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Evaluating Likelihood Estimation Methods in Multilevel Analysis of Clustered Survey Data

 

Table Of Contents


Thesis Abstract

Abstract
Public health researchers often lay little or no emphasis
on multilevel structure of clustered data and its likelihood estimation techniques.
This has led to improper inferences. The aim of this research is to evaluate tradi-
tional methods and the different multilevel likelihood estimation procedures so as
to compare their computational efficiencies.
Key words Clustered survey; Likelihood; Adaptive Gaussian Quadrature; Penal-
ized quasi likelihood, Modern contraception; Akaike’s information criteria.

Thesis Overview

1. Introduction
Likelihood plays important roles in parameter estimation and it is synonymous with
probability. It defines the function of parameters included in a statistical model.
That is, a set of parameter value given outcome y is the probability of those observed
outcome given the parameter values (l(θ/y) = p(y/θ)). Likelihood is one of the tools
used in estimating parameters of multilevel models, including multilevel binary
logistic models.
Multilevel model is a statistical model of parameter that varies at more than one
level (Leyland & Goldstein (2001); Sampson et al. (1997)). This model can be seen
as generalization of linear model, although they also extend to nonlinear models.
Multilevel model are ideal for research design where the data is collected from
study participants who were organized at two or more levels (Maas & Hox (2005);
Srikanthan & Reid (2008)). In which case, one level is nested in the other. Usually,
the unit of analysis are the individuals (at a lower level) who are nested in within
an aggregate unit (at higher level) (Klotz et al. (1969); Li et al. (2011)). Multilevel
(hierarchical) data structure causes correlation among observations within same
clusters (Li et al. (2011)). Multilevel models present alternative analysis procedures
to the famous univariate and multivariate analysis of measures that are collected
repeatedly from same individuals. Over the years, the use of multilevel analysis
to investigate public health problems has gained significant prominence (DiezRouz
& Mair (2011); Leyland & Goldstein (2001)). This growth can be attributed to the
need to understand how individuals are related to each other within groups and im-
portance of such in understanding the distribution of health outcomes (DiezRouz
& Mair (2011);Oye-Adeniran et al. (2004)). The growth has also been aided by in-
creased use of multilevel methods in statistical methods and their applicability to a
broad range of scenarios that have multilevel data. However, its use has been fully
embraced in most public health research (Bingenheimer & Raudenbush (2004)).
The percent of total variance in the individual-level health outcome and the cluster
effects which represent unobserved cluster characteristics that has potentials of
affecting individuals outcomes could be large. (Li et al. (2011)). This must be viewed
in light of the fact that the relevant ”levels” are generally grossly mis-specified. So
far, the methods of parameter estimation have led to several problems in the best
way to carry out multilevel analysis, including under estimation of parameters and
biased estimates (John et al. (2012)). In this study different methods of estimating
multilevel binary logistic model parameters were considered and the best method
was determined.
Cluster sampling, whereby samples are not taken randomly from entire population
but from clusters, often introduces multilevel dependency and correlation among
measurements taken from individuals within same cluster which could substan-
tially affect parameter estimates. The structure of clustered survey data are usu-
ally nested and can be analysed using multilevel techniques. Challenges are of-
ten encountered when multistage sampling is used in data collection without the
use of multilevel analysis. The description of most of ”the theoretical and method-
ological challenges facing contextual analysis” has been made by Blalock (1984).
The dependence among observations in multistage-clustered samples often comes
from several levels of the hierarchy (Maas & Hox (2005)). In this case, the use of
single-level statistical models is no longer valid and reasonable (Leyland & Gold-
stein (2001) ; Li et al. (2011)). The traditional standard logistic regression, that is
single-level logistic regression, usually requires a sort of independence among the
observations conditional on the independent variables and uncorrelated residual
errors. To ensure that appropriate inferences are drawn and that reliable conclusions from clustered survey data is made, it has therefore become necessary to use
more effective and more involving modeling techniques like multilevel modeling.
Also, underlying assumptions of ordinary logistic regression are violated when an-
alyzing nested data, hence the best option is multilevel logistic regression analysis
(Maas & Hox (2005); Srikanthan & Reid (2008)). This is due to the fact that it con-
siders the variations due to multilevel structure in the data and allows the simul-
taneous assessment of effects of different levels in the data used in this study. The
number of levels, the variance of the random effects and the size of the correlation
between random effects may affect the performance of the parameter estimation
method. Some methods of estimation could be biased. Therefore, there is need to
evaluate these methods and determine the best method. The commonest methods
used are Penalized Quasi-Likelihood (PQL), Non-Adaptive Gaussian Quadrature
(NAGQ) and Adaptive Gaussian Quadrature (AGQ) and the Maximum Likelihood
Estimates (MLE). Early methodology work on multilevel logit model includes use of
data from 15 World fertility survey (Goldstein (2003);Hox, J. J. (2002)). Further
documentations on multilevel models especially the type of data it allows, sam-
pling, outliers, repeated measures, institutional performance, and spatial analysis
have been made (Leyland & Goldstein (2001)).
The robustness, sample sizes and statitical power in multilevel modeling for both
categorical and continuous outcome variables has been studied earlier (Bingen-
heimer & Raudenbush (2004); Goldstein (2003); Li et al. (2011); Maas & Hox
(2005); Portnoy (1971)). Monte Carlo simulation has been used to ”assess the im-
pact of misspecification of the distribution of random effects on estimation of and
inference about both the fixed effects and the random effects in multilevel logis-
tic regression models” by Austin (2005). The authors concluded that inferences
aboutg fixed effects estimate were not affected by the inherent misspecification of
random effects distributions. However, the authors opined that inferences about
random effects estimate were influenced by model misspecifications. Simulation
studies indicated that increasing number of levels yield better estimates than larger
number of individuals per level (Goldstein (2003); Goldstein & Rasbash (1996);
Mason et al. (1983)). It was concluded in these studies that for second level units
with a small sample size, while the estimates of the regression coefficients are
unbiased, the standard errors and the variance components are sometimes un-
derestimated (<30)Maas & Hox (2004). This is not envisaged in the current study
since we are using a large dataset.
The use of these statistical methods allows public health researchers to correctly
identify factors and causes of disease at different levels. The approach provides op-
portunity and serves as a tool to investigate disease causation in complex settings.
Contraceptive Use in Nigeria
In 1988, the Nigeria Federal Ministry of Health adopted the ”National Policy on Pop-
ulation for Development, Unity, Progress and Self-Reliance” (Essien et al. (2010)).
It consequently adopted a revised policy in 2004.

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