Design and Analysis of Experiments on the Methods of Estimating Variance Components
Chapter One
AIMS AND OBIECTIVES OF THE STUDY
- To know the most efficient and precise method between the f% methods of variance component estimation in the case unbalanced
- To study the merits and demerits of each method of variance component estimation when dealing with cases of balanced and unbalanced designs when normality
- To know the most efficient method between the four methods of variance component estimation in the case of a balanced
CHAPTER TWO
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.
CHAPTER THREE
DATA COLLECTION AND RESEARCH METHODOLOGY SOURCE OF DATA
The data used in this study are secondary data generated from the institute of agricultural research for development (IARD), Maiduguri, Borno state; which was collected from Mr. Kabiru Mohammed of the lake chad research institute (LRI) Bomo state.
DESCRIPTION OF DATA
The data were collected from 1992 to 2008 from compiled herd books. The herd book contains many information such as; calf record sheet, bull record sheet, cow record sheet, disease record sheet, sire identity, calf identity, weaning weight, etc. Data were interestedly collected on the weaning weight of the progeny, which is the weight of the calf about 5 to 8 months of birth, depending on the Ndama cattle breed.
METHOD OF ANALYSIS
This research work was analyses using two-stage nested design with and without equal replications. In this analysis, four methods among the numerous method of variance component estimation were used to estimate the variance component.
Minimum variance quadratic unbiased estimation (MIVQUE) was the best among the five methods of variance components considered in this research work, this is because as the name implies it has the minimum variance when compared to the remaining three methods.
These four methods of variance component estimation was picked by the researcher based on no criteria, these methods were chosen as a comparative measure to know the one among the methods that will give the minimum variances in both cases of balanced and unbalanced data, hence the best of all methods. The overviews of these four methods of variance component estimation are being considered below:
METHOD OF MOMENTS OR THE ANALYSIS OF VARIANCE METHOD:
This method of estimating variance component was introduced by Henderson 1953 this method is normally used in cases of balanced data. But in 1953 Henderson developed a special technique for unbalanced data.
CHAPTER FOUR
THE ANOVA METHOD
Yij = µi + ∑ij I = 1, 2, …., t and j = 1, 2, ….., ni
Where the µi are uncorrelated with mean O and variance Ơ2Ԑ and µi and the ∑ij are correlated. The two sums of squares that can be used are the sums of squares within, Q0 or SSw and the sums of squares between Q1 or SSB where,
CHAPTER FIVE
CONCLUSION
The following deductions can be made from the estimates obtained using the ANOVA, MAXIMUM LIKELIHOOD AND RESTRICTED MAXIMUM LIKELIHOOD METHODS
- The estimates of the MAXIMUM LIKELIHOOD and RESTRICTED MAXIMUM LIKELIHOOD are
- The estimates of the three methods of estimating variance component used are within the parameter
- The maximum likelihood and restricted maximum likelihood method are the best methods of estimating variance components in animals for case of unbalanced data and one-way
RECOMMENDATION
As a result of the findings made from the work, the following recommendation can be made.
When researchers are confronted with cases of unbalanced data the maximum likelihood or restricted maximum likelihood method should be used since they provide relatively small variance components when compared with the ANOVA method of estimating variance components.
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- Handerson, C. R., 1975: comparison of alternative sire evaluation methods. J. Anim. Sci. 45, 760 – 770.
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