ABSTRACT
A two-stage nested design with unequal replications was used in this study with an intention to capture the variability effects in the model. Moreover, five different methods of variance component estimation which were randomly chosen from the frequency approach were compared in this study. These methods which were Analysis of Variance method (ANOVA), Quasi-maximum-likelihood method (QML), Modified likelihood method (ML), Restricted maximum-likelihood method (REML), and Modified maximum likelihood method (MML), recommended “modified maximum likelihood method as the best since it has the smallest minimum variances”.
CHAPTER ONE
INTRODUCTION
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 is 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 2014 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.
Fourtly, 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 (2010). 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 2017), 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
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 RESEARCH QUESTIONS
1. What is the most efficient and precise method between the f% methods of variance component estimation in the case unbalanced data?
2. What are the merits and demerits of each method of variance component estimation when dealing with cases of balanced and unbalanced designs when normality holds?
3. What is the most efficient method between the four methods of variance component estimation in the case of a balanced design?
1.4 SIGNIFICANCE OF THE STUDY
This study will be of immense benefit to other researchers who intend to know more on this topic and can also be used by non-researchers to build more on their work. This study contributes to knowledge and could serve as a bench mark or guide for other work or study.
1.5 SCOPE/LIMITATIONS OF THE STUDY
This study is on immorality in churches will cover all forms of immoral activities that exist in churches today with a view of finding a lasting solution to the problem.
Limitations of Study
Financial constraint: Insufficient fund tends to impede the efficiency of the researcher in sourcing for the relevant materials, literature or information and in the process of data collection (internet, questionnaire and interview).
Time constraint: The researcher will simultaneously engage in this study with other academic work. This consequently will cut down on the time devoted for the research work.
1.6 DEFINITION OF TERMS
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.
