A MONTE-CARLO INVESTIGATION OF THE ROBUSTNESS OF THE WALD, LIKELIHOODRATIO, AND MEDIAN TESTS WITH SPECIFIED SYMMETRICAL AND ASYMMETRIC MARGINAL DISTRIBUTIONS

The effect of nonnormality on the Type I (tau) error when comparing tw o independent binomial proportions (P) or the nonparametric alternativ es, the Median (Me), Wald (W), and Likelihood Ratio (LR), has not been investigated. If these selected tests are overly conservative the imp lied loss of power would moderate their practical use. The purpose of the present study was to investigate the impact of nonnormality on sma ll to moderate sample sizes on the estimated tau for alpha = 0.10, 0.0 5, and 0.01 for the P, Me, W, and LR tests. Samples were generated fro m nine long-tailed symmetric and asymmetric distributions using a mult iplicative congruential generator. For each marginal distribution and for a variety of sample sizes, the proportion of samples for which the test statistic exceeded the 10, 5, and 1 percentage points was tabula ted. For data that mimic a symmetric distribution, the median test uni formly yields an empirical alpha considerably less than tau, while the likelihood ratio test consistently overestimates tau for small sample s (n less-than-or-equal-to 15) over all symmetric distributions and em pirical alpha levels. For asymmetric distributions, the median test ag ain yields an empirical alpha significantly less than tau. Similar und erestimates of tau were found for the chi-square (2 df), chi-square (4 df) log normal, and gamma (2, 1) distributions. The likelihood ratio test consistently overestimates tau for small samples (n less-than-or- equal-to 15) over all asymmetric distributions and empirical alpha lev els. The independent proportions test produces an empirical alpha clos est to tau for n = 10 for all asymmetric distributions. The most consi stent test procedure, disregarding sample size or distribution charact eristics, is the independent proportions test procedure.