Figure 1.
Results of a simulation study to compare power and sample sizes of F-test in One-way ANOVA with the constrained inference Williams’ type test where the critical values are derived using 10,000 bootstrap samples.
The power of the Williams test was estimated by averaging 1000 simulated where the critical values are estimated using 10,000 bootstrap samples. The power for F-test was determined using PROC POWER in SAS (9.0). The null hypothesis was that the means of the four dose groups were equal (and zero) and the alternative hypothesis was that the means of the four dose groups have an increasing trend with dose. Data representing the four dose groups were simulated from normal populations with dose means taken to be 0, 0.1, 0.5 and 1, respectively. The actual values of the doses are irrelevant for the two methods described here. The population standard deviation for the four populations was taken to be 1. Corresponding to the 14 different patterns of total sample sizes, namely, 20, 24, 28, 32, 36, 40, 44, 48, 52, 60, 76, 80, 100, 116, the powers of the two methods are plotted. The Type I error was set to 0.05.
Figure 2.
Flow chart for constructing test statistic.
Figure 3.
Flow chart for deriving Bootstrap data under the null hypothesis.
Table 1.
Type I errors for homoscedastic normally distributed data.
Table 2.
Power for homoscedastic normally distributed data.
Table 3.
Type I errors for heteroscedastic normally distributed data.
Table 4.
Power for heteroscedastic normally distributed data.
Table 5.
Type I errors for log-normally distributed data.
Table 6.
Power for log-normally distributed data.
Figure 4.
Normal quantile-quantile plots of studentized residuals from regressing log organic mercury in the placebo and succimer groups.