Analysis of experimental designs with unequal group variances

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.