The comparison of nonparametric statistical tests for interaction effects in factorial design

Abstract

Correct application of the classical factorial F-test depends on normality and homogeneity of variance assumptions. If these assumptions are violated the type I error rate will be inflated and power of the test will be decreased. Therefore nonparametric statistical tests have been proposed to analyze the interaction effects in factorial designs. A simulation was conducted to investigate the effect of non-normality on type I error rate and power of the test of the classical factorial F-test and five nonparametric tests namely rank transformation (FR), Winsorized mean (FW), modifies mean (FM), adjusted rank transform (ART) and adjusted median transform (AMT) using program SAS 9.4 with 1,000 replications. The study used 2×2 factorial design with replications of 3, 4 and 6 making sample sizes of 12, 16, and 24, respectively and 3×3 factorial designs with replication of 3 making a sample size of 27 studied at 0.05 level of significance. As a results, when the normality of assumption is satisfied all six statistical tests have the ability to control type I error in all situations. The ART test cannot control type I error rate for 3×3 factorial design when sample size is 27 when normality assumption is violated. For power of the test, the F-test provided the highest test power when the normality of assumption is met. The ART and AMT tests provided approximately the same test power. The AMT and ART tests can be effectively used to analyse the interaction effect between factors A and B in 2×2 factorial design when the sample size is 12 and 16 or 24 respectively and the normality of assumption is not met. Moreover, the results showed that when sample sizes increased, all six statistical tests tended to increase the power of the test.

Keywords

• Factorial design
• Rank transformation
• Modified mean
• Adjusted rank transform test
• Winsorized mean
• Adjusted median transform