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Analysis of experimental designs with unequal group variances

Talukder, M. A. H. (1976) Analysis of experimental designs with unequal group variances. Doctor of Philosophy (PhD) thesis, University of Kent. (doi:10.22024/UniKent/01.02.94683) (KAR id:94683)

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This thesis deals with weighted (generalised) least squares estimation and analysis for some common experimental designs with the error variance heteroscedastic with respect to the levels of one factor, namely, the treatments or (for split-plot designs) sub-plot treatments. The simple regression model with error variance heteroscedastic with respect to the values of the independent variable, is also considered briefly. The observations in any of the analyses considered are grouped in such a way that the error variance is constant within groups but varies from group to group.

On the assumption that the group variances are known, the weighted least squares estimators of the linear parameters and the corresponding analysis (Aitken, 1934-35; Plackett, I960, pp. 47-49) are provided for each design or model. An expression for joint confidence intervals of parametric contrasts for the heteroscedastic models is also obtained. The estimators of the linear parameters and other statistics usually involve actual weights, the reciprocals of the group variances.

The actual weights are not usually known. The estimators of the group variances are therefore derived for each design or model. for some designs, the minimum norm quadratic unbiased estimators (Rao, 1970; 1973, pp. 303-305) of group variances are independently distributed as multiples of x2. For other designs, almost unbiased estimators (Horn et al., 1975) of group variances have negligible bias and are approximately independently distri-buted as multiples of x2 Reciprocals of these estimators are used as the estimated weights.

The weighted least squares estimators of the linear parameters or variance components and other statistics including test-statistics using estimated weights, are generally biased. It is shown in the thesis how a major part of the bias can be removed; the procedure stems from a theorem due to Meier (1953). The estimators and other statistics using estimated weights are adjusted accordingly. A modified form of this theorem is also proved for correlated estimators of the group variances. A small Monte Carlo study conducted for completely randomised designs showed that the performances of the adjusted statistics are more or less satisfactory.

The designs and models covered in this thesis are: completely randomised designs, the general two-way model with proportional cell frequencies, general block designs, randomised complete block designs, latin square designs, split-plot designs with two treatment factors and the linear regression model. For the first three designs, both the fixed-effects models and random or mixed models are considered whereas only the fixed-effects models are dealt with for the remaining three designs.

Item Type: Thesis (Doctor of Philosophy (PhD))
DOI/Identification number: 10.22024/UniKent/01.02.94683
Additional information: This thesis has been digitised by EThOS, the British Library digitisation service, for purposes of preservation and dissemination. It was uploaded to KAR on 25 April 2022 in order to hold its content and record within University of Kent systems. It is available Open Access using a Creative Commons Attribution, Non-commercial, No Derivatives ( licence so that the thesis and its author, can benefit from opportunities for increased readership and citation. This was done in line with University of Kent policies ( If you feel that your rights are compromised by open access to this thesis, or if you would like more information about its availability, please contact us at and we will seriously consider your claim under the terms of our Take-Down Policy (
Uncontrolled keywords: Statistical analysis
Subjects: Q Science > QA Mathematics (inc Computing science) > QA276 Mathematical statistics
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
SWORD Depositor: SWORD Copy
Depositing User: SWORD Copy
Date Deposited: 25 Nov 2022 10:06 UTC
Last Modified: 28 Nov 2022 11:06 UTC
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