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

**CHAPTER ONE**

**INTRODUCTION**

**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 (Lande & Shannon, 1996). 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 (Lynch & Walsh, 1998). 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 (Boag, 1983) 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 (Ritland, 1996; Mousseau *et al*., 1998).

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;

The ANOVA method The Maximum likelihood method The Restricted maximum likelihood method The Quasi maximum likelihood method.

**STATEMENT OF THE GENERAL PROBLEM**

They 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.

**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.

**SIGNIFICANCE OF THE STUDY**

A major significance of this study is to unravel 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 .

**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.

**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.

**HYPOTHESIS TO BE TESTED** : there is no significant difference between the methods of estimating variance components. : 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.

**REFERENCES**

Anderson R.L., Bancroft T.A. (1952): Statistical Theory in Research. McGraw-Hill, New York.

Federer W.T. (1968): Non-negative estimators for components of variance. Appl. Stat., 17, 171–174.

Fisher R.A. (1925): Statistical Methods for Research Workers, Oliver and Boyd.

Gamerman D. (1997): Markov Chain Monte Carlo. Chapman and Hall, New York.

Gelman A., Carlin J.B., Stern H.S., Rubin D.B. (1995): Bayesian Data Analysis. Chapman and Hall, New York. Herbach L.H. (1959): Properties of model II type analysis of variance tests A: Optimum nature of the F-test for model II in balanced case. Ann. Math. Statist., 30, 939–959.

Klotz J.H., Milton R.C., Zacks S. (1969): Mean square efficiency of estimators of variance components. J. Am. Stat. Assoc., 64, 1383–1402.

LaMotte L.R. (1973): Quadratic estimation of variance components. Biometrics, 29, 311–330. Rao C.R. (1971a): Estimation of variance and covariance components: MINQUE theory. J. Multivar. Anal., 1, 257–275.

Rao C.R. (1971b): Minimum variance quadratic unbiased estimation of variance components. J. Multivar. Anal., 1, 445–456.

Rao C.R. (1972): Estimation of variance and covariance components in linear models. J. Am. Stat. Assoc., 67, 112–115.

Rasch D. (1995): Mathematische Statistik. Joh. Ambrosius Barth and Wiley, Berlin, Heidelberg. 851 p. Rasch D., Tiku M.L.,

Sumpf D. (1994): Elsevier’s Dictionary of Biometry. Elsevier, Amsterdam, London, New York. Rasch D., Verdooren L.R.,

Gowers J.I. (1999): Fundamentals in the Design and Analysis of Experiments and Surveys – Grundlagen der Planung und Auswertung von Versuchen und Erhebungen. Oldenbourg Verlag, München,

Wien. Reinsch N. (1996): Two Fortran programs for the Gibbs Sampler in univariate linear mixed models. Dummerstorf. Arch. Tierz., 39, 203–209. Sarhai

H., Ojeda M.M. (2004): Analysis of Variance for Random Models, Balanced Data. Birkhäuser, Boston, Basel, Berlin. Sarhai H.,

Ojeda M.M. (2005): Analysis of Variance for Random Models, Unbalanced Data. Birkhäuser, Boston, Basel, Berlin.

Sorensen A., Gianola D. (2002): Likelihood, Bayesian, and MCMC Methods in Quantitative Genetics. Springer, New York.

Stein C (1969): Inadmissibility of the usual estimator for the variance of a normal distribution with unknown mean. Ann. Inst. Statist. Math. (Japan), 16, 155–160. Tiao G.C.,

Tan W.Y. (1965): Bayesian analysis of random effects models in the analysis of variance I: Posterior distribution of variance components. Biometrika, 52, 37–53. Verdooren L.R. (1982): How large is the probability for the estimate of a variance component to be negative? Biom. J., 24, 339–360.