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-

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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 fixed-effect 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, Non-interactive 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

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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

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where,

Xijk is the kth observation (for k =1,2,…,nij )in the ith level of Factor A and jth 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:

c

σ²A  +

c

b

b

σ²B

a b

-

a b

σ2AB  +

E (SSA) =  n.. -  Σni.2

∑ Σnij - Σn.j2

+ ∑ Σnij2

Σ Σnij2

i

j

j

j

i  j