Simultaneous Confidence Bands for Regression With Unknown Unequal Variances

Abstract

The usual normal-theory Scheffé bands in regression situations are invalid when variances are not equal across the levels of explanatory variables. I derive several simultaneous confidence bands appropriate for this situation when independent estimates of variances are available. The bands are designed to deal with a range of situations from very small to large sample sizes and from no difference to large differences in variances. The bands are also appropriate for combining several regression lines with different variances. The bands are centered at the usual least squares estimator or the weighted least squares estimators with the weights determined from data. Robustness of Scheffe's procedure is assessed and various comparisons of the proposed bands are made by a simulation experiment. Procedure C _MOLS centered at the usual least squares line, is recommended over Scheffe's method when a moderate heterogeneity is suspected. For more extreme heterogeneity, another procedure, C_MWLS centered at the weighted least squares line, is recommended as long as the degrees of freedom for each variance estimate are more than 2. My results also suggest that, even in the case of quite unequal variances, the usual least squares centering is preferable to the weighted least squares centering when most of the variances are estimated with only I df. The results are illustrated in the context of a real-data example involving accelerated life testing.

KEY WORDS:

• Behrens–Fisher problem
• Combination of regression lines
• Common mean estimation
• Heteroscedastic variance regression
• Meta-analysis
• Welch's approximation