# Design And Analysis Of Experiments On The Methods Of Estimating Variance Components In Farm Animals

Design And Analysis Of Experiments On The Methods Of Estimating Variance Components In Farm Animals

## Abstract

There have been general contradictions on the appropriate method to use in the estimating the variance components of animals. So this problem has led us into this research to ascertain the relatively best or appropriate method to be used in estimating these variance components in farm animals. Nested design, otherwise called hierarchical design, is a form of experimental design with a prominent characteristic such that each level of every factor occurs with all levels of the other factors, without interactions. For instance, in certain types of studies, the levels of one factor B, will not be identical across all levels of factor A, each level of factor A will contain different levels of factor B, therefore, levels of B are said to be nested within the levels of factor A. 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) were analyzed with intention of finding the best to use in estimating variance components. This study recommended “modified maximum likelihood method as the best since it has the smallest minimum variances”.

## Chapter One

### Introduction

#### 1.1 Background of the Study

Variance measures the variability or difference from a mean or response. A variance value of 0 indicates that all values within a set of numbers are identical. Statisticians use variance to see how individual numbers or values relate to each other. Estimating variance components in statistics refers to the processes involved in efficiently calculating the variability within responses or values. Variance component are estimated when a new improved trait is discovered, variances or variability changes or alternate overtime due to environmental or genetic changes, a new trait is about to be defined or explained

A cardinal objective of many genetic surveys is the estimation of variance components associated with individual traits. Heritability, the proportion of variation in a trait that is contributed by average effects of genes, may be calculated from variance components. The heritability of a trait gives an indication of the ability of a population to respond to selection, and thus, the potential of that population to evolve (Alwin, D. F. (2006). Estimates of variance components are common in the discipline of animal breeding and production, where this information on the variance components is used in the development of selection regimes to improve economically important traits (Anderson et al, 2002). A requirement for estimating variance components is knowledge of the relationship structure of the population. In a natural population, variance components are also of considerable interest for evolutionary studies (Bainbridge, T. R. 2005) and also for conservation purposes. In natural populations, however, information on relationships may be unreliable or unavailable. These estimates of relationships may be combined with phenotypic information gathered from the same individuals, allowing inferences to be made about variance components Bush, N. (2001).

Molecular data are used to infer relationships between animals on a pair-wise basis, because this provides the least complex level at which relationships may be estimated, while still allowing a population to be divided into several relationship classes. Estimates of pair-wise relationships are then combined with a pair-wise measure of phenotypic information. Several methods of estimating variance components have been studied, but for the purpose of clarity four different methods of estimating these variance components will be evaluated in this research work. They are;

1. The ANOVA method
2. The Maximum likelihood method
3. The Restricted maximum likelihood method
4. The Quasi maximum likelihood method.

#### 1.2 Statement of the Problem

The appropriate method to use in the estimating the variance components of animals have been a general issue. So this problem has led us into this research to ascertain the relatively best or appropriate method to be used in estimating these variance components in farm animals.

#### 1.3 Objective of the Study

The major objective of this study is to determine the best method to be used between the methods enumerated above in estimating variance components of farm animals. To achieve the stated objective, the following specific objectives were laid out:

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.

#### 1.4 Significance of the Study

A major significance of this study is that it unravels the relatively best methods among the methods highlighted above with a view to advising animal breeders, producers and animal researchers on the best method to be used in estimating variance components as which relatively better method has generated a lot of controversies over time.

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 of the Study

The scope of the study is centered on the methods of estimating variance components in farm animals, to know which of the methods is relatively better in estimation.

#### 1.6 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?

#### 1.7 Limitations of the 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.

#### 1.8 Hypothesis to be Tested

• H0: there is no significant difference between the methods of estimating variance components.
• H1: there is a significant difference between the methods of estimating variance components. Level of significance: 0.05

Decision rule: reject H0 if p-value is less than the level of significance. Accept H0 if otherwise.

#### 1.9 Definition of Terms

##### Variance:

The amount by which something changes or is different from something else.

##### Estimation:

A judgment or opinion about the value or quality of somebody or something.

##### Traits:

A particular quality in someone’s personality.

##### Genetic:

The units in the cells of livings that controls its physical characteristics.

##### Components:

One of several parts of which something is made.

## Chapter Five

### Conclusion and Recommendation

#### 5.1 Conclusion

In this work where application of two-stage nested design unbalanced case was applied on a population of Gudali beef Cattle, observing weight of the progeny under dams nested with the sires with a view to observe the significant effects of variability and the variance components, conclusion was that the variability effects of sires was significant and that of dams within sire was not. Moreover, modified maximum likelihood method of variance component estimation was recommended as the one with the smallest minimum variance. This smallest minimum variance, which Modified maximum likelihood method has placed it first before the other considered methods, in line with the properties of estimators.

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