A DESIGN AND ANALYSIS
OF EXPERIMENTS ON THE METHODS OF ESTIMATING VARIANCE COMPONENTS IN FARM ANIMALS.
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 (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;
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 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.
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.
1.4.
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 .
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.
DEFINITION
OF TERMS
Ø Variance:
the amount by which something changes or is different from something else.
Ø Estimation:
a judgment or opinion about the value or quality of somebody or something.
Ø Traits:
a particular quality in someone’s personality.
Ø Genetic:
the units in the cells of livings that controls its physical characteristics.
Ø Components:
one of several parts of which something is made.
1.7.
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.
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.