

sired change can be achieved through the intervention. The "implementation gap" is

the challenge for translating research evidence into routine practice. The dynamics

of the five domains of the Implementation Science in Nutrition (ISN) framework [17]

are crucial for addressing this "implementation gap". The five domains are:

(i) The object of implementation.

(ii) Implementation organizations and staff.

(iii) Enabling environment.

(iv) Participants.

(v) Implementation process.

Bundling and hierarchical structure can enhance the effectiveness of bundled in-

terventions at the individual level through engagement with the different components.

These innovations also present statistical challenges for the intervention’s evaluation

that requires a critical analysis of the whole implementation process for appropriate

conclusions to be drawn [18].

The bundling of interventions has been shown to be an efficient technique [19]

which has been applied for public health as care bundles, community-based, and

nutrition-sensitive agriculture interventions. They have been effective for acute health

problems in high resource settings [20]. The personalized nutrition care bundle that

was created by the American Society of Parenteral and Enteral Nutrition (ASPEN)

in conjunction with the Society of Critical Care Medicine (SCCM), sought to opti-

mize patients’ nutrition statuses during acute care admissions [21]. It consisted of

the following six components:

(i) Malnutrition assessment.

(ii) Initiation and maintenance of enteral feeding.

(iii) Reduction of aspiration.

(iv) Implementation of enteral feeding protocols.

(v) Avoidance of gastric residual volumes use for tolerating enteral nutrition.

(vi) Non-initiation for early parenteral nutrition when enteral feeding.

Its effectiveness depended on patients’ demographics and the involvement of di-

versified professional staff handling varying components of the bundle. The additive,

and synergistic effects of the components need to be acknowledged for the bundled

intervention to be viewed as a single entity [22], for aggregate beneficial effects on the

outcome [23].

As interventions are scaled, their hierarchy structures promote planning for easy

and efficient use by implementers thereby facilitating intervention effectiveness [24].

The Realigning Agriculture for Improved Nutrition (RAIN) was a hierarchically im-

plemented bundled intervention focusing on child nutrition in rural Zambia. RAIN’s

structure involved a primary level (infants at baseline and their parents), a secondary

level (women’s groups), and a tertiary level (implementing organizations) [25]. Strong

implementation emphasis and effective monitoring were significant for RAIN’s effec-

tiveness. Understanding the change process within hierarchy helps in the identifica-

tion of factors that promote the development and implementation of interventions [26].

The agriculture to nutrition (ATONU) bundled intervention was implemented in

Ethiopia and Tanzania to improve the nutrition status for subsistence farmers through

behavioral change communication [27]. It consisted of the following five thematic

components:

(a) Family nutrition.

(b) Dietary diversity.

(d) Women empowerment.

(e) Home gardening.

Figure 2.1 shows the hierarchical structure designed for ATONU implementation.

Individual, unit of analysis

Household, unit of engagement with intervention

Village, unit of randomization and treatment application

Region/District, unit of monitoring

Country, unit of administration

Figure 2.1. Hierarchy structure for ATONU implementation

2.3 Statistical considerations and recommendations

The purpose of this paper is to highlight the statistical considerations for the im-

plementation and process evaluation of bundled nutrition interventions and to make

recommendations. These statistical considerations assist in explaining the conduct

of bundled nutrition interventions to ensure that precise and unbiased intervention

effectiveness are obtained for the possibility of the transition of research evidence to

nutrition practice and policy. This underscores the need for implementation effective-

ness that can help to ascertain the effectiveness of bundled intervention and identify

their success factors.

2.3.1 Bundling innovation

The ToC is a tool for developing and evaluating complex interventions [28], and

there is little knowledge about its use for public health interventions [29]. The Medical

Research Council (MRC) evaluation guidelines fail to incorporate theory-driven ap-

proaches [30]. We posit that the ToC for bundled interventions is complex, involving

additive, synergistic, and potentially antagonistic contributions from the components.

The Engaging Fathers for Effective Child nutrition and development in Tanzania

(EFFECTS), is a bundled nutrition intervention that seeks to assess the impact of

father’s involvement on children’s nutrition. Its ToC consists of nutrition and par-

enting pathways that link and explain child nutrition and morbidity outcomes with

the intervention’s components. They causally connect the messages and activities on

water and sanitation hygiene (WASH), infant and young children feeding (IYCF),

women empowerment, parenting knowledge and practices, and nutrition knowledge

to the target outcomes. The ToC exhibits an integrated and iterative linkage of the

components capturing their additive or synergistic effects.

The following are the statistical concerns for bundling, highlighted by the resource

constraints to testing the components individually. The intermediate outcomes de-

rived from the components are often not measured yet the ToC suggests additive and

synergistic links of the components. The main objective for intervention research is

treatment effects, and a measure of how the bundled components evolve to the target

outcome needs to be captured. Based on Rubin’s motto "no causation without ma-

nipulation", the outcome change should be associated with the bundle components’

manipulations [31].

The robustness of the causal links needs to consider the impact of implementation

quality for bundled interventions. The participants may not get all the bundle com-

ponents and under such circ*mstances, the additive and the synergistic effects may

not be fully realized. The complex causal structure for bundled interventions may

require appropriate process data for their assessment.

We recommend the need to develop ToC based on literature with hypothesized

interactions among the bundle components to address the above-mentioned statistical

concerns for bundling. Power, effect size, and sample size calculations and justifica-

tions should form an integral part of the ToC. Intermediate outcomes need to be

measured to explain the implementation dynamics associated with bundling. Process

outcomes can potentially offer more evidence than observational or perception mea-

surements. Metrics about the components delivered and received by the participants

help to show the extent of interaction with the bundled intervention. They would

allow for the contribution of the bundle’s components to the effect and variation in

the target outcomes.

2.3.2 Heterogeneous implementation

The delivery and reception of bundled interventions can vary in terms of content,

capacity, timing, and participants’ motivation. Population level risk factors such as

poor sanitation, lack of education, infrastructure, illiteracy, and poverty can affect

engagement with the bundle components. Given heterogeneity in the target popu-

lation, implementation may purposely be varied to reach the targeted participants.

Implementation heterogeneity can be intentional when adapting to local context for

food culture, availability, affordability, and seasonality of diverse foods. It can also

be unintentional when there is poor delivery, and variability in the competence of the

implementation staff.

The locally adapted ATONU’s implementation was heterogeneous on content de-

livery, delivery timing, and staff retention due to varying socio-economic, climatic

conditions, and staff turnover. These factors may have negatively impacted on the

delivery decisions and necessitated implementation heterogeneity. In the evaluation

for bundled interventions, unintentional implementation heterogeneity would bias us

towards the null hypothesis, while intentional heterogeneity would bias away from

the null hypothesis. The non-rejection of the null hypothesis may not entirely be

due to failure of the intervention theory, but also implementation challenges. Process

metrics need to be considered as they can adequately capture the mechanism of im-

pact through tracking the engagement dynamics. Compliance metrics can be used to

capture the retention levels, i.e. the extent of intervention reception. However, small

sample size and effects attributed to depressed values on these alternative metrics can

exacerbate the low statistical power challenge, and statistical significance may be due

to false positive results. Caution must be taken in delivering conclusions for decisions

on the adoption of bundled interventions.

Observational assessment of adoption bundle activities or messages may fail to

capture the effect of unmeasured confounding variables. These may undermine their

contribution in the evaluation of the intervention by biasing towards the null hypoth-

esis. On the contrary, they can heighten the Hawthorne effect in conjunction with

the social desirability bias.

Implementation heterogeneity can impact uptake and the effectiveness of the in-

tervention [32]. Adequate sample size and statistical power are needed to improve

uptake of bundle components and enhance effectiveness. The delivery and reception of

the bundle components maybe heterogeneous which may limit their overall effective-

ness [23]. In such scenarios, statistically insignificant conclusions may yield vital trial

trends, for which post-hoc power computations are needs to inform future research

for sample size considerations to facilitate the detection of significant differences [33].

We recommend the need for comprehensive data collection and development of

metrics based on the implementation dynamics, for use in the analysis. A consider-

ation of the intention to treat analysis for evaluating the effects of bundled interven-

tions is a viable alternative to comparison with a control group, or in the presence of

treatment heterogeneity [34]. This however, maybe inadequate as spillovers and con-

tamination may be present. Tracking the individuals will capture their compliance to

the protocol of the intervention in relation to their randomization assignment. Fur-

thermore, we recommend the establishment of guidelines for monitoring participation

levels for the bundle components which allows for composition data analysis.

2.3.3 Hierarchical/vertical implementation

Bundle components may have additive, synergistic or antagonistic effects at all or

some of the hierarchical levels [35]. Hierarchy can limit bias and promote the internal

validity of the bundled intervention studies through facilitating for adjusting for clus-

tering in evaluations. This reduces the likelihood of spillovers, and can help capture

the sources of variation for bundled interventions.

Participation dynamics can effect variation in the target outcomes, which often

respond to the implementation framework defined by the hierarchy structure. Eco-

logical fallacy is an inherent misconception on causal inference [36] that is shown

in the assumption that what is true for a group holds for the individuals. Group

positions can be influenced by stereotypes attributed to research lag typified by fe-

male disadvantage on education [37], which may not translate to individual females.

The ATONU bundled intervention’s implementation varied in terms of ecology, gov-

ernance, socio-culture characteristics, which mirrored its hierarchical structure.

The main statistical concern for hierarchy structured bundled intervention is the

need to adjust for clustering. This helps in capturing and explaining the sources of

variation both in the implementation and outcome metrics. The failure to account

for clustering can lead to spurious conclusions [38]. We need to have appropriate

power/sample size for the experimental design to facilitate for the objective assess-

ment of bundled interventions’ effectiveness. The analysis of bundled interventions,

calls for the use of multilevel models that adjust for clustering at all necessary levels.

Mediation analysis can help identify the facilitators and inhibitors for their effective-

ness.

Design effects and variability obtained at the appropriate hierarchical levels can be

used to correct for statistical inference as cluster randomization is prone to spillover

effects that bias towards the null hypothesis due to social interference [38]. There

is need to define, identify, and estimate spillover effects, and control for them in the

process evaluation for bundled interventions.

2.3.4 Varying context

Contextual variation in the target population or the physical, social, or institu-

tional environment, becomes visible in the interactions of context with the bundle

components in their implementation. Context shows the prevalence or severity of

the challenges under investigation [11]. An understanding of intervention adaptation

to context can help elaborate the processes leading to the target outcome [39]. An-

other challenge in understanding effect modification is the population heterogeneity

for which personal attributes are the stand-out factors [40]. The key characteristics

of individuals tend to vary in the clusters, and may confound on the observational

data on the intervention [41]. Internal validity and causal pathways consolidation

requires a consideration of the potential socio-economic inhibitors in the intervention

clusters [42].

Social drivers of causality in interventions cannot be controlled under different

contexts and they also accentuate the variation in intermediate outcomes for the

bundle components. The intervention’s effects on the outcomes can be heterogeneous

and context-specific and dependent on the quality of implementation [43].

Contextual implementation research should endeavor to define the acceptable

methodological rigor for sound results under real world conditions [17]. ToC is vital for

intervention planning through identifying the underlying conditions and assumptions

and acknowledging contextual effects [44]. Varying contexts allow for the presence of

unmeasured confounding variables that can affect the causal statements for bundled

interventions. These could lead to the occurrence of Type III error in the conclusions

drawn under such competing factors.

Type III error is correctly rejecting the null hypothesis but for the wrong reason,

which needs to be avoided [45]. This is exemplified by a situation where another

program being operated within our treatment group had the positive effect, and our

intervention had no effect. This error can be a consequence of contextual factors

beyond the control of the intervention.

We recommend that key stakeholder and formative research input be incorporated

to gain insight into the contextual attributes for addressing public health issues. Con-

text must be measured hence data collection and appropriate metrics should be at all

levels of the hierarchy. Analytical methods should adjust for background characteris-

tics of the heterogeneous population and the process-driven participation metrics. To

address the confounding problem associated with heterogeneous populations there is

a need to measure as many variables as possible and adjust for them in the evaluation

of the intervention [41]. Heterogeneous target population requires that the sample

size be sufficiently large for significant conclusions to be drawn [46]. To avoid Type

III error, documentation of competing events and the interactions of the intervention

with context and the minimization of contamination are fundamental.

2.4 Discussion

This study highlighted the statistical considerations for the implementation and

evaluation of hierarchically implemented bundled nutrition interventions. We ac-

knowledged bundling and hierarchy as implementation innovates for nutrition inter-

ventions. Four statistical issues were identified requiring careful statistical thought for

the betterment of evaluation for bundled interventions. They were about bundling,

implementation heterogeneity, implementation hierarchy, and varying contexts. Their

significance lies in the facilitation for contingency measures to ensure that implemen-

tation and intervention effectiveness remain the goals for bundled interventions.

We recommended sound ToC, the development of process-driven participation

metrics, power and sample size optimization within the bundled components. Con-

text and spillover measurement and the use of analytical methods that adjust for

clustering and implementation dynamics covariates were also recommended. The rig-

orous data collection proposed may seem to be a burden for bundled interventions, but

new technologies can help alleviate it. Tools such as the open data kit (ODK) allows

for real-time monitoring, and corrective action to be undertaken on implementation.

These tools are becoming ubiquitous even for developing countries due to improved

internet access and exposure to smartphones and electrical gadgets. Documentation

of the processes involved in the implementation can aid in the specification of the

causal pathways for bundled interventions.

Adaptability of bundled interventions to local contexts while minimizing contam-

ination, and ensuring comparability enhances the generalization of their findings.

Statistical modeling should adjust for contextual and hierarchical level-specific co-

variates in causal inference [47]. There is need for delivery capacity and reception

optimization metrics for bundle components under constrained resources to ascertain

their feasibility.

In order to ensure the translation of research to routine practice, implementation

effectiveness should be distinguished from intervention effectiveness [48]. This helps

in separating intervention failure from implementation failure which impact on the

adoption for potentially effective bundled nutrition interventions in real world.

These highlighted statistical considerations may need to be addressed for the contri-

bution of bundled nutrition intervention to ISN research. They can serve to improve

on their implementation quality, evaluation, adoption, sustainability, and scaling.

3. PROCESS-DRIVEN METRICS AND PROCESS EVALUATION

OF BUNDLED INTERVENTIONS: THE AGRICULTURE TO

NUTRITION (ATONU) TRIAL

Abstract

Background

Bundled nutrition interventions examine causes of nutritional deficiencies through

the additive and synergistic effects of their components. Their implementation is

often heterogeneous due to contextual confounders that impact their effectiveness.

Objective

We propose process-driven participation metrics to capture implementation dy-

namics and apply them to a bundled nutrition intervention, the Agriculture to Nu-

trition (ATONU) intervention. We generate specific recommendations to improve

implementation quality and evidence for impact of the intervention the primary out-

come, women’s dietary diversity.

Methods

A cluster randomized experimental design was used for the agriculture to nutrition

(ATONU) intervention in Ethiopia and Tanzania. Villages formed the clusters. The

aim of the intervention was to improve the nutritional welfare of vulnerable members

in subsistence farming communities. The metrics were compliance, bundled interven-

tion components received (BICR), and gender-specific engagement for men, women

and joint. Beta-logistic and logistic models were used to determine the sources of

variation in the process-driven metrics. Further, generalized mixed models were ap-

plied to link the intervention and the outcome, the dietary diversity for women of

reproductive age (WRA) at the end of the intervention.

Results

The implementation of ATONU among the villages in Ethiopia and Tanzania

was heterogeneous in terms of content delivery and timing of delivery. Variation in

compliance was greater within villages, and variation for BICR was greater between

the villages. To improve compliance, focus should be on participants’ mobilization

and for BICR, the administration of the research staff must be revamped. The linear

mixed model was a better fit than the Poisson mixed model for the dietary diversity

score for WRA. Compliance was a significant determinant of the mechanism of impact

of bundled intervention on the WRA’s dietary diversity. Adjusting for clustering,

compliance, livestock diversity, baseline dietary diversity score, and contextual factors

is important for the process evaluation of the bundled nutrition intervention.

Conclusion

Bundled interventions are needed to improve nutritional outcomes. Their evalua-

tion requires a focus on the individual participants and accounting for implementation

heterogeneity in different settings. A considerable amount of participation variation

is due to inter-household and intra-household factors. The linear mixed model with

adjustments for clustering, process-metrics and contextual covariates can significantly

explain the change in women’s dietary diversity scores.

3.1 Introduction

Malnutrition is a multifactorial problem that requires holistic and multidimen-

sional interventions [49]. Bundled nutrition interventions are nutritional methodolo-

gies for solving complex nutrition problems in communities. Their implementation

in varying geographical locations introduces a hierarchical structure that impacts on

the delivery and reception of bundled components. Observational studies based on

aggregate metrics have been shown to be effective ways to improve on nutrition out-

comes in women and children [50].

Public health interventions are frequently implemented at the cluster level to min-

imize costs and contamination, and for administrative convenience. Their metrics are

often aggregated, however they seek to address population level changes of outcomes

that are captured at the disaggregate level. Aggregate metrics though simple, neglect

information on the implementation dynamics for bundled interventions. Decisions

based on aggregate metrics for changes in populations are prone to the ecological

fallacy problem, where inferences about individuals are deduced from inference about

the group to which those individuals belong. Observational studies fail on the estab-

lishment of causal statements to link the interventions to the nutrition outcomes [50],

because of the presence of unmeasured confounding variables. On the other hand,

bundled interventions conducted in communal settings lack clear evidence of im-

pact as they focus on distal instead of proximal measures for women’s nutritional

outcomes [51]. Gender inequities on food decisions and participation dynamics are

potential causes for such effects. We posit that the engagement of participants with

the components of bundled nutrition interventions is essential for their effectiveness.

The dietary diversity score is a key proximal indicator for women’s nutritional

adequacy and quality. It is defined as a function of several food groups eaten within

the previous 1 or 7 days. The women’s minimum dietary diversity (MDD-W) is de-

fined in terms of the following ten food groups, (i) staples, (ii) pulses, (iii) seeds, (iv)

dairy produce, (v) meats, (vi) poultry produce, (vii) green vegetables, (viii) fruits and

vegetables containing Vitamin-A, (ix) non-green vegetables, and (x) non-Vitamin A

rich fruits [27]. Rural communities in developing countries perennially face the prob-

lem of poor dietary diversity [52]. The MDD-W score for rural Ethiopia farming

communities has been shown to be poor and beyond the solution of home gardening

interventions [27, 53]. Diets for WRA are typically monotonous and of low quality

for low and middle income countries (LMICs) [54], and have been found to be low

on diversity [55]. Increasing dietary diversity could potentially reduce the burden of

malnutrition [56].

Randomized controlled trials (RCT) for socially complex interventions have been

acknowledged to be problematic in their evaluation [57]. Bundled nutrition inter-

ventions involving nutrition behavior change communication can be characterized as

complex. They lack blinding, involve heterogeneous participants and may be im-

plemented heterogeneously, and have difficulty in controlling for confounders [58].

These attributes may violate the conditions for them to be assessed as standard clus-

ter randomized controlled trials (cRCTs) and thereby fail to guarantee attribution of

causation to the interventions [59].

We examine determinants of how and why change occurs through the process,

dynamics and conditions of intervention implementation. A lack of intervention ef-

fectiveness can be attributed to imprecise measurement [60], and poor evaluation

makes evidence of interventions effect inconclusive [61]. As a result it may be diffi-

cult to get information to improve processes, to ascribe causality; and to establish

ecological validity, i.e. to generalize research findings. There is a need for metrics

that can be captured for contextual effects. Defining concepts and developing mea-

surement tools are crucial for ascertaining causal relationships and generalization of

bundled interventions’ findings. These are crucial for their adoption, sustainability,

and scalability.

There is need for appropriate metrics and relevant methodologies to monitor and

evaluate bundled nutrition intervention [62]. These can illuminate their mechanism

of impact and thus provide an evidence base for the generalizability of their findings.

The understanding of their change process can offer feedback for the consolidation

of their complex theory of change (ToC) framework and hypotheses for the determi-

nants for positive nutrition outcomes. There is a need in delivery-system research

for the understanding of the process underlying the intervention [63]. This requires

process data which is difficult to collect. The availability of smart technologies such

as open data kit (ODK) especially in developing countries can facilitate the capture

and management of process data.

Process methods and metrics are needed for capturing participant engagement

given heterogeneous implementation where compliance confounds intervention deliv-

ery and participant engagement. Metrics are needed for participant engagement dis-

tinguishing it from delivery. We propose process-driven participation metrics that (a)

allow for the individual tracking of participants, (b) quantify compliance of a package

of components, and (c) quantify gender inequities in participation. We demonstrate

that these novel process-driven participation metrics can be used to improve the im-

plementation and establish the process-outcome link of heterogeneously implemented

bundled interventions. Our objectives are to:

(i) develop metrics that capture participation dynamics for bundled interventions.

(ii) identify factors that explain variation in household participation metrics for the

Agriculture to Nutrition (ATONU) bundled intervention.

(iii) identify contextual factors that define the change process linking ATONU bun-

dled intervention to WRA’s dietary diversity.

3.2 Methods

3.2.1 ATONU intervention

The Food, Agriculture, Natural Resources Policy Analysis Network (FANRPAN)

initiated ATONU to promote nutritional security for the vulnerable WRA and young

children in sub-Saharan smallholder farming families. This was implemented as a

cRCT in Ethiopian and Tanzanian villages during the period February 2017 to April

2018. It focused on behavior change communication and had the following five the-

matic components: family nutrition, dietary diversity, maternal infant and young

children feeding (IYCF), women’s empowerment, and home gardening. These were

administered through group discussion meetings, home visits, and practical activities.

3.2.2 Study area

We did not have access to the outcome data for Tanzania, hence we focused our

study on Ethiopia. The study area was a low resource smallholder farming rural

area with varying agro-ecological zones, and social norms. Data was collected from

20 villages from the 4 study regions and in each village 40 households were targeted.

The regions served as strata from which villages were randomly sampled and assigned

to the treatment arms. Our focus here is on the treatment arm only.

Figure 3.1. ATONU study regions in Ethiopia (circled)

3.2.3 Implementation dynamics for ATONU intervention

Heat maps were used to visualize the implementation dynamics of ATONU be-

tween the two countries and among the regions and villages.

Figure 3.2. Implementation dynamics of the bundled components forATONU in Ethiopia and Tanzania

Figure 3.2 shows that the implementation of the five bundle components between

the two countries was heterogeneous. . If the implementation was done hom*oge-

neously, the display of the heat maps would relay a distinct and similar pattern

within all the regions and villages. This however was not so, the delivery in Ethiopia

for Tigray-20 was delayed and thereafter some consistency prevailed yet in Tigray-19

it was delayed and scantily as only two messages were delivered. This pattern is

prevalent both between the two countries and also within the regions and villages,

showing that there were peculiar contextual factors that determined the delivery-

reception dynamics for the bundled intervention. The heterogeneity was in terms of

delivery content and timing which could be attributed to staff turnover, contextual

and background characteristics of the participants. Seasonal variation could not be

identified as the intervention was conducted over a short period of time. Mobilization

incentives of seeds and cooking activities were the dominant home gardening and

maternal IYCF components.

3.2.4 Participation metrics

Conventional participation metrics at cluster level are typically attendance, cover-

age, and dose received. They focus on either but not both dimensions of participation,

i.e. frequency and extent of involvement. They neglect substantial information on

participation dynamics in relation to the bundled intervention as shown for coverage

in comparison to retention in Tables 1 and 2 below.

Table 3.1.Same coverage, good retention scenario.

Participant Time 1 Time 2 Time 3 Time 4 Retention

1 X X X X 100%

2 X X X X 100%

3 0%

4 0%

Coverage 50% 50% 50% 50% 50%

The coverage in Table 3.1 is overall 50% and fails to capture the non-participation

and thereby suppress the variation among participants. The retention metric captures

the non-participation and hence allows for the variation in analysis. The bundled

intervention may have impact on only 50% of the target population.

Table 3.2.Same coverage, poor retention scenario

Participant Time 1 Time 2 Time 3 Time 4 Retention

1 X X 50%

2 X X 50%

3 X X 50%

4 X X 50%

Coverage 50% 50% 50% 50%

Table 3.2 shows 100% coverage but does not capture the extent of involvement

thereby failing to reveal the non-participation in the paired time slots. On the other

hand, retention levels of 50% reveal that there was non-participation but cannot dis-

tinguish it in terms of delivery times. The interventions may not have the intended

impact on all the participants as they each received half of the bundle components.

The other dimension of disparity on coverage is when we factor in the gender

inequities prevalent in patriarchal communities which impacts decisions to participa-

tion. Suppose the target group consists of 20 households in which an intervention is

targeting participation of both the husband and wife. We propose that behavioral

change in household on nutritional status requires mutual participation of the adults.

We can have the participation scenarios depicted in Table 3.3 below.

Table 3.3.Same household coverage, different gender composition scenarios

Case Female only engagement Male only engagement Joint engagement Coverage

1 0 0 10 50%

2 5 5 0 50%

3 10 0 0 50%

4 0 10 0 50%

The case 1 is ideal but shows that the bundle intervention would impart only 50%

of the target populations, and the other cases shows no impact as only half of the

target audience is receiving. The coverage situations shown in Tables 3.1 to 3.3 forms

the basis for our argument for individualized process-driven participation metrics.

Process metrics are valuable for the description of the functioning of interventions in

real world. They support causal statements for bundled interventions, and inform and

improve implementation quality. These metrics will allow for tracking of participation

over the continuum of the intervention’s lifespan, their engagement with the different

components, and gender disparities. We proposed the metrics for compliance, bundle

intervention components received (BICR), male participation, female participation,

and joint participation.

The compliance metric tracked the individual participants’ engagement with the

intervention over its lifetime, i.e. retention.

Compliance =Count of messages received

Count of messages delivered(3.1)

Compliance is a function of the process dynamics of delivery and context which

influence decisions to participate. It captures the frequency of attendance and the

extent of involvement in relation to delivery.This metric has similar traits to those

of the compliance metric for clinical trials. The implementation heterogeneity shown

in Figure 3.2 can be revealed through this compliance metric. However, it does not

retain the timing of engagement with the bundle components.

The bundled intervention components received (BICR) metric quantifies the ex-

tent to which individual participants engaged with the bundle components. It is a

function of content, the contextual effects on implementation, and the background

characteristics of the participants. However, it does not preserve participation time

order.

BICR =Count of bundle components received

Expected count of bundle components delivered(3.2)

The gender coverage metrics are binary measures for joint, male, and female en-

gagements with the bundle intervention. These ascertain the social drivers for par-

ticipation. They are functions of the frequency dimension of participation and they

do not capture retention and participation time. The female participation metric in

3.3 below illustrates the gender participation metrics.

Female participation =

1 if woman attended in household attended at least one meeting,

0 otherwise.(3.3)

3.2.5 Variance decomposition and Mediation analysis

Errors bars were used to describe the variation in the process-driven metrics of

compliance and BICR. Based on the model in Figure 3.3 below, we sought to develop

a framework for the process evaluation for ATONU bundled nutrition. We sought to

demonstrate the causal relationships among the intervention, context and outcomes,

facilitating for the no confounding assumption [64] through utilizing as much data

as possible from the intervention, context and background characteristics of the par-

ticipants and adjustments for clustering. Conventional mediation analysis often use

regression models that do not adjust for clustering. We argue for its accommodation

because of the hierarchy structure for bundled nutrition interventions implementa-

tion. We identified the determinants of the process-driven participation metrics and

the WRA dietary diversity scores for 24 hours and 7 days recall.

Figure 3.3. Mechanism of impact for hierarchical structured bundledATONU intervention

Beta-logistic and traditional logistic models were used to investigate the link be-

tween context and demographic variables with the process-driven metrics. Linear

and Poisson mixed models were used for the mediation analysis of the WRA dietary

diversity outcomes.

The proposed metrics of compliance and BICR are proportions at the individual

level and do not represent independent trials; they are not binomial variables. Trans-

formation of these data such as the logit for standard linear analysis have the short-

comings in terms of parameter interpretation and such data are often heteroskedastic

and deviate from normality [65]. We ascertained their contextual determinants and

variance decomposition through the Beta-logistic model. This model treats the pro-

portions for selecting options as dependent on exogenous variables with heterogeneous

variance [66]. The Levene’s and Brown-Forsythe test for hom*ogeneity of variance

amongst the villages for the compliance and BICR test were conducted. These are

robust techniques that are insensitive to heavy-tailed and skewed distributions in con-

trast to the Bartlett test that depends on the normality assumption.

For our analysis we use the beta distribution, whose mean lies within (0, 1,) with

a logistic link function. The scale parameter of the Beta-logistic model is inversely

related to the variance of the response variable. A limitation of the model is that it

does not allow for proportions equal to zero or one.

f(yijk) =Γ(aijk + bijk)

Γ(aijk)Γ(bijk)yaijk−1

ijk (1− yijk)bijk−1 + εijk (3.4)

where yijk is the compliance and BICR response, aijk = eα′lg(X), bijk = eβ

′lh(X) and

X = [[X]ijk, Z] = (Regioni, V illage(region)j(i), Householdijk, Covariate)

Participation in bundled nutrition interventions can be affected by individual,

household, and communal level factors. We examine the following hypotheses on

the contextual determinants the process-driven participation metrics of compliance,

BIRC, and gender engagement. Gender participation in bundled interventions is

defined in terms of communication, decision inequities within households, and com-

munal values. The logistic model with adjustments for clustering was used to identify

the determinants of gender participant metrics. The logit model has a binomial dis-

tribution and a logistic link function and seeks to model the logarithm of the odds

ratio.

Logit(πijk) = µ+ αi + βj(i) + τZ + εijk (3.5)

where πijk = p1−p , µ is the grand mean, αi is the fixed region factor, βj(i) is the

village nested in region random factor, Z is the contextual covariate defined at either

i, j, k level and a random error εijkl ∼ N(0, σ2).

We used the metrics to identify the determinants of participation by the target

households, men and women. We examined the following hypotheses for the con-

textual determinants of the process-driven metrics and also for the outcomes for

hierarchically implemented bundled intervention.

H1: High baseline livestock household wealth can either promote or im-

pede participation in bundled nutrition interventions

Under rural and low resource settings, wealth is often associated with access to

infrastructure and resources. Households in the higher wealth quintiles tend to have

diverse foods to incorporate in their diets, thus they are less motivated to participate

in behavioral change communication that promote nutrition status of their members.

Livestock and crop diversity at home [27] often associated with wealthier households

promotes dietary diversity, which may negate their participation levels. On the con-

trary wealthier folks might have more time to attend, as bundled components for

behavioral change communication may be beneficial for them.

H2: Larger baseline family size can hinder participation in bundled inter-

ventions

Larger family size may involve more sharing of food and less resources per person.

Under such conditions there are challenges on welfare priority and time management

for household decision makers making their participation in interventions with mul-

tiple components to be inconsistent.

H3: High education for woman in household can hinder participation in

bundled nutrition interventions.

When there is variation in the education status for women in the households,

their uptake and importance of messages on nutritional needs for their families may

be divergent. This can be an indirect measure of their self-efficacy, which can measure

how they perceive the nutritional content delivered in line with their already acquired

knowledge and experience.

H4: Female headed households are less likely to participate in bundled

nutrition interventions.

Women are less inclined to seek out nutritional resources for their households for

rural and limited resource settings due to marginalization and the social structure.

Women-headed households have one less person to take care of household responsi-

bilities, so their time burden is way too much to allow for their participation.

H5: Remoteness hinders participation in bundled nutrition interventions

Households that are located faraway from meeting places and markets tend to be

low in their engagement with bundled nutrition interventions.

H6: Agro-ecological zones can both promote and hinder participate in

bundled nutrition interventions.

Agro ecological zones measures the elevation from sea level of the settlements for

subsistence farmers which influence their agro-produce. Those at high elevation have

commercial produce and are more susceptible to restricted diverse food production.

They may have a high dependence on the market’s availability, affordability and di-

versity for food, and may be more inclined to seek knowledge on nutrition education

and behaviors.

H7: Baseline parity can hinder participation in bundled nutrition inter-

ventions.

Baseline parity is the number of infants within a household. They require ade-

quate care-giving and stimulation for food consumption. These time constraints limit

their caregivers’ participation in bundled nutrition interventions. Baseline parity is

associated with maternal age and hence can indirectly influence participation.

H8: Farm size can promote or hinder participation in bundled nutrition

interventions.

This can be an indirect measure of wealth, household productivity, and food secu-

rity. This economic indicator may allow for low participation when outsourcing labor

is expensive for those with bigger farms. It can also lower participation among those

with small farms as they are more inclined to offer labor to those with big farms when

harvests have been adverse.

H9: Age of household head can hinder participation in bundled nutrition

interventions

Old age tends to hinder participation in interventions and this can also be at-

tributed to distance traveled and gender factors [67].

3.2.6 Determinants of change in female dietary diversity scores for ATONU

bundled intervention

The response of women to nutrition interventions has been shown to vary along

contextual factors [68]. Livestock ownership and market participation of WRA are

associated with the adequacy of dietary diversity [55]. Gender has been shown to be a

significant factor on dietary diversity and agro-ecological zones are insignificant [53].

Husbands support and more participation of women in household financial decisions

enhances women’s adequate dietary diversity [55]. Home vegetable gardening and

food preparation and nutrition knowledge are positively associated with household

dietary diversity [69]. Ownership of livestock and female headed households improve

on dietary diversity for rural communities [70]. Family food security and farm pro-

duction diversity facilitate dietary diversity [71]. Linear regression models for dietary

diversity have also indicated low R2 values showing that there are other potential

determinants of dietary diversity that need to be discovered [72]. The hierarchical

structure introduced in the ATONU bundled interventions calls for hierarchical de-

fined mixed models use in analyzing their target outcome. We seek to compare linear

and Poisson mixed models on how they ascertain the change process on WRA’s di-

etary diversity outcomes in relation to ATONU intervention’s participation dynamics

and contextual factors.

The literature described above suggests that contextual and background charac-

teristics of the participants are related to the WRA dietary diversity scores. We

address these factors with measurements from lower hierarchical levels (disaggregate

metrics) and also adjust for process-driven participation metrics for the assessment

of dietary diversity scores for bundled nutrition interventions. We model the effects

of these covariates on the bundled ATONU intervention outcomes using linear and

Poisson (with a log link function) mixed models. The generalized mixed model is

given by

Yijk = µ+ αi + βj(i) + τZ + εijk (3.6)

where yijk is the women’s dietary diversity, µ is the grand mean, αi is the fixed

region factor, βj(i) is the village nested in region random factor, Z is the contextual

covariate defined at either i, j, k level and βj(i) ∼ N(0, σ2β(α)), εijkl ∼ N(0, σ2).

3.3 Results

3.3.1 Variation decomposition for process-driven participation metrics

Figure 3.4. Error bars for the compliance and BICR metrics for ATONUintervention

Figure 3.4 shows that there is variation among the four regions of this study for the

process-driven metrics of compliance and BICR. This variation is further distinguished

both between and within the villages in the regions. Compliance metrics have a wider

range in comparison to the BICR metric. Generally there was good compliance and

a poor BICR among the villages.

Table 3.4.Variance decomposition for the compliance and BICR metrics for ATONU

Source of Variation Compliance BICR

Between villages(nested in regions) .047 (25.0%) .027(82.7%)

Within Villages(nested in regions) .141 (75.0%) .006(17.3%)

Table 3.4 shows that the variation in the compliance metric was larger within

villages and that for BICR was larger between villages. Compliance improvement

requires focus on the participantsâĂŹ engagement and addressing disparities in par-

ticipation. For the BICR there is need for supervision improvement for the research

staff to ensure that they deliver all the bundle components in all the villages and for

minimization of staff turnover.

The hom*ogeneity of variance tests for both compliance and BICR gave p-values

<.0001, indicating presence of heterogeneity among the villages. This supported the

use of Beta-logistic model in the identification of the determinants of the participation

dynamics for ATONU.

3.3.2 Determinants of participation and WRA dietary diversity scores

for ATONU

Statistical models with adjustments for clustering were utilized to identify the

covariates that influenced participation and the target outcomes of dietary diversity

scores for WRA for ATONU.

Table 3.5.Determinants of compliance for ATONU bundled intervention

Determinant Estimate Standard Error p-value

Baseline wealth quintile (ref=5) 0 - -

4 .1187 .0941 .2073

3 -.26685 .0956 .0054*

2 -.0120 .109 .9127

1 -.2980 .1304 .0226*

Family size .0537 .0156 .0006*

Women’s education (years) .0092 .0129 .4733

Women headed household (ref=0) 0 - -

1 .1391 .0899 .1219

Remoteness(minutes) .0002 .0010 .8555

Baseline parity (ref=1 infant) 0 - -

2 - 4 infants .0300 .1237 .8084

More than 4 infants .1575 .1180 .1824

Farm size (1 timad = 4 ha) -.0163 .0126 .1943

Agro-ecological zone (ref=low altitude) 0 - -

Medium altitude .1117 .5147 .8282

High altitude 1.1936 .6349 .0603

Age of household head (years) .0060 .0070 .3942

Table 3.5 shows the results based on the beta-logistic model with adjustments

for contextual factors. It shows that family size and baseline wealth were significant

determinants for compliance. The relative increase in the odds of compliance for a

unit increase in family size was 1.0552; and that for the first and third quintiles for

baseline wealth were 0.766 and 0.742, respectively.

Table 3.6.Determinants of BICR for ATONU bundled intervention

Determinant Estimate Standard Error p-value

Baseline wealth quintile (ref=5) 0 - -

4 -.0790 .0333 .0339*

3 -.1649 .0344 <.0001*

2 -.1523 .0371 .0008*

1 -.1172 .0379 .0020*

Family size .0015 .0050 .7580

Women’s education (years) -.0046 .0042 .2765

Women headed household (ref=0) 0 - -

1 -.0324 .0293 .2693

Remoteness(minutes) -.0003 .0004 .4124

Baseline parity (ref=1 infant) 0 - -

2 - 4 infants .0362 .0444 .4146

More than 4 infants .0384 .0427 .36794

Farm size (1 timad = 4 ha) .0011 .0026 .6776

Agro-ecological zone (ref=low altitude) 0 - -

Medium altitude -.0514 .1055 .6263

High altitude .1718 .1300 .1863

Age of household head (years) .0047 .0022 .0349*

Table 3.6 shows that baseline wealth and age of household head were significant

factors in determining the number of bundle components that the participants re-

ceived. The relative increase in the odds of BICR for a unit increase in the age of

the household head was 1.0047; and for the first up to the fourth quintile of baseline

wealth were 0.924, .848, .859, and 889, respectively.

Table 3.7.Determinants of men’s participation for ATONU bundled intervention

Determinant Estimate Standard Error p-value

Baseline wealth quintile (ref=5) 0 - -

4 .2891 .1937 .1417

3 .2956 .2112 .1676

2 -.1770 .2234 .4316

1 .3581 .2262 .1194

Family size .1591 .0306 <.0001*

Women’s education (years) -.0127 .0244 .6043

Remoteness(minutes) -.0028 .0022 .1989

Baseline parity (ref=1 infant) 0 - -

2 - 4 infants .8914 .2607 .0020*

More than 4 infants .8343 .2505 .0025*

Farm size (1 timad = 4 ha) -.0446 .0180 .0131*

Agro-ecological zone (ref=low altitude) 0 - -

Medium altitude -.1939 .7171 .7924

High altitude 2.6986 .8982 .0132*

Age of household head (years) -.0135 .0129 .2930

Table 3.7 above shows that determinants for men’s participation were family size,

high elevation, farm sizes, and the number of infants in households. The relative

increase in the odds of men’s participation for a unit increase in family size was

1.1725, for high altitude agro-ecological zone (12.7458), farm size (0.9564); families

with between 2 and 4 infants (2.4385), and for families with more than 4 infants

(2.3032).

Table 3.8.Determinants of women’s participation for ATONU bundled intervention

Determinant Estimate Standard Error p-value

Baseline wealth quintile (ref=5) 0 - -

4 -.3940 .1884 .0414*

3 -.1111 .1992 .5795

2 -.0408 .2166 .8479

1 .0825 .2167 .7048

Family size -.0069 .0285 .0151*

Women’s education (years) .0128 .0239 .5934

Remoteness(minutes) .0043 .0020 .0373*

Baseline parity (ref=1 infant) 0 - -

2 - 4 infants -.5048 .2574 .0603

More than 4 infants -.5880 .2479 .0251*

Farm size (1 timad = 4 ha) -.0392 .0154 .0110*

Agro-ecological zone (ref=low altitude) 0 - -

Medium altitude .4953 .5828 .4152

High altitude -.5989 .7218 .4260

Age of household head (years) .0025 .0124 .8383

Table 3.8 shows that women were driven to participate in ATONU bundled nutri-

tion intervention because of the demands of their family size, distance to the meeting

place (distance to the market was the proxy), farm sizes, and the number of infants in

their families. The relative increase in the odds of women’s participation for a unit

increase in family size was .9303, for distance to meeting place was 1.0041, farm size

(1.0378), and for families with more than 4 infant children (.5554).

Table 3.9.Determinants of joint participation for ATONU bundled intervention

Determinant Estimate Standard Error p-value

Baseline wealth quintile (ref=5) 0 - -

4 -.2591 .2890 .3741

3 .2429 .2898 .4058

2 -.3870 .3062 .2119

1 .6184 .2910 .0384*

Family size .1359 .0388 .0005*

Women’s education (years) .0017 .0335 .9606

Remoteness(minutes) .0037 .0026 .1574

Baseline parity (ref=1 infant) 0 - -

2 - 4 infants .7514 .3929 .0665

More than 4 infants .4779 .3782 .2172

Farm size (1 timad = 4 ha) -.0047 .0173 .7854

Agro-ecological zone (ref=low altitude) 0 - -

Medium altitude 1.4384 1.8759 .4609

High altitude 4.7004 2.4273 .0816

Age of household head (years) -.0298 .0172 .0847

Table 3.9 above shows that the relative increase in the odds for joint participation

were 1.8560 for the first quintile of baseline wealth and 1.1456 for a unit increase in

family size.

The Poisson mixed model was not a good fit for the WRA’s dietary diversity

scores with adjustments for clustering, associated covariates and the process-driven

metrics. Its χ2

dfstatistic was not approximately equal to one. The linear mixed model

was a good fit based on the AIC.

Table 3.10.Determinants of WRA end of the intervention 24-hour recall dietary di-versity score for ATONU bundled intervention

Determinant Estimate Standard Error p-value

Baseline wealth quintile (ref=5) 0 - -

4 -.1297 .0961 .1772

3 -.0473 .0987 .6316

2 .0157 .1071 .8833

1 .1264 .1095 .2487

Family size .0210 .0139 .1322

Women’s education (years) .0161 .0094 .0859

Remoteness(minutes) -.0024 .0010 .0193*

Baseline parity (ref=1 infant) 0 - -

2 - 4 infants .0211 .1258 .8670

More than 4 infants .2026 .1204 .0928

Farm size (1 timad = 4 ha) .0206 .0069 .0029*

Agro-ecological zone (ref=low altitude) 0 - -

Medium altitude -.0008 .2152 .9971

High altitude -.2439 .2651 .3792

Age of household head (years) .0025 .0050 .6102

Women headed household (ref=0) 0 - -

1 .1565 .0837 .0618

Compliance .4331 .1557 .0055*

BICR -.1802 .3421 .5985

Women participation(ref=0) .0274 .0630 .6643

Men’s participation(ref=0) -.0258 .0657 .6949

Joint participation(ref=0) .0065 .0850 .9389

Livestock diversity .1082 .0234 <.0001*

Baseline dietary diversity score .4483 .0301 <.0001*

Table 3.10 shows that for a unit increase in the distance to the market (a proxy

for remoteness), farm size, compliance, livestock diversity, and baseline 24 hour recall

dietary diversity score, the end of the intervention 24 hour recall dietary diversity

score would change by -.0024, .0069, .4331, .1082, and .4483 units, respectively.

Table 3.11.Determinants of WRA end of the intervention 7-days recall dietary diver-sity score for ATONU bundled intervention

Determinant Estimate Standard Error p-value

Baseline wealth quintile (ref=5) 0 - -

4 -.0547 .1413 .6986

3 .0411 .1451 .7770

2 .0500 .1576 .7513

1 .3322 .1613 .0396*

Family size .0660 .0204 .0013*

Women’s education (years) .0384 .0144 .0079*

Remoteness(minutes) -.0019 .0015 .2159

Baseline parity (ref=i infant) 0 - -

2 - 4 infants .1034 .1830 .5721

More than 4 infants .2930 .1752 .0947

Farm size (1 timad = 4 ha) .0361 .0103 .0005*

Agro-ecological zone (ref=low altitude) 0 - -

Medium altitude -.2150 .3950 .5980

High altitude -.6561 .4867 .2073

Age of household head (years) -.0013 .0076 .8642

Women headed household (ref=0) 0 - -

1 .0977 .1233 .4281

Compliance .5813 .2304 .0117*

BICR -.6435 .5039 .2018

Women participation(ref=0) -.0040 .0929 .9658

Men’s participation(ref=0) -.0067 .0971 .9451

Joint participation(ref=0) -.0184 .1256 .8838

Livestock diversity .1839 .0341 <.0001*

Baseline dietary diversity score .4791 .0285 <.0001*

Table 3.11 shows that for a unit increase in the family size, women’s education,

farm size, compliance, livestock diversity, and baseline dietary diversity score, the end

of the intervention 7 days recall dietary diversity score would change by .0660, .0384,

.0361, .5813, .1839, and .4791 units, respectively. The relative increase in the odds

of end of the intervention 7 days recall dietary diversity score for participants in the

fifth quintile level of baseline livestock wealth was 1.3940.

3.4 Discussion

The ATONU bundled intervention was implemented heterogeneously in terms of

content delivery and timing among the villages. This impacted the participation

dynamics over the course of the intervention and the number of the bundled com-

ponents received by the participants. The process-driven metrics of compliance and

BICR had heterogeneous variance. The variance decomposition showed greater vari-

ation within villages for compliance and greater variation between villages for BICR.

Compliance improvement requires participants’ mobilization, and BICR improvement

requires staff retention and delivery of all the bundled components.

The significant determinants for men’s participation were farm size, high agro-

ecological zone, having at least two infants in the household, and farm size. Women’s

participation was determined by the fourth quintile for baseline wealth, family size,

remoteness, having at least 5 infants in household, and farm size. The joint partici-

pation was determined by the first quintile baseline livestock wealth and family size.

The Poisson mixed model was not a good fit for the end of the intervention WRA

dietary diversity scores compared with the linear mixed model. The determinants for

compliance were the third quintile of baseline livestock wealth, and family size; while

those for BICR were the age of the household head and baseline livestock wealth

for the range second to fifth quintiles. Remoteness, farm size, compliance, baseline

dietary diversity score were the determinants for the end of the intervention dietary

diversity score for 24 hour recall. The first quintile of baseline livestock wealth, family

size, women’s education, farm size, compliance, and baseline dietary diversity score

were the determinants for the end of the intervention dietary diversity score for 7 days

recall. The 24 hour recall and the 7 days recall dietary diversity scores are proxies

for household food access and consumption, measured in terms of the variety of food

types consumed.

The common determinants for both dietary diversity measures shows a positive

contribution. Distance away from the meeting place (remoteness) has a negative ef-

fect on the dietary diversity for the 24 hour recall, while family size, baseline wealth

and women’s education have a positive effect on the 7 days recall dietary diversity

metric. Compliance contributed positively to both dietary diversity measures while

the gender and BICR metrics had insignificant effects.

The mediation analysis conducted for the ATONU bundled intervention showed

that compliance was a significant determinant for both measures of dietary diversity

for WRA. Adjustment for clustering, compliance, baseline WRA dietary diversity

scores, livestock diversity, and the contextual and background characteristics’ are

important for linking the intervention to the end of the intervention WRA dietary

diversity scores.

3.5 Conclusion

There are different context-sensitive profiles of engagement for bundled nutrition

interventions. Process-driven metrics capture aspects of implementation that are

missed by traditional metrics. Identifying at which level of the hierarchical implemen-

tation variation exists for these process metrics allows for the differentiation among

strategies and decisions to improve implementation quality. Poor implementation

can be attributed to staff turnover, supervision, context, and participants’ decisions.

We applied the metrics to identify the determinants of greater participation by the

target households, men and women. The determinants of greater participation by tar-

get households included farm size, baseline parity, baseline wealth, and family size.

These should be targets for the improvement of implementation for future bundled

interventions. These attributes are important for the establishment of the bundled

nutrition interventions’ complex ToC that can substantiate their causal statements.

Compliance had a significant effect on the WRA’s dietary diversity scores, showing

that to effectively ascertain the impact of bundled interventions on outcomes com-

pliance has to be adequately measured and monitored. In spite of the BICR being

an insignificant factor for the effects of the intervention on the WRA’s dietary diver-

sity scores, the low values shows that there is a need to promote adherence to the

implementation of the intervention by the research staff to ensure delivery of all the

bundled components for the full realization of its impact.

4. SIMULATION STUDY OF TIME SERIES MODELS GENERATED

BY UNDERLYING DYNAMICS

4.1 Introduction

Time series analysis has been successfully applied in many areas of science and

engineering. This has been necessitated when data records met strong statistical

assumptions underlying traditional methods and were long enough for the results ob-

tained by these methods to be reliable. In atmospheric and climate studies, however,

observed records are often prohibitively short with only one record typically available,

and the underlying assumptions for time series modeling are rarely met [14].

4.2 Motivating Example

Figure 4.1 below shows a typical atmospheric record - the vertical velocity of wind

in a convective boundary layer, taken 29km across lake Michigan, 50m above the lake.

Figure 4.1. Record of 20-Hz vertical velocity measurements over LakeMichigan. Figure from [73]

For this realization of data, the routinely computed sample mean, variance, skew-

ness, and kurtosis were -0.04, 1.06, 0.83, 4.10, respectively. The elevated skewness

and kurtosis (from values 0 and 3 specific for a normal distribution) were attributed

to the occurrence of coherent structures in turbulent flows [74], but to learn the ex-

tent one can trust such statistics, confidence intervals (CI) are needed. The extent to

which sample statistics estimate the underlying population parameters is on its own

an open-ended research problem [75]. To make inference on such numbers, a measure

for precision would be required to account for the associated random error [76]. The

establishment of the measure of precision depends on the assumptions made on the

data generating mechanism for the underlying population.

The other challenge is the attainment of the accuracy level (coverage probability,

say 0.90). This is attained only if the assumptions underlying the CI construction are

met, a common one being that the model generating the series is linear. Atmospheric

time series are produced by inherently nonlinear systems, hence the linearity assump-

tion fails to be met. The actual coverage probability may differ from the target level

(0.90), sometimes considerably. Moreover, the CIs for the skewness cannot be based

on linear models, which imply zero skewness, inference made from such models would

be unreliable. Thus, there is a need for nonlinear models, but finding an appropriate

one among the conventional time series models is problematic.

We aimed at improving the reliability of statistical inference on atmospheric data

through time series models generated by atmospheric underlying dynamics. The fol-

lowing were the objectives of our study;

1. estimating the subsampling confidence interval for the skewness of the vertical

velocity of wind using time series generated from the underlying dynamic of

atmospheric systems (G-models) and comparing them with those from conven-

tional nonlinear time series models.

2. expanding on the G-models to incorporate more atmospheric mechanisms and

compare and contrast their associated subsampling confidence intervals with

the basic G-model at varying confidence levels.

4.3 Literature Review

4.3.1 Modern Statistical inference

The progress in Statistics has been stimulated and can be traced to the real-

ization of what statisticians can provide to address the problems in real world ap-

plication areas. This offers an indication for a mutual and symbiotic relationship

between statistical theory and statistical applications [77]. Theory on one hand offers

the framework, guidelines and arguments for statistical methodologies development,

while applications aid in the justification of the postulated assumptions and rele-

vance of inference derived through the statistical methods. A considerable number of

statistical methodology have been developed through endeavors to solve problems in

physical sciences and engineering. Response surface design was developed by George

Box in his collaboration with chemical engineers, exploratory data analysis (EDA)

was developed by John Tukey alongside telecommunication engineers, and sequential

testing was postulated by Abraham Wald in his work with military engineers [77].

Interestingly, most of these developments relied on both Fisher and Neyman’s con-

siderations on statistical modeling [78].

R.A Fisher identified that specification, estimation, and distribution were the

fundamental problems for modern statistical inference, but little attention has been

directed towards addressing the specification challenge [79, 80]. Statistical models

are fundamental for Statistics, hence the issue of their specification require utmost

attention. In particular, the role of subject matter in statistical modeling is crucial

to address the relevance of inference in statistical applications [81]. This is so be-

cause the specification problem centers on the choice of the mathematical form of the

population from which the sample originates, i.e. addressing the question of how the

observed data was generated [79].

One of the foundational problems on frequentist inference is the role of subject

matter information in statistical modeling, in terms of their theoretical explana-

tion [82]. According to Fisher, data generating mechanism (DGM) are important

for addressing the specification problem, and that often may require knowledge be-

yond Statistics [79]. If there is some form of subject matter information available for

a phenomenon of interest, statistical models should incorporate it [82]. Neyman’s ex-

planatory models make an attempt to explain the mechanism underlying the observed

phenomena [81]. The two schools of thought on statistical model building of Fisher

and Neyman creates an interesting position on how to address the DGM, the former

tend to make assumptions on it and the latter acknowledges its presence and contri-

bution. This scenario highlights the need for statistical theory and subject matter

expertise to be considered in statistical modeling to enhance applications especially

in situations where the data is generated under complexities, simplification of which

may belittle the inference obtained.

Statistical models are often data driven, which may fall short in their relevance

for application areas when expertise knowledge is not considered. The latter aspect is

the basis for the argument for data-centric statistical model building, which require

the incorporation of the scientific understanding of the application area and pertur-

bations allowing for randomness to improve the relevance and reliability of inference

for statistical applications. When statistical models are fully pre-specified, some of

the deficiencies in statistical inference can be resolved [83].

4.3.2 Dynamical systems theory and nonlinear time series analysis

Dynamical systems theory is a branch of mathematics that consists of principles

and tools for studying serial changes in physical or artificial systems. Lorenz em-

phasized on the importance of understanding the nonlinearity of atmospheric motion

in modeling procedures [84]. Most of the systems in nature can best be described

through nonlinear models [85]. The complexity of geophysical phenomena can be

exemplified by temperature which requires high-dimensional physics-based models of

the atmosphere instead of AR(1) models to accurately describe it [86]. Nonlinear

time series utilizes dynamical systems theory in the analysis of univariate observa-

tional data [86]. Our knowledge of the underlying systems is often restricted to the

information we have from a single realization of data from a variable in the system,

called a time series [85]; thus the state-space reconstruction of the underlying attrac-

tors for the system forms the foundation for nonlinear time series analysis [86]. The

later can be co-opted in the time series model to explain the underlying dynamics, in

the system under study, responsible for the generation of the time series data.

Attempts to model nonlinear non-normal time series has led to the development

and utilization of new models such as the newer exponential auto-regressive (NEAR)

and product auto-regressive models (PAR) [87], which depend on the AR(1) model’s

characterization that is often used in atmospheric modeling. Such models may ac-

knowledge the nonlinearity and non-normality of the observed data, and consequently

can give similar statistics but do not exhibit the fundamental theory underlying the

operations effecting the generation of the observed data in dynamic systems. This

overshadows the reliability of their statistical inference. The building of phenomenon-

specific models as derivatives of the governing physical laws and associated properties

and controlling variables can enhance the modeling of systematical variables [88]. We

seek to postulate nonlinear statistical models that can explain the underlying pro-

cesses for atmospheric phenomena, which are typically complex in their comprehen-

sion as they are generated from the interaction of nonlinear atmospheric processes.

4.3.3 Atmospheric systems and statistical inference

Mathematical models underlying phenomena in physical science and engineer-

ing are a source of prior knowledge about the problems that are in need of being

solved [89]. They help in the description of the science of the problems we intend to

address using statistical methods. The amalgamation of such mathematical models

in statistical procedures and the use of statistical techniques in estimating the param-

eters of the mathematical models can aid the statistical modeling and interpretation

of data realizations on physical phenomena [89]. In essence, scientifically justified

statistical methodology are pivotal for understanding the often complex underlying

dynamics responsible for the generation of the observed physical science data. Sta-

tistical research aims to develop tools for use at the frontiers of science which can be

heightened through collaborations, as statisticians acquire/comprehend application

area knowledge and offer statistical expertise [90]. These endeavors can help facil-

itate agreement between statistical significance and substantive significance which

thereby aid the relevance of statistical applications in scientific research.

The atmosphere is a complex nonlinear system with mechanisms such as rotation,

topography, shear, and stratification, constituting its underlying dynamics. A dy-

namical system can be mathematically defined by the triple, (Ω, φ, T ), where Ω ⊆ Rd

is the state space, φ is an evolution operator, and T denotes the set of possible

times [91]. Atmospheric processes are essential to the determination of the state

of the climate, and to climate change studies. Statistical inference are conclusions

drawn on unknown population parameters based on probability models of data gen-

erating processes, based on sampled data [83]. On the contrary scientific inference

depend on the accumulated subject-matter knowledge acceptable by members of the

field, which plays a crucial role in their acceptance of new findings [46]. Inferential

problems for atmospheric data can be attributed to the need for both their deter-

ministic and statistical properties to be incorporated in their modeling. This allows

for an understanding of the atmospheric system through physical thinking applied to

statistical analysis of the observed data. Statistical modeling of climate phenomena

should be preceded by consideration of the nonlinearity property of their underlying

dynamics. Classical time series models hinge on unrealistic assumptions on the data

generating mechanisms (DGMs) for atmospheric data, yielding misleading inference.

In particular, the usual assumptions for time series of linearity and stationarity are

often violated in practice [91]. Alternative time series models for classical time series

models should capture the underlying theory and provide potentially better forecasts

for the observed series [92]. According to [92] such models must exhibit the following

features;

(i) They must be interpreted and based on potentially realistic theory.

(ii) Must exhibit the stability condition that is necessary for the stationarity of their

associated time series.

(iii) All their components must at least be potentially observable.

Atmospheric data are non-normal and high-order moments such as skewness and

kurtosis are required for their description [8]. Skewness measures the asymmetry of

distribution, while kurtosis measures the peakedness of a distribution function. Much

of the information that has been acknowledged as missing from the first and second

moments maybe found in the third and fourth moments, and especially if they are

tied to the physics underlying the observed data [93]. Higher order moments can be

used to ascertain the levels of normality for atmospheric data, in particular, their

skewness has been shown to be significantly different from the zero value for normal-

ity [94]. These nonlinearities in the underlying data generating mechanisms (DGM)

for atmosphere data promotes misleading inference from traditional time series mod-

els that assume linearity [95]. On the other hand, statistical advances have shown

that slight deviations from normality are a source of great concern [96]. Subsampling

methods which work under weak assumptions are a useful option for finding the stan-

dard errors for high-order moments [73]. The variability of non-normal data depends

on their underlying distributions [8].

Sampling distribution is fundamental to statistical inference as it allows for relat-

ing sample statistics to population parameters [97]. Hence, efforts to make inference

on atmospheric data using subsampling methods with approximating models that do

not infuse the physics of the original data can be questionable. On the other hand,

resampling methods though flexible may under-perform in handling atmosphere data

whose observed realizations are commonly too short for asymptotic inference. Knowl-

edge of higher order statistical moments plays a crucial role in validating the approxi-

mating models for extreme events [8], a chief characteristic of atmosphere data. They

also serve to assist in the analysis of the coherent structures (CSs) of atmosphere

and climate data that are characteristically non-normal [95]. A coherent structure

is said to be a connected turbulent fluid mass with phase-correlated vorticity over

its spatial extent [98]. The CSs are responsible for the heat and moisture exchange

that is responsible for the transportation of mass and momentum, which heightens

the measures for skewness and kurtosis [95]. Coherent structures occur in localized

regions of persistent vorticity, and they strongly influence heat exchange and turbu-

lent flows between locations [74]. Fully developed turbulence is prevalent at boundary

layers [93], and investigations of atmospheric phenomena there need to take into ac-

count its presence and impact on the assumptions for their DGM.

Confidence interval (CI) provides information on the amount of random error as-

sociated with an observed statistic (precision) and on the probability of how it relates

to the corresponding parameter in the population from which the sample under in-

vestigation was drawn (accuracy) [99]. The trade-off between precision and accuracy

is that an increase in precision entails a decrease in accuracy, and vice versa. Some of

the advantages of confidence intervals include their link with p-values for hypothesis

testing, they give information about precision, and estimates are in units that are

readily comprehensible with the research context [100].

Statistical modeling seeks to complement mathematical modeling of atmospheric

phenomena, as their forecasting ability hinge on the computing power, quality of

data, and the challenge of initial conditions for the complex equations. Many strides

are being made for the treatment of physical processes in atmospheric models and

the exploration of advanced statistical methods. We seek to highlight the importance

of subsampling methods for inference on atmospheric data using G-models as time

series models whose DGM are inherited from their governing equations.

4.3.4 Subsampling Confidence intervals

It is a resampling procedure without replacement from the original sample n,

yielding samples of smaller size b, where b n [101]. This techniques works in

complex situations without asserting unverifiable assumptions on the data generating

mechanisms (DGM). The record at hand of length n is divided in n− b+ 1 subsam-

ples or blocks of consecutive observations, all of the same length b, that retains the

dependence structure of the series [102]. The technique of randomization which is at

the heart of most simulation and resampling techniques can affect the resultant in-

ference based on their assumption of the randomness of the data. In order to capture

physical meaningful relationships, there is a need for procedures such as subsampling

that allow for the capture of the complex dependent structure between observations.

Subsampling allows for samples to be taken from the true unknown distribution func-

tion F of the original sample. This technique contrasts Efron’s bootstrap method

in that it uses b instead of n on sample size, and also that bootstrap samples are

from an empirical distribution F associated with the original sample. Subsampling

can be used on dependent data which are identically distributed (ID) and for ex-

treme events that are independent and identically distributed (IID). In contrast,

bootstrap requires distribution of data to be both identically and independently. The

scenario above, does not give subsampling any superiority over bootstrap, but rather

opportunities for us to experiment with it in more varied situations. The blocking in

subsampling can capture the dependence in the original data, which allows it to work

for stationary time series data.

Subsampling has been proposed to be a method for estimating parameters for the

sampling distribution of statistics based on sub-series [103]. The performance of such

parameter estimates for fixed n depends on the sub-series length b. Suppose we are

interesting in inference on a parameter θ, typically a summary or shape measure for an

observed time series realization, using the subsampling procedure. We postulate that

θn is an arbitrary statistic that is consistent for θ at the convergence rate of τn, then

for large n, τn(θn−θ) tends to some well-defined asymptotic distribution, say J [104].

The distribution of J needs not to be normal or its shape to be known but that

its existence be acknowledged, and the main hypothesis in subsampling is that the

subsampling empirical distribution converges weakly to J , the limiting/asymptotic

distribution. The subsampling estimator for J will be the associated empirical distri-

bution of τb(θi,b − θn), where θi,b is the subsampling value for the statistic of interest

that was obtained from the i subsample of size b.

Subsampling confidence intervals were developed in [101], and in particular for

stationary time series to address the problem of estimating variance of a statistics

based on values of that statistic computed from sub-series. The use of overlapping

blocks is more efficient, in comparison to non-overlapping sub-series, but they are

both L2 consistent and almost sure convergent [102]. Biased reduction ensures that

the estimate is closer to the parameter of interest, at time series statistics are often

heavily biased. The asymptotic consistency of the subsampling estimator of J has

been shown [101] and it allows for the construction of confidence intervals for θ using

its quantiles instead of those for the unknown J [104].The following assumptions fa-

cilitates the construction of subsampling confidence intervals for unknown parameters

θs of time series of asymptotically correct coverage when.

(i) b→∞

(ii) bn→ 0

(iii) τb →∞

(iv) τbτn→ 0

Under these assumptions, the weak convergence in distribution hypothesis is sat-

isfied, where τn is the convergence rate, given by nβ, for 0 < β < 1. The sampling

distributions for the sub-samples and that of the the original sample are close to each

other. If β is 0.5, this satisfies the "square root law" for the standard error. This is

not so for atmosphere data as their limiting distribution is non-normal. Variance is of

order O( bn), hence it requires that the first two assumptions above, be satisfied. The

following weak conditions, which can be relaxed, to work alongside the assumptions

for subsampling confidence interval above;

(i) The observed time series is strictly stationary.

(ii) The observed time series is strong mixing.

(iii) The rate τn is known.

Upon relaxation the first condition allows for asymptotic stationarity, the second

whittles to the weak dependence condition [104]; while the last condition is important

for the practical considerations for subsampling confidence interval construction. The

use of subsampling methodology in the derivation of a consistent estimator for τn, has

facilitated the relaxation the third condition above [105]. The latter estimate is then

used for the subsampling confidence interval construction with the actual coverage

that is as near to the target coverage as possible. Overall, the subsampling method

does not require any specific knowledge of the structures of the time series other than

its attributes of asymptotic stationary and strong mixing.

The estimator for the statistic of interest Tn, θ depends on the unknown distri-

bution F . The difficult part to subsampling procedure is the determination of the

underlying subsampling distribution, F . Monte Carlo simulations for time series data

require models that can preserve the dependence structure in the data for reliable in-

ference to be made. In the case under review, valid confidence intervals for skewness

and kurtosis for nonlinear time series cannot be obtained using linear models [95].

The two issues that have to be addressed concurrently for subsampling confidence

intervals’ efficacy are the short record of realizations and the approximation of the

underlying data generating mechanisms (DGMs) for atmosphere data.

4.3.5 The challenge of short record length for atmosphere data

Subsampling confidence interval construction depends on the block size for their

accuracy, and they also have to contend with the challenge of shortness of atmosphere

data realizations. They tend to fail to satisfy the conditions for the assumptions for

subsampling confidence interval, and so in practice, approximating models are needed

(those sharing statistical properties with the series under study) to assess the actual

coverage of the subsampling confidence intervals. In order to satisfy the convergence

in distribution assumption for subsampling methods, a convergence rate is needed.

It ensures that the target coverage is attained in the computations of the confidence

interval. The empirical convergence rate τn = nβ was introduced [106], where the

value of the exponent β was different from the theoretical one.

Atmospheric data records are usually short in length, and single realizations that

can contain very specific attributes. Monte Carlo simulation has been used to address

the challenge of the short record length and models with similar statistical properties

help in the selection of the optimal block size [107]. Plots of block size b against

coverage are profound in the determination of the optimal fixed b for subsampling

confidence interval construction. The use of approximating models that exhibit some

of the statistical properties of the original data as sampling distributions for the

subsampling procedure has also been shown to be helpful in ensuring that the target

coverage is attained [107].

4.3.6 Time series modeling challenge for atmospheric data

The primary purpose for time series analysis is to develop statistical models that

can describe the sampling data, which is an often data-driven endeavor. The "confu-

sion factor" postulated in [108] shows the challenge of model computation agreement

with observations, at the expense of the sufficiency of the model’s representation of

the physical processes underlying the data. The distribution of τb(θi,b − θn) in sub-

sampling confidence interval is empirically derived from the subsamples data, that

has nothing to do with the original data. In such instances, it may yield some of

the statistical properties of the data, but falls short in accounting for the influence of

the physics of the atmospheric data under investigation. [108] proposed that models

of low complexity would be appropriate in geophysical simulations to reach scientific

conclusions.

We seek to employ a new form of time series models that retain the physics of

atmosphere data in the construction of the confidence interval for their skewness.

They are characteristically simple, have the conservative property, and are able to

incorporate mechanisms peculiar to atmospheric dynamics for their expansion which

further retain the atmospheric reality.

Time series serve to offer some information about the systems that generate them,

whose comprehension is pivotal for predictions to be made on the time-dependent

variables under consideration. The assumptions made on the underlying DGM goes

a long way in giving credit on the inference made in time series analysis. The govern-

ing equations and field records helps in advancing our understanding of atmospheric

dynamics [13]. The assumption of normality do not hold for atmospheric data, which

is often non-normal and non-linear, hence inference made from classical time series

analysis can be misleading. The underlying dynamics for atmospheric data are non-

linear [73], which has to be captured in the approximating models. AR(1)-based

nonlinear models satisfying some of the statistical properties have been employed,

but they have nothing to do with the physics of atmospheric data.

Low-order models (LOMs), are a system of finite ordinary differential equations

(ODEs) popularized by [109] that approximates the partial differential equations

(PDEs) underlying the DGM for atmospheric data. These however fail to retain the

conservative properties of the original PDEs in their endeavor to realistically model

atmospheric dynamics, due to mathematical problems encountered in their establish-

ment. This problem was solved through the establishment of G-models, which are

physically sound, proposed by [13]. G-models have been shown to capture some of the

statistical properties of atmospheric data, and their allowance for the incorporation

of more mechanisms peculiar to them to improve their capture of the reality of the

original data have been documented.

4.3.7 Related Works

We seek to investigate the nonlinear atmospheric data on vertical velocity of wind

in a convective boundary layer, Figure 4.1 data. The convective boundary layer is

the part of the atmosphere that is most directly affected by the solar heating of the

earth’s surface. Buoyancy is an atmospheric mechanism that is generated by the

heating from the surface, and it is responsible for the vertical transportation of heat,

pollutants, moisture and momentum. Buoyancy is responsible for the generation of

convective turbulence which is an important aspect for global climate modeling and

for the dynamics of many atmospheric phenomena. The treatment of turbulence as

a random process raise profound statistical questions [110]. Efforts to construct the

90% subsampling confidence interval for the skewness parameter of these data have

brought eye-opening results depending on the underlying approximating model and

tuning parameters involved.

Subsampling confidence intervals were developed [101] for stationary time series

to address the problem of the estimating variance of a statistic based on its values

computed from sub-series. This procedure allows for the construction of confidence

intervals from single records of time series. The use of overlapping blocks was found to

be more efficient, in comparison to non-overlapping sub-series, but they are both L2

consistent and almost sure convergent [102]. Bias reduction ensures that the estimate

is closer to the parameter of interest, as time series statistics are often heavily biased.

The estimator for the statistic of interest Tn, θ depends on the unknown distribution

F . The difficult part to subsampling procedure is the determination of the underly-

ing subsampling distribution, F . Monte Carlo simulations for time series data require

models that can preserve the dependence structure in the data for reliable inference.

Valid confidence intervals for skewness and kurtosis for nonlinear time series cannot

be obtained using linear models [95].

Subsampling confidence interval construction depends on the block size for their

accuracy, which in turn depends on the coverage level. In order to satisfy the con-

vergence in distribution assumption for subsampling methods, a convergence rate to

is needed to ensure convergence in distribution for the estimate of the parameter of

interest. This ensures that the target coverage is attained in the computations of

the confidence interval for accurate interpretation of the results. Plots of block size b

against coverage have been used to determine the optimal fixed b for use in subsam-

pling confidence interval construction. The sampling distributions for the sub-samples

and that of the the original sample are close to each other. If β is 0.5, this satisfies

the "square root law" for the standard error. This is not so for atmosphere data as

their limiting distribution is non-normal. Variance is of order O( bn), requiring that

the first two assumptions above, be satisfied.

The main problem encountered in subsampling confidence intervals (CIs) construc-

tion for the higher order moments, in particular for skewness of atmosphere data has

been on coverage probabilities. It has been noted that the actual coverage tend to

be considerably different from the target coverage, attributed to the availability of

a single record of data of limited length. A single record cannot adequately answer

a scientific question on its own, calling for at least meta-analytic thinking [111]. A

calibration function h : 1−α→ 1−λ, where 1−α is the nominal confidence level and

1− λ is the actual confidence level [107] can be applied. Attempts to use non-linear

time series models face the daunting task of choosing models that can adequately

capture non-linearity that is inherent in the DGMs of atmospheric data. Initially, the

nonlinear approximating models were borrowed from traditional time series analysis,

which allowed for the construction of subsampling CIs with the required coverage

using calibrations [106]. Their data generating mechanisms (DGMs), however were

considerably different from those of real atmospheric dynamics (though some statisti-

cal properties might be similar, thus motivating the choice of the models). The model

4.1 postulated by [112] was used in subsampling confidence interval construction

Xt = Yt + a(Y 2t − 1) (4.1)

where Yt is an AR(1) process, and for a = 0.145, the first four moments of Xt

were close to those of the observed vertical velocity of wind data [13]. The AR(1)

with φ = 0.83 served to fairly imitate the dependence structure as characterized by

autocorrelation functions. Model 4.1 is an AR(1)-based nonlinear model, and us-

ing the calibration h(0.95) = 0.9, gave a 90% subsampling confidence interval for

skewness of (0.41, 1.24) [73]. The need to ensure that the actual coverage meets the

target coverage led to the incorporation of a convergence rate function in the non-

linear models used to approximate the underlying dynamics of atmospheric data to

improve inference, using subsampling methods. Consideration of the convergence rate

τn = nβ, β ∈ (0, 1) [102] on model 4.1 (referred below as approximating Model A) for

β = 0.42 gave a markedly improved 90% subsampling confidence interval (0.56, 1.10)

for the skewness of the vertical velocity of wind data, in terms of precision. Both

methods served to show that there was nonlinearity in the vertical velocity of wind

time series, through indicating a positive skewness.

One could then presume that Model A might be adequate for fixing subsampling

confidence intervals, but there is no guarantee that other statistical properties of the

data and the model do not differ to considerably affect the intended applications.

The "confusion factor" postulated in [108] shows the challenge of model computation

agreement with observations, at the expense of the sufficiency of the model’s repre-

sentation of the physical processes underlying the data. The confusion factor is the

probability that an insufficient theory leads to similarities between model results and

observational data. In particular, for nonlinear time series model the justification

for model selection can be limited to the satisfaction of some and not necessarily all

statistical properties of concern for an investigation to be generalized. The use of

nonlinear time series methods to field measurements has been marred by controversy

because of their exclusion of the fundamentals of dynamical systems theory from their

theoretical basis [113]. The model in equation 4.1 was utilized because of the simi-

larity of the first four moments from it to those in the observed data set, which may

not hold in different data sets of the same variable under consideration. This may

create a disconnection between model and the underlying theory of the application

areas for time series as model parameters maybe subjective to the observational data,

i.e. data-driven, for the output to be consistent. The distribution of τb(θb − θn) for

the modified model (4.1) in subsampling confidence interval is empirically derived

from the subsamples data, that has nothing to do with the original data. In such

instances, it may yield some of the statistical properties of the data, but falls short

in accounting for the influence of the physics of the atmospheric data under inves-

tigation. Using Model A at a = 0.145, and β = 0.5 the theoretical convergence

rate, for various block sizes indicates under-coverage in the constructed subsampling

confidence intervals [106]. Estimating the skewness does require long records, and a

simple way to improve coverage is to increase the record length, which is possible via

Monte Carlo simulations with approximating models. This can be lead to the actual

coverage probabilities being closer to the target when the empirical convergence rate

of β = 0.42 is applied, in comparison to β = 0.5 as shown in Figure 4.2.

Figure 4.2. Actual coverage probabilities of 90% subsampling CIs withβ = 0.42 (in red) and β = 0.5 (in black) using Model A for the skewnessof nonlinear time series. Figure is adjusted and adopted from that in [106]

It has been proposed that models of low complexity can be appropriate in geophys-

ical simulations to reach scientific conclusions [108]. We seek to employ a new form of

time series models that retain the physics of atmosphere data in the construction of

the subsampling confidence interval for their skewness. G-models have the property of

retaining the physics of the underlying atmospheric dynamics and hence the statistics

obtained from them are a near reflection of the reality for the vertical wind velocity

under investigation. They have been used as physically sound low-order models in

problems of atmospheric dynamics [13, 114], and have drawn increasing attention in

various physical and mathematical studies [115–119].

4.4 G-Models and subsampling confidence interval for atmosphere data

Turbulent dynamical systems occur in systems exemplified by the atmosphere and

the ocean, and have large dimensional phase space [120]. These are responsible for

the behaviors exhibited by atmospheric and oceanic phenomena, i.e. the underlying

dynamics determining the measures on such phenomena. Atmospheric dynamics offer

an important advantage in providing the governing equations that generates the data

on phenomena we seek to model [13]. This is a reservoir of subject matter knowledge

that we can potentially tap into for statistical modeling. The governing equations for

atmospheric dynamics consists of partial differential equations (PDEs) [121], that are

problematic to solve due to the butterfly effect attributed to sensitivity to initial and

boundary conditions.

Simple models can advance our understanding of the atmosphere, but there is little

hope of establishing such models that can simulate all atmospheric processes from the

global to the micro-physical scale, at least in the foreseeable future [122]. Attempts to

handle them through approximations have led to the establishment of finite systems of

ordinary differential equations (ODEs) called low-order models (LOMs) [109]. Here,

we are seeking to represent a high dimension model with a simple model, and we are

transitioning from PDEs, to ODEs. LOMs have been used for studying atmospheric

phenomena, and their nonlinear Volterra gyrostats equivalence possess fundamental

properties of the PDEs, promoting their use as a basis for the the development of

G-models in particular [123]. Inasmuch as ODEs are a subset of PDEs, the reverse

does not hold as they are derivatives in multiple variables, the curse of dimension-

ality maybe apparent in this endeavor. This is so, because the nonlinearity in the

governing PDEs causes LOMs to contain more unknowns that equations, which cre-

ates a need for increasing the LOMs’ row dimension [124]. LOMs are an important

tool for geophysics fluid dynamics and need to retain the following features of the

original system, quadratic nonlinearity, and in the absence of forcing and dissipation,

conservation of energy and of space phase volume [125].

Gyrostat models (G-models) are a form of LOMs that exhibit sound physical be-

havior that were developed to solve the problem of loss of upholding the conservative

properties of the PDEs by the ODEs [13]. The loss of conservative properties is due to

the truncation employed in the Galerkin method for the construction of the LOMs.

The statistical properties of dynamical systems have been noted to be simple and

predictable, in particular the geometric Lorenz flow satisfies the almost sure invari-

ance principle (ASIP) because of the attractor present in it, which in turn implies

that they satisfy the central limit theorem [126]. Consequently, G-models satisfy the

central limit theorem, and exhibit the physical ergodic invariant probability measure

possessed by the Lorenz model [13], asserting their prospect as alternative time series

models for atmospheric dynamics [106]. The latter attribute of invariant probability

measure can be due to the fact that the flows described by Lorenz equations have a

basin that covers Lebesgue almost every point of the topological basin of attraction,

and are expansive [127].

A gyrostat is a mechanical system of bodies whose motion is explained by Volterra

equations, without changing the mass distribution of the system [125]. The Volterra

gyrostat is the basic G-model that consists of a system of mechanical and allowing for

fluid dynamical components of atmospheric dynamics [128] as shown below in (4.2).

x1 = px2x3 + bx3 − cx2,

x2 = qx1x3 + cx1 − ax3, (4.2)

x3 = rx1x2 + ax2 − bx1,

where p+q+r = 0, and the linear terms called linear gyrostatic terms do not affect

the conservation of energy or the conservation of phase space volume. They exhibit

some form of energy, the quadratic integral motion, that ensures that they retain the

physical behavior of the underlying atmospheric dynamics upon increasing the order of

approximation for the Galerkin method [128]. These models are simple, and unlike the

large numerical models often in use in climate numeric modeling, can also be used in

data simulations which allows for their potential use in resampling methodologies [13].

They have been used in problems of atmospheric dynamics [13,114], and have drawn

increasing attention in various physical and mathematical studies [115–119].

Subsampling procedures are extremely flexible making them one of the most intu-

itive method for statistical inference [129]. They can handle dependent data as they

hinge on a weak set of assumptions. The convergence in distribution assumption for

subsampling, allows for the use of the often consistent estimator of the asymptotic

distribution for the subsampling confidence interval construction using its associated

quantiles for the parameters of interest to one’s investigation [129]. The adoption

of such models as alternatives for time series analysis may allow for the realistic

representation of the underlying dynamics generating the data under investigation.

The simplest G-model (r=b=c=0) with added forcing and linear friction terms is

the G-model equivalence of the Lorenz model. The state vector X, [Xi], i = 1, 2, 3

for the Lorenz model consists of fluid velocity, horizontal and vertical temperature

gradients for modeling thermal convection [110]. The Lorenz gyrostat given below

fails to be a suitable approximating model for subsampling confidence interval for

atmosphere data in spite of its well-defined statistical properties, and possession of

the Rayleigh-Bénard convection (RBC), responsible for the generation of the original

data [13].

x1 = −x2x3 − α1x1 + F,

x2 = x1x3 − x3 − α2x2, (4.3)

x3 = x2 − α3x3,

Model (4.3)’s simulated records gave a skewness value of zero, which points to Gaus-

sian distribution, but the observed sample’s skewness value was 0.83. This result

shows the inadequacy of the Lorenz systems of equations as approximations for the

data generating mechanism for nonlinear atmospheric time series data.

Time series model specification must allow for the capture of the underlying data

generating mechanism’s salient features to facilitate relevance of inference made from

them [130]. One intricate feature of G-models is their allowance for the incorpora-

tion of mechanisms of atmospheric such as stratification, rotation, topography, shear,

magnetohydrodynamic effects as linear gyrostatic terms, to capture the physics of the

underlying dynamics [13]. These facilitates their capture of the atmospheric reality,

and the use of such models for time series heightens their usefulness for this particular

application, and appeal amongst atmospheric scientists in particular and physical sci-

entists in general for their scientific inference on atmospheric data. The introduction

of one pair of linear gyrostatic terms in model (4.3) as shown in model (4.4), herein

called Model B, below for a value of 0.35 for the constant c, yielded the values of 0.81

and 4.2, for skewness and kurtosis, respectively. The term X3 represents the vertical

velocity of wind time series in Figure 4.1. The summary statistics were closer to those

for the observed data, and were a considerable improvement from those obtained from

the nonlinear AR(1) derived Model A in 4.1.

x1 = −x2x3 + cx3 − α1x1 + F,

x2 = x1x3 − x3 − α2x2, (4.4)

x3 = x2 − cx1 − α3x3,

Further an introduction of another pair of linear gyrostatic terms in model (4.4)

resulted in a G-model (4.5), herein called Model C, with a value of 1 for the constant

d, yielded the values of 0.83 and 4.3, for the skewness and kurtosis, respectively. This

new G-model retains the physics of the observed data with more mechanisms ex-

plaining it, which is an exclusive advantage of G-models, gaining from the knowledge

already available from the underlying governing equations for atmospheric dynamics.

This was facilitated by the fact that in addition to the Rayleigh-Bernard convec-

tion principal mechanism, the dynamics over Lake Michigan involves a hoist of other

mechanism accounted for through terms associated coefficients c and further d in

the model below. G-models allows for the incorporation of these mechanisms which

serve to make them capture the reality of the underlying dynamics without loss of

the physical properties.

x1 = −x2x3 + cx3 − dx2 − α1x1 + F,

x2 = x1x3 − x3 + dx1 − α2x2, (4.5)

x3 = x2 − cx1 − α3x3,

The first four statistical moments from the two G-models were similar to those

from the original data and asserts the nonlinearity exhibited in them. The results ob-

tained using model (4.4), shows that G-models allows for the incorporation of mecha-

nisms explaining the underlying dynamics for the observed data, hence increases their

explanation from the atmospheric science theory and their capturing of the reality of

the associated physical behavior.

We proceed to incorporate these models in the construction of the subsampling

confidence intervals for the vertical wind velocity data, as the data generating mech-

anism approximations. Firstly, the models were used to determine the block sizes

that would ensure that the coverage was as close as possible to the accuracy level

we intend to make inference at. Once, the best possible block size was determined,

we proceeded to ensuring that the actual coverage was as close as possible to the

target coverage. Upon attaining proximity of the target coverage, the corresponding

subsampling intervals were constructed and investigated for their behavior in terms

of precision, and accuracy as we changed the confidence level for the inference.

Figure 4.3. Actual coverage probabilities of 90% subsampling CIs withβ = 0.65 using Model B for the skewness of nonlinear time series

Figure 4.4. Actual coverage probabilities of 95% subsampling CIs withβ = 0.61 using Model B for the skewness of nonlinear time series

Figure 4.5. Actual coverage probabilities of 99% subsampling CIs withβ = 0.57 using Model B for the skewness of nonlinear time series

Figure 4.6. Actual coverage probabilities of 90% subsampling CIs withβ = 0.74 using Model C for the skewness of nonlinear time series

Figure 4.7. Actual coverage probabilities of 95% subsampling CIs withβ = 0.71 using Model C for the skewness of nonlinear time series

Figure 4.8. Actual coverage probabilities of 99% subsampling CIs withβ = 0.67 using Model C for the skewness of nonlinear time series

Using the plots in figure 4.3 - Figure 4.8, we were able to determine the block sizes

b that would lead to the construction of subsampling confidence intervals with actual

coverage that were almost the same as the target coverage. These values occurred

in distinctly short ranges. The constructed subsampling confidence intervals for the

vertical wind velocity based on simulations of its data from the G-models in 4.4 and

4.5 were shown in Table 4.1 below. These were compared with the 90% intervals

constructed from subsampling nonlinear AR(1) modified model with a convergence

rate function.

4.5 Discussion

Table 4.1.Subsampling confidence intervals

Model Confidence Level b β subsampling CI

A 90% 100 0.42 (0.560, 1.100)

B 90% 170 0.65 (0.650, 1.000)

B 95% 190 0.61 (0.634, 1.015)

B 99% 175 0.57 (0.621, 1.028)

C 90% 140 0.74 (0.677,0.972)

C 95% 145 0.71 (0.670, 0.980)

C 99% 160 0.67 (0.654, 0.995)

Table 4.1 shows that the interval obtained for the 90% confidence level using the

G-model became narrower than the one obtained using the classical nonlinear time

series model with a convergence rate tweak. This indicated that it became more pre-

cise as the random error associated with it had become smaller. For models B and C,

there was a general downward trend in the convergence rate as the confidence levels

increased. Model C also showed that as the confidence level increased, the block size

increased, but for Model B there was no obvious trend. Upon adding more mecha-

nisms underlying vertical wind velocity in model C, through a pair of linear gyrostats,

the precision of the subsampling confidence interval for skewness increased, i.e. be-

came narrower. The above observation was similar for each confidence level between

the two models B and C. The actual coverage were closer to the target coverage for

Model C in comparison to those for Model B, as the confidence levels increased.

The simulation study conducted in this research served to demonstrate that sub-

sampling techniques may be developed to obtain valid statistical inference in a vari-

ety of problems, where traditional time series analyses are hindered due to nonlinear

data generating mechanisms and limited records. This involved the incorporation of

G-model approximations to the underlying DGM for atmospheric dynamics in the

subsampling procedure.

5. SUMMARY

5.1 Handling complexity through Statistics

The statistical challenge of complexity is not only limited to the big data, and

small and difficult to obtain data, but also to the subject-matter knowledge, develop-

ments in application areas, and the dynamics associated with data generation. Simple

linear thinking though relevant is proving to be limited in the reliability of inference

in the presence of the complexity trait of data. This calls for a revamp of both statis-

tical theory and methodology as we acknowledge and seek to model it. Complexity is

a phenomenon not limited to specific application areas but also to the interactions of

emerging scientific fields, which require inference to be made. Most of these emerg-

ing fields have an organic/systematic view for which simple models based on rigid

assumptions may fail to objectively address the scientific problems they are trying

to solve. This can create a gap between statistical and scientific inference, hence the

need for an objective endeavor for Statistics to comprehend complexity.

5.2 Statistical input for bundled interventions implementation and eval-

uation

Acknowledging the statistical concerns on bundling, hierarchy, heterogeneity of

population and implementation, and the varying contexts in which bundled interven-

tions can be used to resolve public health issues is critical for their evaluation. These

careful statistical thoughts serve to streamline the focus for statistical inference on

bundled interventions, so as to avoid the challenges of ecological fallacy, and Type III

error which impact on the decisions for practice from research findings on adoption,

sustainability, and scaling. Such input aid in widening the theory of change for the

mechanism of impact, a crucial component of process evaluation that has not been

well documented for in the Medical Research Council (MRC) guidelines, pivotal for

linking the intervention to the outcome. Contextual considerations alongside adjust-

ments for confounders in the analysis for bundled interventions will improve on the

inference reliability, and allow for the replicability of such interventions with a con-

sideration for their adaptability for their comparability.

In the evaluation of the ATONU bundled intervention, which consisted of five

behavioral messages to improve on women’s dietary diversity, its implementation

was shown to be heterogeneous. Heterogeneity of implementation was attributed

to contextual factors and the background characteristics of the participants, hence

a suggestion to capture the interaction of the participants with the intervention at

individual level was proposed. This would help in tracking their retention, contact

with the various bundle components, and measure gender coverage to ascertain the

source of the variations in the outcome in relation to the implementation dynam-

ics. This focus on the participants from the implementers’ based fidelity widens the

measurement of process dynamics on delivery-reception interactions. Bundling em-

phasizes the importance of participants engagement for the success of the intervention

on effectiveness. We measured how the individual participants responded to the het-

erogeneous scheduling of the bundle components in the presence of the intervention’s

hierarchy structure. We developed the process-driven metrics for compliance, BICR,

and gender engagement. These metrics captured the implementation dynamics that

were missed by traditional metrics. They offer an insight into how the participants

interact with the bundled intervention, were emphasis for successful implementation

is on engagement with the bundled components over the continuum of the interven-

tion study.

Identify at which level in the hierarchical implementation variation existed for

the process-driven metrics facilitated for the differentiation of the strategies or deci-

sions to improve implementation quality. A considerable amount of variation in the

postulated quantitative metrics of compliance and intervention components received

(BICR) were attributed to within villages source, indicating the presence of both

inter- and intra-household variation. The latter is not adequately accounted for in

cluster level analysis as public health interventions are often conducted at communal

level. There is a need to mobilize participants to improve on BICR proportions while

ensuring that all the bundled components are adequately delivered by the research

staff. There was a domination of between village variation over the within village

for compliance. This could be attributed to the impact of training and retention

of staffers,delivery, and administration on implementation of the bundle components

over the course of the intervention’s lifespan. Staff retention, competence, and adher-

ence to the implementation protocol needs to be emphasized in the theory of change

framework for bundled interventions. The moderately high amount of within villages

variations points to both inter- and intra-household variations bring to light that some

of the noise in the outcome could be attributed to the individuals’ characterization

within the villages. The engagement with the components of the bundle metric ICR,

showed the reverse composition, with a domination of the within village variation,

which can be attributed to the individual background characterization, and decision-

making inequities within patriarchal communities.

The process-driven metrics were used to identify the determinants for greater par-

ticipation by target households, men and women. These determinants were defined

at the hierarchical levels of the bundled intervention showcasing the need to adjust

beyond the clustering through inclusion of covariates that are context-defined. These

findings were crucial for the improvement of future implementations of such interven-

tions.

Participation has been known to mediate the effects of interventions on outcomes,

so in this research we proposed to utilize the new metrics as mediators alongside

other demographic factors in evaluating the impact of the intervention on the dietary

diversity score for women of reproductive age (WRA) in Ethiopia. The accuracy of

inference from bundled interventions is steeped in the critical analysis of the imple-

mentation dynamics to avoid Type III error where the implementation is not properly

done. Adaptation of bundle components facilitate for potential context-intervention

interactions that should be accounted for in the interpretation of the findings, and

where possible recourse be found to avoid them. Sample size and power issues need to

be addressed at the lowest level of the hierarchical structure of bundled interventions

to ensure the validity of inference across the structures in order to avoid ecological fal-

lacy challenge. Compliance had a significant effect on the women’s dietary diversity.

In spite of the BICR being an insignificant factor for the effects of the intervention

on the WRA’s dietary diversity, its observed low values showed the need to promote

adherence to the implementation of the intervention by the research staff to ensure

delivery of all the bundled components for the full realization of its impact. Similarly

the gender metrics gave different factors influencing participation which are crucial

for the need differential mobilization for participation by gender on the different bun-

dled components to accentuate the success of the implementation effectiveness which

translate to intervention effectiveness.

Data management and warehousing is crucial when implementation heterogeneity

is evident as much noise can be introduced into the data through revisits, delivery

decisions and strategies and the intervention process evolve. The use of technologies

for data collection such as the open data kit (ODK) allows for good management of

data in low resource settings, and also for the collection of more data at disaggregate

levels. These data bring out as much information as maybe deemed necessary for the

evaluation of bundled interventions especially on the often unmeasured confounding

variables which need to be adjusted for in their statistical analysis. The process met-

rics captured were able to capture implementation dynamics, which have been missed

by the conventional participation metrics. On the other hand, such innovations for

data management as ODK allows for the capturing data on as many intermediate

outcomes and contextual factors whose adjustments for are paramount to enhancing

the process-outcome link for bundled interventions. This is more defined under low

resource settings where there are competing factors that promotes time diversion in

the participation dynamics of the respondents and if not adjusted for may lead to

Type III error which is consequential on the adoption, sustainability and scaling of

such interventions for practice and policy.

The linear mixed model with adjustments for the process-driven metrics and con-

textual measures was used to explain the link between the women’s dietary diversity

outcomes and ATONU bundled interventions. The dietary diversity scores measured

after 24 hours and 7 days, respectively, for the women of reproductive age in Ethiopia

can be mediated by distance to market, a measure of how far the participants had

to travel to attend the intervention’s activities, baseline dietary diversity scores, and

compliance. Almost all the variation in the outcomes of interest were attributed to

within village sources, which include inter- and intra-household variation. The latter

agrees well with the evidence shown that a considerable amount of the variation in the

process-driven metrics was due to inter- and intrahousehold sources. The linear mixed

models shows that adjusting for both clustering, handling the hierarchical structure

complexity in the bundled intervention, and process dynamics, handling implemen-

tation heterogeneity attributed to the bundling complexity through process-driven

metrics can enhance the process evaluation of bundled interventions. This may give

an impetus for reliable effectiveness assessment, which may allow for adoption, sus-

tainability, and scaling of bundled interventions into practice. In spite of women’s

dietary diversity measure being a perceived count, the Poisson mixed model could not

adequately fit the data from the bundled ATONU intervention. This can be attributed

to the underlying nonlinear attributes of the data generating mechanisms during the

implementation process which heightens unpredictability in the intervention-outcome

linkage.

5.3 G-models and inference on atmospheric data

The findings in this research give pointers to the potential for G-models as substan-

tive time series models, that are appropriate for handling atmospheric data. Firstly,

we were able to extend the basic G-model with one pair linear gyrostatic terms, and

were able to obtain estimates for skewness and kurtosis that were very close to those

for the original data. Further, we computed the subsampling confidence intervals for

the wind velocity data using simulated data from the G-models, and obtained more

precise intervals than those previously computed.

All the intervals constructed confirmed the nonlinearity of the underlying atmo-

spheric dynamics responsible for the generation of the observed vertical wind velocity

data under study. As the confidence level increased, the block size did not exhibit a

distinct trend for Model B, while for Model C it exhibited an upward trend. On the

other hand, the convergence rate exhibited a steady downward trend for both models

as the confidence level increased. Upon adding an extra pair of linear gyrostatic terms

to model B, we obtained model C that also yielded similar statistical properties with

the observed data for the extra parameter d = 1.00. Comparing the subsampling

confidence intervals obtained through the use of these two approximating models, the

precision increased from those obtained with model B to those from model C. The

addition of an extra pair of linear gyrostatic terms to model B ensured that the block

size dropped considerably, while the convergence rate increased substantially. The

two attributes showed a potentially inverse relationship for G-models as approximat-

ing models for subsampling confidence intervals.

G-models help avoid the dilemma of choosing among the many data-driven non-

linear time series models which though giving a semblance of the statistical prop-

erties, have nothing to do with the physics underlying the data being investigated.

G-models share some fundamental physics with the original system which helps to

(a) better align statistical properties of series generated by the model with those of

observed series beyond the first moment and autocorrelation function, (b) avoid the

difficult task of finding an appropriate approximating model based entirely on sta-

tistical characteristics estimated with questionable accuracy, and (c) run meaningful

Monte Carlo simulations, particularly when estimators are more sensitive to prop-

erties of the DGMs. The subsampling confidence intervals are narrower than those

previously computed, showing that these models allow for the improvement of preci-

sion in the estimation. This heightens the reliability of the inference on atmospheric

data. The fact that G-models are derived from the governing equations for atmo-

spheric dynamics necessitate their potential appeal and uptake amongst geosciences

researchers. This also opens opportunities for wider applications of the subsampling

procedure in climate and weather inference.

5.4 Limitations

The data used in the assessment of bundled interventions was of low quality due

to profound implementation challenges in the study regions, to the extent that we had

to limit our analysis to data from Ethiopia and not incorporating data from Tanzania.

Denominator challenges were a cause of concern in the development of the metrics

to avoid biases in inference.Expected values were used for the BICR metric instead

of the actual values to minimize on the variance distortion on it. How a metric

handles variation is key to decision making based on it. The composition of the

bundle message was also not objectively documented for prior to the evaluation stage

of the intervention, whose composition and administration could also have been a

source of the heterogeneity experienced in the implementation. Many implementation

decisions were made during the course of the intervention’s lifespan due to contextual

factors but were not well documented to be acknowledged on the interpretation of the

conclusions that can be obtained for ATONU. This can have the consequential effects

of Type IV error where the interpretations may be based on wrong parameters that

may impact adversely on the adoption of ATONU and decisions on its sustainability

and scaling for practice and policy.

5.5 Future research on bundled interventions

Extensive assessments of bundled interventions under low resource settings is

needed to strengthen their theory of change (ToC). The input of all stakeholders

in the ToC should be amalgamated to ensure that the evaluation of bundled inter-

ventions will meet their respective aims, and ultimately address the public health

issues at stake. This will help provide the guidelines for their replicability in other

settings and for comparisons to be made. Identification of key and redundant bundle

components is essential for the sustainability of bundled interventions on adoption,

and can be of profound economic benefit for the implementers. The statistical con-

siderations highlighted point to the need for ethics to be upheld accordingly in the

administration and implementation of such interventions to enhance their relevance

in the emerging fields such as implementation science in nutrition, translation, and

scaling.

We seek to evaluate the bundled intervention, Engaging Fathers For Effective

Child nutrition in Tanzania (EFFECT) that has a well-documented ToC and utilize

process participation metrics and contextual factors. This interventions seeks to ad-

dress the household power dynamics and decision-making on the roles that fathers

can play in infant and young child feeding (IYCF) to address the challenges of mal-

nutrition and further to investigate their role in early child development alongside

nutritional engagement through the EFFECT+ bundled intervention. We seek to

utilize composition data analysis technique in the explanation of the effect of bundled

nutrition interventions on a host of outcomes based on process-driven participation

metrics. We seek to further develop the process-driven participation metrics to have

a time-dimension for longitudinal studies analysis.

5.6 Future research on subsampling and G-models

We seek to expand our exploration of the viability of G-models as alternative time

series models through using expanded G-models with additional linear gyrostats, and

other G- models for the construction of the subsampling confidence interval of skew-

ness for nonlinear atmospheric time series. These G-models allows for the aligning

of the statistical properties of the observed data with those from themselves beyond

the second order moments. We would also want to investigate how subsampling con-

fidence intervals come up for kurtosis using G-models in comparison to the AR(1)-

derived nonlinear time series model. This allows for the assessment of the peakedness

of their distribution, a crucial aspect for tail distribution of nonlinear data.

We seek to investigate the limiting behavior of subsampling CIs as b→ n, a vital

assumption for the subsampling procedure application, taking note of the fact that

b is integral for the accuracy of the intervals.We seek to incorporate other G-models

for inference on atmospheric data for skewness and kurtosis. We would also want

to investigate the limiting behavior of the models as the value of β → 0.5 for the

determination of the block size and the corresponding precision in the subsampling

confidence intervals.

The perceived inverse relationship between the block size and the convergence

rate in the G-models as approximating models for subsampling confidence interval

estimation may allow for holding the convergence rate at β = 0.5, and obtaining the

block size that ensures that similar statistical properties are obtained and then use

it for the interval computation. This will help on the investigation of the behavior of

the confidence intervals as b→∞.

Vertical wind velocity is a function of both time and position, spatial considera-

tions for its modeling may allow for alternative perspectives of making inference on

it based on location. The data has been shown to be stationary, there is a possibility

for the use of G-models as intrinsic models for spatial modeling.

Methodology ties in with computational considerations, we seek to develop a time-

efficient program that will facilitate for subsampling confidence intervals using G-

models.

We seek to employ G-models for statistical inference on other atmospheric phe-

nomena accounting for bay, ocean and land effects of atmospheric dynamics in the

generation of their time series. Shear aids our understanding of sediment dynamics

in coastal areas and beaches for sediment analysis. The underlying dynamics con-

tributing to the generation of such phenomena data are nonlinear, accounting for this

attribute in their modeling affords reliable inference to be obtained.

Future works will explore other types of G-models as approximating models for

inference on atmospheric data. We will adopt this technique for inference on linking

theory and data for Astro-Statistics and modeling pharmaco*kinetics for the absorp-

tion, metabolism, and excretion of drugs in living organisms.Biological systems and

diseases have been explained through mathematical modeling based on systems of

PDEs, we can also employ the concept that was used in the development of G-models

to develop time series models for analyzing phenomena that includes HIV-AIDS, Tu-

berculosis, and malaria with adjustments for confounding variables on interventions

to address them in low and middle income countries (LMIC).

REFERENCES

[1] Trisha Greenhalgh and Chrysanthi Papoutsi. Studying complexity in healthservices research: desperately seeking an overdue paradigm shift, 2018.

[2] Xiao-Li Meng. A trio of inference problems that could win you a nobel prize instatistics (if you help fund it). Past, present, and future of statistical science,2014.

[3] Chad M Schafer. A framework for statistical inference in astrophysics. AnnualReview of Statistics and Its Application, 2:141–162, 2015.

[4] Susan Holmes and Wolfgang Huber. Modern statistics for modern biology. Cam-bridge University Press, 2018.

[5] Thomas Homer-Dixon. Complexity science. Oxford Leadership Journal, 2(1):1–15, 2011.

[6] John Protevi. Deleuze, guattari and emergence. Paragraph, 29(2):19–39, 2006.

[7] Kate Churruca, Chiara Pomare, Louise A Ellis, Janet C Long, and JeffreyBraithwaite. The influence of complexity: a bibliometric analysis of complexityscience in healthcare. BMJ open, 9(3):e027308, 2019.

[8] Maxime Perron and Philip Sura. Climatology of non-gaussian atmosphericstatistics. Journal of Climate, 26(3):1063–1083, 2013.

[9] Purnima Menon, Namukolo M Covic, Paige B Harrigan, Susan E Horton,Nabeeha M Kazi, Sascha Lamstein, Lynnette Neufeld, Erica Oakley, and DavidPelletier. Strengthening implementation and utilization of nutrition interven-tions through research: a framework and research agenda. Annals of the NewYork Academy of Sciences, 1332(1):39–59, 2014.

[10] Reinette Tydeman-Edwards, Francois Cornel Van Rooyen, and Corinna MayWalsh. Obesity, undernutrition and the double burden of malnutrition in theurban and rural southern free state, south africa. Heliyon, 4(12):e00983, 2018.

[11] Neil C Campbell, Elizabeth Murray, Janet Darbyshire, Jon Emery, AndrewFarmer, Frances Griffiths, Bruce Guthrie, Helen Lester, Phil Wilson, andAnn Louise Kinmonth. Designing and evaluating complex interventions to im-prove health care. Bmj, 334(7591):455–459, 2007.

[12] Rabiul Karim, Lene Lindberg, Sarah Wamala, and Maria Emmelin. MenâĂŹsperceptions of womenâĂŹs participation in development initiatives in ruralbangladesh. American journal of men’s health, 12(2):398–410, 2018.

[13] Alexander Gluhovsky and Kevin Grady. Effective low-order models for atmo-spheric dynamics and time series analysis. Chaos: An Interdisciplinary Journalof Nonlinear Science, 26(2):023119, 2016.

[14] Michael Ghil, MR Allen, MD Dettinger, K Ide, D Kondrashov, ME Mann,Andrew W Robertson, A Saunders, Y Tian, F Varadi, et al. Advanced spectralmethods for climatic time series. Reviews of geophysics, 40(1):3–1, 2002.

[15] Taren Swindle, Geoff M Curran, and Susan L Johnson. Implementation scienceand nutrition education and behavior: Opportunities for integration. Journalof nutrition education and behavior, 51(6):763–774, 2019.

[16] Graham F Moore, Rhiannon E Evans, Jemma Hawkins, Hannah Littlecott,GJ Melendez-Torres, Chris Bonell, and Simon Murphy. From complex socialinterventions to interventions in complex social systems: Future directions andunresolved questions for intervention development and evaluation. Evaluation,25(1):23–45, 2019.

[17] Alison Tumilowicz, Marie T Ruel, Gretel Pelto, David Pelletier, Eva C Mon-terrosa, Karin Lapping, Klaus Kraemer, Luz Maria De Regil, Gilles Bergeron,Mandana Arabi, et al. Implementation science in nutrition: concepts and frame-works for an emerging field of science and practice. Current Developments inNutrition, 3(3):nzy080, 2018.

[18] Karl E Peace, Anthony V Parrillo, and Charles J Hardy. Assessing the validityof statistical inferences in public health research: An evidence-based,âĂŸbestpracticesâĂŹ approach. Journal of the Georgia Public Health Association, 2008.

[19] Hugo Sax, Benedetta Allegranzi, Ilker Uckay, E Larson, J Boyce, and Didier Pit-tet. âĂŸmy five moments for hand hygieneâĂŹ: a user-centred design approachto understand, train, monitor and report hand hygiene. Journal of Hospitalinfection, 67(1):9–21, 2007.

[20] Marin Schweizer, Eli Perencevich, Jennifer McDanel, Jennifer Carson, MichelleFormanek, Joanne Hafner, Barbara Braun, and Loreen Herwaldt. Effectivenessof a bundled intervention of decolonization and prophylaxis to decrease grampositive surgical site infections after cardiac or orthopedic surgery: systematicreview and meta-analysis. Bmj, 346:f2743, 2013.

[21] Stephen A McClave, Beth E Taylor, Robert G Martindale, Malissa M Warren,Debbie R Johnson, Carol Braunschweig, Mary S McCarthy, Evangelia Davanos,Todd W Rice, Gail A Cresci, et al. American society for parenteral and enteralnutrition. guidelines for the provision and assessment of nutrition support ther-apy in the adult critically ill patient: Society of critical care medicine (sccm)and american society for parenteral and enteral nutrition (aspen). JPEN JParenter Enteral Nutr, 40(2):159–211, 2016.

[22] SW Aboelela, PW Stone, and EL Larson. Effectiveness of bundled behaviouralinterventions to control healthcare-associated infections: a systematic review ofthe literature. Journal of Hospital Infection, 66(2):101–108, 2007.

[23] Amisha V Barochia, Xizhong Cui, David Vitberg, Anthony F Suffredini,Naomi P OâĂŹGrady, Steven M Banks, Peter Minneci, Steven J Kern, Robert L

Danner, Charles Natanson, et al. Bundled care for septic shock: an analysis ofclinical trials. Critical care medicine, 38(2):668, 2010.

[24] William H Brown, Samuel L Odom, and Maureen A Conroy. An interventionhierarchy for promoting young children’s peer interactions in natural environ-ments. Topics in early childhood special education, 21(3):162–175, 2001.

[25] Neha Kumar, Phuong Hong Nguyen, Jody Harris, Danny Harvey, Rahul Rawat,and Marie T Ruel. What it takes: evidence from a nutrition-and gender-sensitive agriculture intervention in rural zambia. Journal of Development Ef-fectiveness, 10(3):341–372, 2018.

[26] Dimitri Batras, Cameron Duff, and Ben J Smith. Organizational change theory:implications for health promotion practice. Health promotion international,31(1):231–241, 2016.

[27] Ramya Ambikapathi, Nilupa S Gunaratna, Isabel Madzorera, Simone Passarelli,Chelsey R Canavan, Ramadhani A Noor, Tshilidzi Madzivhandila, SimbarasheSibanda, Semira Abdelmenan, Amare Worku Tadesse, et al. Market food di-versity mitigates the effect of environment on womenâĂŹs dietary diversity inthe agriculture to nutrition (atonu) study, ethiopia. Public health nutrition,22(11):2110–2119, 2019.

[28] Dixon Chibanda, Ruth Verhey, Epiphany Munetsi, Frances M Cowan, and CrickLund. Using a theory driven approach to develop and evaluate a complex mentalhealth intervention: the friendship bench project in zimbabwe. Internationaljournal of mental health systems, 10(1):16, 2016.

[29] Erica Breuer, Lucy Lee, Mary De Silva, and Crick Lund. Using theory ofchange to design and evaluate public health interventions: a systematic review.Implementation Science, 11(1):63, 2015.

[30] Mary J De Silva, Erica Breuer, Lucy Lee, Laura Asher, Neerja Chowdhary,Crick Lund, and Vikram Patel. Theory of change: a theory-driven approachto enhance the medical research council’s framework for complex interventions.Trials, 15(1):267, 2014.

[31] Dean Rickles. Causality in complex interventions. Medicine, Health Care andPhilosophy, 12(1):77–90, 2009.

[32] Kristen M Hurley, Aisha K Yousafzai, and Florencia Lopez-Boo. Early childdevelopment and nutrition: a review of the benefits and challenges of imple-menting integrated interventions. Advances in nutrition, 7(2):357–363, 2016.

[33] Frederick J Dorey. In brief: Statistics in brief: Statistical power: What is itand when should it be used?, 2011.

[34] Aolin Wang, Roch A Nianogo, and Onyebuchi A Arah. G-computation of aver-age treatment effects on the treated and the untreated. BMC medical researchmethodology, 17(1):3, 2017.

[35] Jessica Datta and Mark Petticrew. Challenges to evaluating complex interven-tions: a content analysis of published papers. BMC public health, 13(1):568,2013.

[36] Sharon Schwartz. The fallacy of the ecological fallacy: the potential misuse ofa concept and the consequences. American journal of public health, 84(5):819–824, 1994.

[37] Jerry A Jacobs. Gender inequality and higher education. Annual review ofsociology, 22(1):153–185, 1996.

[38] Robert L Wears. Advanced statistics: statistical methods for analyzing clus-ter and cluster-randomized data. Academic emergency medicine, 9(4):330–341,2002.

[39] Sumedha Sharma, Olalekan O Adetoro, Marianne Vidler, Sharla Drebit, Beth APayne, David O Akeju, Akinmade Adepoju, Ebunoluwa Jaiyesimi, John So-tunsa, Zulfiqar A Bhutta, et al. A process evaluation plan for assessing acomplex community-based maternal health intervention in ogun state, nigeria.BMC health services research, 17(1):238, 2017.

[40] Michelle L Bell, Antonella Zanobetti, and Francesca Dominici. Evidence onvulnerability and susceptibility to health risks associated with short-term expo-sure to particulate matter: a systematic review and meta-analysis. Americanjournal of epidemiology, 178(6):865–876, 2013.

[41] Thomas A Glass, Steven N Goodman, Miguel A Hernán, and Jonathan MSamet. Causal inference in public health. Annual review of public health, 34:61–75, 2013.

[42] Sabu S Padmadas. Community-based bundled interventions for reproductiveand child health in informal settlements: evidence, efficiency, and equity. TheLancet Global Health, 5(3):e240–e241, 2017.

[43] Aulo Gelli, Elisabetta Aurino, Gloria Folson, Daniel Arhinful, ClementAdamba, Isaac Osei-Akoto, Edoardo Masset, Kristie Watkins, Meena Fernan-des, Lesley Drake, et al. A school meals program implemented at scale in ghanaincreases height-for-age during midchildhood in girls and in children from poorhouseholds: A cluster randomized trial. The Journal of nutrition, 2019.

[44] Rishma Maini, Sandra Mounier-Jack, and Josephine Borghi. How to and hownot to develop a theory of change to evaluate a complex intervention: reflec-tions on an experience in the democratic republic of congo. BMJ global health,3(1):e000617, 2018.

[45] Charles E Basch, Elena M Sliepcevich, Robert S Gold, David F Duncan, andLloyd J Kolbe. Avoiding type iii errors in health education program evaluations:a case study. Health education quarterly, 12(3):315–331, 1985.

[46] Raymond Hubbard, Brian D Haig, and Rahul A Parsa. The limited role offormal statistical inference in scientific inference. The American Statistician,73(sup1):91–98, 2019.

[47] Ana V Diez Roux. Estimating the neighborhood health effects: The challengesof casual inference in a complex world. 2004.

[48] Enola Proctor, Hiie Silmere, Ramesh Raghavan, Peter Hovmand, Greg Aarons,Alicia Bunger, Richard Griffey, and Melissa Hensley. Outcomes for implemen-tation research: conceptual distinctions, measurement challenges, and researchagenda. Administration and Policy in Mental Health and Mental Health ServicesResearch, 38(2):65–76, 2011.

[49] R Rukmani, R Gopinath, G Anuradha, R Sanjeev, and Varun Kumar Yadav.Women as drivers of change for nutrition-sensitive agriculture: Case study of anovel extension approach in wardha, india. Agricultural Research, pages 1–8.

[50] Marie T Ruel, Agnes R Quisumbing, and Mysbah Balagamwala. Nutrition-sensitive agriculture: What have we learned so far? Global Food Security,17:128–153, 2018.

[51] Elizabeth L Fox, Claire Davis, Shauna M Downs, Werner Schultink, and JessicaFanzo. Who is the woman in women’s nutrition? a narrative review of evidenceand actions to support women’s nutrition throughout life. Current developmentsin nutrition, 3(1):nzy076, 2018.

[52] Marie T Ruel. Is dietary diversity an indicator of food security or dietaryquality? a review of measurement issues and research needs. Food and NutritionBulletin, 24(2):231–232, 2003.

[53] Gamuchirai Chakona and Charlie Shackleton. Minimum dietary diversity scoresfor women indicate micronutrient adequacy and food insecurity status in southafrican towns. Nutrients, 9(8):812, 2017.

[54] Seid Aliwo, Melkitu Fentie, Tadesse Awoke, and Zemichael Gizaw. Dietarydiversity practice and associated factors among pregnant women in north eastethiopia. BMC research notes, 12(1):123, 2019.

[55] Melaku Desta, Mohammed Akibu, Mesfin Tadese, and Meskerem Tesfaye. Di-etary diversity and associated factors among pregnant women attending ante-natal clinic in shashemane, oromia, central ethiopia: A cross-sectional study.Journal of nutrition and metabolism, 2019, 2019.

[56] Ali Sié, Charlemagne Tapsoba, Clarisse Dah, Lucienne Ouermi, Pascal Zabre,Till Bärnighausen, Ahmed M Arzika, Elodie Lebas, Blake M Snyder, CaitlinMoe, et al. Dietary diversity and nutritional status among children in ruralburkina faso. International health, 10(3):157–162, 2018.

[57] Bruce Lindsay. Randomized controlled trials of socially complex nursing in-terventions: creating bias and unreliability? Journal of advanced nursing,45(1):84–94, 2004.

[58] Feras Ali Mustafa. Notes on the use of randomised controlled trials to evaluatecomplex interventions: Community treatment orders as an illustrative case.Journal of evaluation in clinical practice, 23(1):185–192, 2017.

[59] Richard OâĂŹReilly and Evelyn Vingilis. Are randomized control trials thebest method to assess the effectiveness of community treatment orders? Ad-ministration and Policy in Mental Health and Mental Health Services Research,45(4):565–574, 2018.

[60] Hopin Lee, Alix Hall, Nicole Nathan, Kathryn L Reilly, Kirsty Seward,Christopher M Williams, Serene Yoong, Meghan Finch, John Wiggers, andLuke Wolfenden. Mechanisms of implementing public health interventions: apooled causal mediation analysis of randomised trials. Implementation Science,13(1):42, 2018.

[61] Marie T Ruel, Harold Alderman, Maternal, Child Nutrition Study Group,et al. Nutrition-sensitive interventions and programmes: how can they helpto accelerate progress in improving maternal and child nutrition? The lancet,382(9891):536–551, 2013.

[62] Patrick Webb and Eileen Kennedy. Impacts of agriculture on nutrition: natureof the evidence and research gaps. Food and nutrition bulletin, 35(1):126–132,2014.

[63] Jeffrey A Alexander and Larry R Hearld. Methods and metrics challenges ofdelivery-system research. Implementation Science, 7(1):15, 2012.

[64] Hopin Lee, Robert D Herbert, and James H McAuley. Mediation analysis.Jama, 321(7):697–698, 2019.

[65] Francisco Cribari-Neto and Achim Zeileis. Beta regression in r. 2009.

[66] Min Yi, Nancy Flournoy, and Eloi Kpamegan. A ballooned beta-logistic model.Statistics in Biopharmaceutical Research, 9(1):98–106, 2017.

[67] Jacob J Oleson, Patrick J Breheny, Jane F Pendergast, Sandi Ryan, and RuthLitchfield. Impact of travel distance on wisewoman intervention attendance fora rural population. Preventive medicine, 47(5):565–569, 2008.

[68] Hitomi Komatsu, Hazel Jean L Malapit, and Sophie Theis. Does womenâĂŹstime in domestic work and agriculture affect womenâĂŹs and childrenâĂŹsdietary diversity? evidence from bangladesh, nepal, cambodia, ghana, andmozambique. Food policy, 79:256–270, 2018.

[69] Justus Ochieng, Victor Afari-Sefa, Philipo Joseph Lukumay, and ThomasDubois. Determinants of dietary diversity and the potential role of men inimproving household nutrition in tanzania. PloS one, 12(12):e0189022, 2017.

[70] A Taruvinga, V Muchenje, and A Mushunje. Determinants of rural householddietary diversity: The case of amatole and nyandeni districts, south africa. IntJ Dev Sustainability, 2(4):2233–2247, 2013.

[71] Laura Adubra, Mathilde Savy, Sonia Fortin, Yves Kameli, Niamké Ezoua Kodjo,Kamayera Fainke, Tanimoune Mahamadou, Agnès Le Port, and Yves Martin-Prével. The minimum dietary diversity for women of reproductive age (mdd-w)indicator is related to household food insecurity and farm production diversity:Evidence from rural mali. Current developments in nutrition, 3(3):nzz002, 2019.

[72] Liv E Torheim, Fatimata Ouattara, Modibo M Diarra, FD Thiam, IngridBarikmo, Anne Hatløy, and Arne Oshaug. Nutrient adequacy and dietary di-versity in rural mali: association and determinants. European journal of clinicalnutrition, 58(4):594, 2004.

[73] A Gluhovsky. Statistical inference from atmospheric time series: detectingtrends and coherent structures. Nonlinear Processes in Geophysics, 18(4):537–544, 2011.

[74] Jori E Ruppert-Felsot, Olivier Praud, Eran Sharon, and Harry L Swinney. Ex-traction of coherent structures in a rotating turbulent flow experiment. PhysicalReview E, 72(1):016311, 2005.

[75] Avijit Hazra. Using the confidence interval confidently. Journal of thoracicdisease, 9(10):4125, 2017.

[76] Abdelhamid Attia. Why should researchers report the confidence interval inmodern research. Middle East Fertil Soc J, 10(1):78–81, 2005.

[77] Guest Edited By Diane Lambert. Statistics in the physical sciences and en-gineering. In Statistics in the 21st Century, pages 212–287. Chapman andHall/CRC, 2001.

[78] Johannes Lenhard. Models and statistical inference: The controversy betweenfisher and neyman–pearson. The British journal for the philosophy of science,57(1):69–91, 2006.

[79] Aris Spanos et al. Where do statistical models come from? revisiting theproblem of specification. In Optimality, pages 98–119. Institute of MathematicalStatistics, 2006.

[80] Ronald A Fisher. On the mathematical foundations of theoretical statistics.Philosophical Transactions of the Royal Society of London. Series A, ContainingPapers of a Mathematical or Physical Character, 222(594-604):309–368, 1922.

[81] Eric L Lehmann. Model specification: the views of fisher and neyman, andlater developments. In Selected Works of EL Lehmann, pages 955–963. Springer,2012.

[82] Aris Spanos et al. Foundational issues in statistical modeling: Statistical modelspecification and validation. Rationality, Markets and Morals, 2(47), 2011.

[83] Christopher Tong. Statistical inference enables bad science; statistical thinkingenables good science. The American Statistician, 73(sup1):246–261, 2019.

[84] Dragutin T Mihailović, Gordan Mimić, and Ilija Arsenić. Climate predictions:the chaos and complexity in climate models. Advances in Meteorology, 2014,2014.

[85] KP Harikrishnan, Ranjeev Misra, and G Ambika. Nonlinear time series anal-ysis of the light curves from the black hole system grs1915+ 105. Research inAstronomy and Astrophysics, 11(1):71, 2011.

[86] Elizabeth Bradley and Holger Kantz. Nonlinear time-series analysis revisited.Chaos: An Interdisciplinary Journal of Nonlinear Science, 25(9):097610, 2015.

[87] Jan G De Gooijer et al. Elements of nonlinear time series analysis and fore-casting. Springer, 2017.

[88] Marian Anghel, Yehuda Ben-Zion, and Ramiro Rico-Martinez. Dynamical sys-tem analysis and forecasting of deformation produced by an earthquake fault.In Computational Earthquake Science Part I, pages 2023–2051. Springer, 2004.

[89] F Din-Houn Lau, Niall M Adams, Mark A Girolami, Liam J Butler, and Mo-hammed ZEB Elshafie. The role of statistics in data-centric engineering. Statis-tics & Probability Letters, 136:58–62, 2018.

[90] Bruce G Lindsay, Jon Kettenring, David O Siegmund, et al. A report on thefuture of statistics. Statistical Science, 19(3):387–413, 2004.

[91] Analisi Statistica dei Sistemi Dinamici. Statistical analysis of dynamic systems.

[92] CWJ Granger. New classes of time series models. Journal of the Royal StatisticalSociety. Series D (The Statistician), 27(3/4):237–253, 1978.

[93] John L Lumley. Coherent structures in turbulence. In Transition and turbulence,pages 215–242. Elsevier, 1981.

[94] Vladimir Petoukhov, Alexey V Eliseev, Rupert Klein, and Hermann Oesterle.On statistics of the free-troposphere synoptic component: an evaluation of skew-nesses and mixed third-order moments contribution to the synoptic-scale dy-namics and fluxes of heat and humidity. Tellus A: Dynamic Meteorology andOceanography, 60(1):11–31, 2008.

[95] Alexander Gluhovsky and Ernest Agee. Estimating higher-order momentsof nonlinear time series. Journal of Applied Meteorology and Climatology,48(9):1948–1954, 2009.

[96] Alexander Gluhovsky and Ernest Agee. On the analysis of atmospheric andclimatic time series. Journal of applied meteorology and climatology, 46(7):1125–1129, 2007.

[97] Nikolaos Pandis. Statistical inference with confidence intervals. American jour-nal of orthodontics and dentofacial orthopedics, 147(5):632–634, 2015.

[98] AKM Fazle Hussain. Coherent structures and turbulence. Journal of FluidMechanics, 173:303–356, 1986.

[99] Julius Sim and Norma Reid. Statistical inference by confidence intervals: issuesof interpretation and utilization. Physical Therapy, 79(2):186–195, 1999.

[100] Geoff Cumming and Sue Finch. Inference by eye: confidence intervals and howto read pictures of data. American Psychologist, 60(2):170, 2005.

[101] Dimitris N Politis and Joseph P Romano. Large sample confidence regionsbased on subsamples under minimal assumptions. The Annals of Statistics,pages 2031–2050, 1994.

[102] Dimitris N Politis, Joseph P Romano, and Michael Wolf. Subsampling. SpringerScience & Business Media, 1999.

[103] Edward Carlstein et al. The use of subseries values for estimating the vari-ance of a general statistic from a stationary sequence. The annals of statistics,14(3):1171–1179, 1986.

[104] Efstathios Paparoditis and Dimitris N Politis. Resampling and subsamplingfor financial time series. In Handbook of financial time series, pages 983–999.Springer, 2009.

[105] Patrice Bertail, Dimitris N Politis, and Joseph P Romano. On subsampling es-timators with unknown rate of convergence. Journal of the American StatisticalAssociation, 94(446):569–579, 1999.

[106] A Gluhovsky and T Nielsen. Brief communication" improving the actual cov-erage of subsampling confidence intervals in atmospheric time series analysis".Nonlinear Processes in Geophysics, 19(5):473–477, 2012.

[107] Alexander Gluhovsky, Martin Zihlbauer, and Dimitris N Politis. Subsamplingconfidence intervals for parameters of atmospheric time series: block size choiceand calibration. Journal of Statistical Computation and Simulation, 75(5):381–389, 2005.

[108] AJ Mannucci, BT Tsurutani, OP Verkhoglyadova, and X Meng. On scien-tific inference in geophysics and the use of numerical simulations for scientificinvestigations. Earth and Space Science, 2(8):359–367, 2015.

[109] Edward N Lorenz. Deterministic nonperiodic flow. Journal of the atmosphericsciences, 20(2):130–141, 1963.

[110] Javier Jiménez. Coherent structures in wall-bounded turbulence. Journal ofFluid Mechanics, 842, 2018.

[111] Melissa Coulson, Michelle Healey, Fiona Fidler, and Geoff Cumming. Confi-dence intervals permit, but don’t guarantee, better inference than statisticalsignificance testing. Frontiers in psychology, 1:26, 2010.

[112] DH Lenschow, Jakob Mann, and Leif Kristensen. How long is long enough whenmeasuring fluxes and other turbulence statistics? Journal of Atmospheric andOceanic Technology, 11(3):661–673, 1994.

[113] Thomas Schreiber. Interdisciplinary application of nonlinear time series meth-ods. Physics reports, 308(1):1–64, 1999.

[114] A Gluhovsky. Energy-conserving and hamiltonian low-order models in geophys-ical fluid dynamics. Nonlinear Processes in Geophysics, 13(2):125–133, 2006.

[115] Alexander Bihlo and Johannes Staufer. Minimal atmospheric finite-mode mod-els preserving symmetry and generalized hamiltonian structures. Physica D:Nonlinear Phenomena, 240(7):599–606, 2011.

[116] Andre N Souza and Charles R Doering. Transport bounds for a truncated modelof rayleigh–bénard convection. Physica D: Nonlinear Phenomena, 308:26–33,2015.

[117] G Chen, Nikolay V Kuznetsov, Gennady A Leonov, and TNMokaev. Hidden at-tractors on one path: Glukhovsky–dolzhansky, lorenz, and rabinovich systems.International Journal of Bifurcation and Chaos, 27(08):1750115, 2017.

[118] Marco Bianucci. Large scale emerging properties from non hamiltonian complexsystems. Entropy, 19(7):302, 2017.

[119] Andrew J Majda and Di Qi. Strategies for reduced-order models for predictingthe statistical responses and uncertainty quantification in complex turbulentdynamical systems. SIAM Review, 60(3):491–549, 2018.

[120] Themistoklis P Sapsis and Andrew J Majda. Statistically accurate low-ordermodels for uncertainty quantification in turbulent dynamical systems. Proceed-ings of the National Academy of Sciences, 110(34):13705–13710, 2013.

[121] Fedor Mesinger and Akio Arakawa. Numerical methods used in atmosphericmodels. 1976.

[122] Penelope Maher, Edwin P Gerber, Brian Medeiros, Timothy Merlis, StevenSherwood, Aditi Sheshadri, Adam Sobel, Geoffrey Vallis, Aiko Voigt, and PabloZurita-Gotor. The value of hierarchies and simple models in atmospheric.

[123] Alexander Gluhovsky and Ernest Agee. An interpretation of atmospheric low-order models. Journal of the atmospheric sciences, 54(6):768–773, 1997.

[124] Edward N Lorenz. Low order models representing realizations of turbulence.Journal of Fluid Mechanics, 55(3):545–563, 1972.

[125] Alexander Gluhovsky and Christopher Tong. The structure of energy conservinglow-order models. Physics of Fluids, 11(2):334–343, 1999.

[126] Vitor Araujo, Stefano Galatolo, and Maria José Pacifico. Statistical propertiesof lorenz-like flows, recent developments and perspectives. International Journalof Bifurcation and Chaos, 24(10):1430028, 2014.

[127] Vitor Araujo, M Pacifico, Enrique Pujals, and Marcelo Viana. Singular-hyperbolic attractors are chaotic. Transactions of the American MathematicalSociety, 361(5):2431–2485, 2009.

[128] Alexander Gluhovsky. A gyrostatic low-order model for the el ñino-southernoscillation. Complexity, 2017, 2017.

[129] Dimitris N Politis. The impact of bootstrap methods on time series analysis.Statistical Science, pages 219–230, 2003.

[130] JMMarriott and AR Tremayne. Alternative statistical approaches to time seriesmodelling for forecasting purposes. Journal of the Royal Statistical Society:Series D (The Statistician), 37(2):187–197, 1988.

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