TABLE OF CONTENTS
TITLE PAGE…………………………………………………………………….………i
DECLARATION ………………………………………………………………………………………………………….. ii
CERTIFICATION ……………………………………………………………………………………………………….. iii
DEDICATION ……………………………………………………………………………………………………………. iv
ACKNOWLEDGEMENTS …………………………………………………………………………………………… v
ABSTRACT ……………………………………………………………………………………………………………….. vi
TABLE OF CONTENTS …………………………………………………………………………………………….. vii
CHAPTER ONE …………………………………………………………………………………………………………… 1
GENERAL INTRODUCTION ………………………………………………………………………………………. 1
1.0 Introduction ……………………………………………………………………………………………………………. 1
1.1 Purpose of the Study ………………………………………………………………………………………………… 2
1.2 Significance of the Study ………………………………………………………………………………………….. 2
1.3 Aim and Objectives of the Study ……………………………………………………………………………….. 3
1.4 Methodology of the Study ………………………………………………………………….3
1.5 Scope of the Study ………………………………………………………………………….4
CHAPTER TWO ………………………………………………………………………………………………………….. 5
LITERATURE REVIEW ………………………………………………………………………………………………. 5
2.0 Introduction ……………………………………………………………………………………………………………. 5
2.1 Polynomial Regression Models for Continuous Covariates …………………………………………… 5
2.2 Fractional Polynomial Modeling ……………………………………………………………………………….. 9
2.3 Categorization of Covariates …………………………………………………………………………………… 16
2.4 Model Adequacy Check …………………………………………………………………………………………. 23
CHAPTER THREE ………………………………………………………………………….. 35
METHODOLOGY ……………………………………………………………………………………………………… 35
3.0 Introduction ………………………………………………………………………………………………………….. 35
3.1 Normal Error Model ………………………………………………………………………………………………. 35
3.2 Fractional Polynomials …………………………………………………………………………………………… 35
3.3 Fractional Polynomials with Multiple Covariates…………………………………………37
3.4 Deviance Measure of Model Fitness ………………………………………………………………………… 38
3.5 Working Rule for using Deviance ……………………………………………………………………………. 40
3.6 Median Method of Categorizing Continuous Covariates …………………………………………….. 41
3.7 Data Description ……………………………………………………………………………………………………. 43
CHAPTER FOUR ………………………………………………………………………………………………………. 44
ANALYSIS AND DISCUSSION OF RESULTS ……………………………………………………………. 44
4.0 Introduction ………………………………………………………………………………………………………….. 44
4.1 Generalized Linear Model Multivariable Fractional Polynomial Regression Results ……… 44
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4.2 Generalized Linear Model Fractional Polynomial Regression Results ………………………….. 51
4.3 Discussion…………………………………………………………………………………………………………….. 55
CHAPTER FIVE ………………………………………………………………………………………………………… 60
SUMMARY, CONCLUSION AND RECOMMENDATION …………………………………………… 60
5.0 Introduction ………………………………………………………………………………………………………….. 60
5.1 Summary………………………………………………………………………………………………………………. 60
5.2 Conclusion………………………………………………………………………………….61
5.3 Recommendation …………………………………………………………………………………………………… 61
5.4 Contribution to Knowledge …………………………………………………………………………………….. 62
5.5 Suggestion for Further Research ……………………………………………………………………………… 62
REFERENCES …………………………………………………………………………………………………………… 63
CHAPTER ONE
GENERAL INTRODUCTION
1.0 Introduction
It is common in Statistics to be interested in a simple approximation for smoothing relationships between variables and such relationships may be known but complicated, or completely unknown. Research work includes the collection and analysis of data on one or more variables. Often multiple regression analyses are used to model such data sets which may include only linear terms in the covariate(s). In most applications, the choice of the model building is based on simple linear effect modeling approach, but the linearity assumption may be questionable. According to Sauerbrei and Royston (2010) to avoid this strong assumption, researchers often apply cutpoints to categorize the variable, implying regression models with step functions and this simplify the analysis and interpretation of the result. In many research studies, covariate(s) encountered are continuous and most regression models constructed for such data type include only linear terms in the covariate(s) but if curvature is suspected between the outcome variable Y and a covariate X, the model may be extended to include a quadratic term.
Royston and Sauerbrei (2008) stated that in most applications, a choice is made between linear and quadratic, with cubic or higher order polynomials being rarely used. It has been recognized that conventional low order polynomials do not always fit the data well. Higher order polynomials tend to fit the data more closely but may fit badly at the extremes of the observed range of X (Royston and Altman, 1994). Also, polynomials do not have asymptotes and cannot fit data where limiting behavior is expected (McCullagh and Nelder, 1989). In an attempt to obtain acceptable models, Box and Tidwell (1962) developed an approximate linearization of each variable in a
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multiple-regression model giving
i
n
i
i
f x
1
. They concentrated on power
transformations of the X‘s and showed how to estimate the powers iteratively. Royston
and Altman (1994) stated that models with more than one X-variable have considerable
difficulties in estimating their powers reliably. They believed that estimation of the
precise power(s) is unnecessary because the likelihood surface is usually nearly flat
near maximum, but in any case Y may not be linear in Xp.
In this research work, fractional polynomial regression model given by Royston and
Altman (1994) will be studied under normal errors. The fractional polynomials are
models whose power terms are restricted to a small predefined set of integer and noninteger
values. The powers selected encompass that of the conventional polynomials
(Royston and Sauerbrei, 2008).
1.1 Purpose of the Study
Most of the existing method on fractional polynomial models focused on fitting models
to psychological and pharmacokinetic experimental data. Little has been done on
agronomic data although; Nelder (1966) introduced and applied the inverse polynomial
model on fertilizer trials, while Salawu (2007) applied the inverse polynomial model at
quadratic variable on fertilizer response of three rice varieties.
This research focuses on fitting all the power set of a fractional polynomial model on
Pmain aim is to observe how well the fractional polynomial model fit the data using
normal errors regression analysis when the covariates are continuous or are grouped.
1.2 Significance of the Study
The main significance of the study is to present how to fit a fractional polynomial
model and assessing the fitted model using deviance difference test. Attempt was made
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to show if the fractional polynomial models are more efficient compared to the conventional polynomial models in fitting data. Furthermore, effort to come-up with a new median algorithms method for grouping continuous covariates was pursued. It will serve as a reference material to researchers and scientist who may wish to undertake similar study.
1.3 Aim and Objectives of the Study
The study is aimed at comparing fractional polynomial models to conventional polynomial models using grouped and ungrouped continuous covariates with a view to achieving the following objectives to;
1. Fit a generalized linear model fractional polynomial models for all predefined set of powers {-2, 1.5, -1, -0.5, 0, 0.5, 1, 1.5, 2, 2.5, 3} with categorized covariate and compared with a non-categorized covariate fit. Powers selection is based on Royston and Altman (1994) Algorithms.
2. Compare the fitted models using the deviance difference measure, and
3. Propose a new median algorithms method for grouping continuous covariates.
1.4 Methodology The methodology used in this research work are fractional polynomial for normal error regression models, median method for grouping continuous covariates and the deviance method for checking model adequacy. 1.5 Scope of the Study
This research focuses on fitting all pre-defined set of power of a fractional polynomial model on an agronomic set of data with continuous covariate(s) and grouped
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covariate(s).The data was obtained from Data Processing Unit, Institute of Agricultural Research, Ahmadu Bello University, Zaria.
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