A New Quantitative Method for Analysing Unreplicated Factorial Designs

Abstract

AbstractThe paper proposes a new method for the analysis of unreplicated factorial designs. The new method does not use an estimate of the error variance and has the potential to identify up to m – 1 active contrasts, where m is the number of contrasts in the study. It can be shown that the proposed test statistic, called MaxUr, is a function of the generalized likelihood ratio test statistic under normality, which was also used by Al‐Shiha and Yang (1999). Our strategy to identify active contrasts using MaxUr, however, is different from the multistage procedure proposed by Al‐Shiha and Yang (1999). Additionally, our simulation study seems to show that the new method is superior to Al‐Shiha and Yang's (1999) method.In order to test the performance of the new method, we did an extensive simulation study based on 10,000 samples to compare the new method with 12 other methods from the literature. To reasonably compare these methods, some of the 12 existing methods had to be slightly modified, such that the probability of falsely rejecting the global null hypothesis of no active factors was 0.05 for all methods. Two types of evaluation standard were used, the empirical power and the loss of decision. The results show that the new method performs very well, especially for large number of active contrasts (say more than 3 out of 15).A second purpose of the simulation study was to compare the performance of two well‐known estimates for the variance. One is the PSE introduced by Lenth (1989), the other the ASE introduced by Dong (1993). The simulation study confirmed the approximations of Kunert (1997) which indicate that Dong's (1993) estimate should perform better for small numbers of active factors. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)