DESIGN AND ANALYSIS OF EXPERIMENTS ON THE METHODS OF ESTIMATING VARIANCE COMPONENTS

CHAPTER ONE

1.1 BACKGROUND OF THE STUDY

ln our everyday life, there are differences or variations in every repeated thing We do. Take for example a student writing a series of examinations, this student is bound to have differences in scores of each of the exams Written by him or her.

Also, a lady preparing a series of meals, there must be also differences in the taste of each of the meals. Another example is in the case of a cement production company, there are bound to be differences in the quantity of cement that is being found in each bag on a daily bases. No matter how negligible these differences may be, the most important thing is that differences abound. Measuring or estimating these differences that occur in the above scenarios are known simply as estimating the measure of differences or estimating variance component.

In this work, we shall narrow down our focus on just agriculture precisely animal breeding. This is because estimation of variance component are usually more practical and of utmost importance in animal breeding. 

In animal breeding, which is one of the major sectors of agriculture, the measure of differences or variability between individual animals or between traits can be measured or estimated. When this is done successfully, we simply say that we have estimated the variance component of the animal under study. (See koltz et al 1974 for details) The relationship between estimation of variance component and animals cannot be overemphasized in that the knowledge of estimating variance component is needed by the researcher for better understanding of the genetic mechanism of animals to aid reproduction.

Secondly, the knowledge of estimating variance components is needed by the researcher in order to be able to predict breeding values With this knowledge of variance component estimation where the differences in traits of animals are studied, a better understanding of the measure of differences in traits would go a long way in making the researcher to be able to predict what the offspring of a particular parent animal under study would look like Thus for effective and efficient breeding, the knowledge of how to estimate the measure of differences in individual animals or traits (estimation of variance component) is inevitable.

Thirdly, the knowledge of estimating variance component is needed for optimization of breeding programs and prediction of response in that with the help of this knowledge of variance component estimation, the breeding programs can be done in the best possible way such that the cost of breeding would be minimal while at the same time maximizing profit. Mostly it is assumed that variances and especially the ratio of both of them (like heritability, correlation) are based on particular biological rules which do not rapidly change overtime.

However, it is well known that the genetic variance changes are consequences of selection. Changes are especially expected in situations with short generation intervals, high selection intensities or high degree of inbreeding or in a situation in which a new trait is determined by only a few genes. Secondly the circumstances under which measurements are taken can change. If conditions are getting more uniform overtime, the environmental variance decreases and consequently the heritability  increase.

Thirdly, the biological interpretation of a trait can change as consequences of a changed environment; feed intake under limited feeding is not the same as feed intake under ad-lib feeding. In conclusion, there are sufficient reasons for regular estimation of variance component. Long et al (1990).

There are different methods that can be used in estimating variance component of genes or traits of animals. Each of these methods has its merits and demerits and scenarios where they can be best applied. In this work We will be considering four methods of estimating variance components, which are; methods of moments or analysis of variance method (ANOVA method), maximum likelihood method (ML method), restricted maximum likelihood method(REML method) and Minimum variance quadratic unbiased estimator method (MVQUE) These methods of variance component estimation will be properly treated in chapter two and three of this Work.

Each of these four methods mentioned above have scenarios. Where they are best applied. ln cases of unbalanced designs (when there are missing values or empty cells) and when the solutions of the variance  component are all positive, then the method of moments, restricted maximum likelihood (REML) method and minimum quadratic unbiased estimators are identical. But in cases of unbalanced designs (no missing values or empty cells) the method of moments are easiest to compute (Eisenhart 1947) , while the other three require iterative algorithms.

The method of moments does not require an assumption of normality in order to obtain the estimators ( see Graybill 1976 for details)

1.2 AIMS AND OBIECTIVES OF THE STUDY Wirral

1) To know the most efficient and precise method between the f% methods of variance component estimation in the case unbalanced data.

2) To study the merits and demerits of each method of variance component estimation when dealing with cases of balanced and unbalanced designs when normality holds.

3) To know the most efficient method between the four methods of variance component estimation in the case of a balanced design. 

1.3 LITERATURE REVIEW

David (1994) used nested design to estimate the plate to plate variance nested Within the day to day variance for a single technician and the laboratory to laboratory unit (main factor). The analysis showed that for a balanced design, the method of moments estimates was more precise when compared with the estimates obtained using the maximum likelihood method.

Meyer (1990) also indicated that the maximum likelihood (ML) and the restricted maximum likelihood (REML) estimators may be an appropriate choice even if normality does not hold. Meyer (1989) described that under the REML method Where the both the first and second derivatives are being used are more precise and efficient in cases of balanced data.

Wang (1967) compared the maximum likelihood and the restricted maximum likelihood estimators with respect to their mean square error criterion, the result of his studies indicate that the maximum likelihood estimator and its modifications are more efficient than other classical and Bayesian estimators.

Koltz et al (1974) used the balanced two way nested random models to show that the maximum Likelihood estimators when compared with the restricted maximum likelihood estimators are more efficient.

Harville et al (1978) used the balanced two way crossed classification model to show that the maximum likelihood estimators have minimum variance when compared to the restricted maximum likelihood estimators.

Li et al (1978) compared the maximum likelihood and the restricted maximum likelihood with the minimum variance unbiased estimators of the variance component with respect to their mean square error criterion using the split plot model, their result showed that the maximum likelihood estimators were more efficient when compared with the other estimators.

Graybill (1954) demonstrated that for the 2-fold nested classification model, the analysis of variance estimators have minimum variance in the  class of all quadratic unbiased estimators of the variance components that is they are uniformly best quadratic unbiased estimators of the variance components.

Gandula et al (1949) used the unbalanced two ways crossed classification to show that analysis of variance method of estimation of variance components is deficit in that it can produce negative estimates and also suggested that such estimates be replaced with zero.

Leone et al (1968) showed using a two way unbalanced model that the negative estimates can occur frequently in the analysis of variance method of estimation, in certain cases up to 25% of the time, he also noted that it is not so much the case of an inappropriate model or invalid assumptions but rather an intrinsic property of the method of estimation.

Smith et al (1984) in an attempt to resolve the problem of negative estimates of the analysis of variance method of estimation, proposed an alternative formulation of the model in which certain variance component are viewed as covariance, the formulation results in two sided test being conducted for these covariance. 

Sahia (1974c) gave analytical expressions for the non-negative maximum likelihood and restricted maximum likelihood estimators of the variance components in two fold nested and two way crossed classification random models.

Herbach (1959) showed that the balanced two way crossed classification random model with or without interaction does not have explicit maximum likelihood estimators.

Rothschild et al (1979) in their work used the balanced nested design to show that the REML method accounts for the degree of freedom used to estimate fixed effects and they also showed that maximum likelihood does not. They also stated that the REML method is robust to certain types of selection bias.

1.4 DEFINITION OF TERM

DAM: IT is referred to as a cow after having her first calf.

PROGENY: it is referred to as genetic offspring or descendant.

SIRE: sire in cattle breeding is referred to as the bull.

WEANING: Weaning is the process of introducing an infant to other food and reducing the supply of breast milk. 

INBREEDING: it is the production from the mating of two genetically related parents, which can increase the chances of the offspring being affected by recessive or deleterious traits.

CALF: it is a young domestic cattle, or it is the term used from birth to weaning.