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 fixedeffect model, Y_ijk = µ + τ_i + β_j + ε_ijk, 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 andβ 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 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  

X_ijk = μ + α_i +β_j  + λ_ij  + ε_ijk

       where, 

X_ijk is the k^th observation (for k =1,2,…,n_ij )in the i^th level of Factor A and j^th level of Factor B; i= 1,2,…,p; j= 1,2,…,q; µ is the general mean;

α_i is the effect due to the i^th level of Factor A; β_j is the effect due to the j^th level of Factor B;

λ_ij is the interaction between the i^th level of Factor A and j^th  level of Factor B; ε_ijk is the observation error associated with X_ijk.

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                  c b        b               a b   ₂      a b   _{2       2}

                             _{2                                                   2}

E (SSA)  =   n.. –   Σni.i    σ²_A   +      ∑  Σnijj     –  Σn.jj    σ²_B  +  ∑ Σnijj      –  Σ Σniji j       σ _AB   +  

                              n..                           ni.          n..                     ni.          n..i          j           

                     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:

                a       2      a                 a b      2     b       2       2 a b       a2 b         2