TABLE OF CONTENTS
Title page…………………………….…….…………………….…i
Certification…………………………………………………….……ii
Dedication………………..………………………………………..iii
Acknowledgement………….……………………………………..iv
Table of contents…………………..……………………………….v
List of tables………….…………….…………………………….vii
Abstract……………………..……………………………………viii
CHAPTER ONE
INTRODUCTION
1.0 Introduction 1
1.1 Background of the study 1
1.2 Problem involved in random models 2
1.3 Problem of mixed effect models 3
1.4 Aim and objective of the study 4
1.5 Significance of the study 4
5
1.6 Scope/ Limitations of the study 6
1.7 Organization of the study 6
CHAPTER TWO
LITERATURE REVIEW
2.0 Introduction 7
2.1 The test for interaction effect 7
2.2 Elimination of the interaction effect 11
CHAPTER THREE
METHODOLOGY
3.0 Introduction 13
3.1 Method of data analysis 13
3.2 The Intra-factor analysis of unbalanced data 13
CHAPTER FOUR
PRESENTATION AND ANALYSIS OF DATA
4.0 Introduction 18
4.1 The least squares method of analyzing Intra-factor design 18
4.2 Test of hypothesis 22
4.3 Illustrative example on the application of the Intra-factor
design 24
4.3.1 The hypothesis 25
4.3.2 Test for fixed-effectness of the data 25
4.3.3 Test for absence of interaction 27
4.3.4 Analysis of variance for the main effects 27
CHAPTER FIVE
SUMMARY, CONCLUSION AND RECOMMENDATIONS
5.0 Introduction 30
5.1 Discussion of the main findings 30
5.2 Conclusion 31
5.3 Recommendations 32
6
References 33
Appendices 36
LIST OF TABLES
Table 4.1 Within-Factor ANOVA (Intra-Factor ANOVA).
Table 4.2 Fixed Effect Non-interactive Unbalanced Data.
Table 4.3 Population Means Without Interaction.
Table 4.4 ANOVA Table for the Model.
Table 4.5 ANOVA Table for Unadjusted Factor A with Adjusted
Factor B.
ABSTRACT
This study examines the analysis of fixed-effect non-interactive
unbalanced data by a method called Intra-Factor Design. And to derive
this design for analysis, mathematically, the matrix version of the fixed-
7
effect model, Yijk = µ + τi + βj + εijk, was used. This resulted to the
definition and formation of many matrices such as the Information Matrix,
L; Replication Vector, r; Incidence Matrix, N; Vector of adjusted factor A
totals, q; Variance–Covariance Matrix, Q, which is the generalized inverse
of the Information Matrix; and other Matrices. The Least Squares Method
which gave birth to several normal equations was used to estimate for the
parameters, τ and β mathematically. Then, an illustrative example was
given to ascertain the workability of this Intra-Factor procedure in testing
for the main effects under some stated hypothesis for significance. But,
before testing for the variance component of the main effects on the
illustrative data, it was necessary to first ascertain that the data is fixedeffect and that interaction is either absent or non-significant since our
interest is on “Fixed-Effect Non-interactive Models”. Thereafter, the
Analysis of Variance Components for Adjusted Factor A with Unadjusted
Factor B effects was carried out. This gave the result that Adjusted Factor
A effect, τ, was not significant; whereas, the Unadjusted Factor B effect,
β, was significant. Also, the Analysis of Variance Components was
performed for Unadjusted Factor A effect with Adjusted Factor B effect
and it yielded similar result as that of Adjusted Factor A with Unadjusted
Factor B effects. We therefore concluded that for a Fixed Effect, Noninteractive Unbalanced Data Analysis, the Method of Intra-Factor Design
can be successfully employed.
CHAPTER ONE
INTRODUCTION
1.0 INTRODUCTION
Estimating variance component from unbalanced data is not as
straightforward as from balanced data. This is so for two reasons. Firstly,
several methods of estimation are available (most of which reduce to the
analysis of variance method for balanced data), but no one of them has yet
been clearly established as superior to others. Secondly, all the methods
involve relatively cumbersome algebra; discussion of unbalanced data can
therefore easily deteriorate into a welter of symbols, a situation we do our
8
best to minimize here. However, we shall review some works on
unbalanced data.
1.1 GENERAL OUTLAY OF UNBALANCED DATA
Balanced data are those in which every one of the subclasses of the
model has the same number of observations, that is, equal numbers of
observations in all the subclasses. In contrast, unbalanced data are those
data wherein the numbers of observations in the subclasses of the model
are not all the same, that is, unequal number of observations in the
subclasses, including cases where there are no observations in some
classes. Thus unbalanced data refers not only to situations where all
subclasses have some data, namely filled subclasses, but also to cases
where some subclasses are empty, with no data in them. The estimation of
variance components from unbalanced data is more complicated than from
balanced data.
In many areas of research such as this, it is necessary to analyze the
variance of data, which are classified into two ways with unequal numbers
of observations falling into each cell of the classification. For data of this
kind, special methods of analysis are required because of the inequality of
the cell numbers. This we shall attempt to solve in this research work.
1.2 PROBLEM INVOLVED IN RANDOM MODELS
The problem associated with the random effect models has been the
determination of approximate F-test in testing for the main effects say, A
and B using F-ratio. In this case there would be no obvious denominator
for testing the hypothesis Ho: σ² = 0; for a level of Factor A crossed with
the level of Factor B in the model such as
Xijk = μ + αi +βj
+ λij + εijk
9
where,
Xijk is the kth observation (for k =1,2,…,nij )in the ith level of Factor A and
j
th level of Factor B; i= 1,2,…,p; j= 1,2,…,q;
µ is the general mean;
αi is the effect due to the ith level of Factor A;
βj is the effect due to the jth level of Factor B;
λij is the interaction between the ith level of Factor A and jth
level of Factor
B;
εijk is the observation error associated with Xijk.
1.3 PROBLEM OF MIXED EFFECT MODELS
According to Henderson (1953), in the customary unbalanced
random model from a crossed classification, the nature of the expected
value of the sum of squares is such that:
E (SSA) = n.. – Σni.2
σ²A + ∑ Σnij – Σn.j2
σ²B + ∑ Σnij2
– Σ Σnij2
σ
2
AB +
n.. ni. n.. ni. n..
α -1 σ2
e –
………………………………………………………………..…(1.1)
This means that with unbalanced data from crossed classification in
random model, the expected value of every sum of squares contains every
variance component of the model.
But in an unbalanced mixed, with α’s being fixed and not random
effects, the expectation would then be:
E (SSA) = ∑ni.αi2
– Σni.αi /n.. + ∑ Σnij2
– Σn.j2
σ
2
β + ∑ Σnij2
– Σ Σnij2
σ
2
βα +
n.. n.. ni. n..
α -1 σ
2
e –
…………………………………………………….…………….(1.2)
i j
c b
j
c
i
i
a
j
j
b b
j j
b
b
i
j j
a
a
i
a
i
a a
i
j
i
b b a b
10
It is observed that the first term of (1.2) which is a function of the fixed
effect is different from that in (1.1); and this occurs in expected values of
all sum of squares terms (except for SSE). And, more importantly, the
function of the fixed effects is not the same from one expected sum of
squares term to the next.
For example, with the α’s fixed, E (SSE) contains the term,
∑ (Σnijαi)2
/ n.j – (Σni.αi)2
/n.. which differs from the first term in E (SSA) of (1.2).
Thus
E [SSA – SSB] does not get rid of the fixed effects even though it does
eliminate terms in µ. This is true generally in mixed models, expected
values of the sum of squares contain functions of the fixed effects that
cannot be eliminated by considering linear combinations of the sum of
squares. This means that the equations E (SS) = Pσ2
+ σ2
ef of the random
model takes the term E (SS) = Pσ2
+ σ2
ef + q in the mixed model, where q
is a vector of the quadratic functions of the fixed-effects in the model.
Hence, σ² cannot be estimated and the analysis of variance method applied
to unbalanced data cannot be used for mixed as well as for fixed-effect
models. It yields biased estimators. Simply put, with unbalanced data, the
analysis of variance method for mixed and fixed effect models lead to
biased estimators of variance components.
Mixed models involve dual estimation problems – estimating both
fixed effects and variance components.
1.4 AIM / OBJECTIVE OF THE STUDY
The aim of this work is to analyze unbalanced fixed effect noninteractive model using the Intra-Factor Design.
1.5 SIGNIFICANCE OF THE STUDY
b
i
a
j
a
i
11
As a result of dual estimation problems of the mixed model with
unbalanced data which accounted for biased estimators of variance
components, Henderson (1953), designed a method to correct this
deficiency. This he does by his method 2 which uses the data first to
estimate fixed effects of the model and then using these estimators to
adjust the data. Variance components are estimated from the adjusted data
by the analysis of variance method. This whole procedure was designed so
that the resulting variance estimators were not biased by the presence of
the fixed effects in the model as they were with the analysis of variance
estimators derived from the basic data. So far as the criterion of
unbiasedness was concerned, this was certainly achieved by this method.
But the general method of analyzing data adjusted according to
some estimator of the fixed effects is open to criticism on other grounds
such as: it cannot be uniquely defined, and a simplified form of it, of
which Henderson’s Method 2 is a special case, cannot be used whenever
the model includes interactions between the fixed effects and the random
effects. As such, the need for the birth of this research “Analysis of
Unbalanced Fixed-Effect Non-interactive Model”.
1.6 SCOPE / LIMITATION OF THE STUDY
This study is basically limited to fixed effect non-interactive
models.
1.7 ORGANIZATION OF THE STUDY
The work was organized in five systematic chapters. Chapter one
is made up of the introduction to the study. Other topics considered in this
chapter include: the problem involved in random models, problem of
mixed effect models, the aim and objective of the study, significance of
12
the study, the scope/limitations of the study and the organization of the
study.
In chapter two, the related literatures were reviewed as to see what
various researchers had said or written about the topic under
consideration. In this chapter, the following subtopics were considered:
the test for interaction effect and elimination of the interaction effect.
In chapter three, the methodology used in the study was
systematically described as to enhance the understanding of the study. The
method of data analysis was also carefully derived mathematically.
The chapter four of this study shows the presentation and analysis
of data. In doing this, the following subtopics were considered: the least
square method of analyzing the unbalanced factor design, test of
hypothesis and illustrative example on the application of the unbalanced
factor design. An example was given to illustrate the analysis of
unbalanced data using the Intra-Factor Design.
Finally, chapter five which is the concluding chapter contains the
discussion of the main findings, conclusion and recommendations.
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