| The purpose of this dissertation was to compare the Type I error and power properties of the Rank Transform Hotelling's T 2 with the parametric Hotelling's T2 in one sample and two sample cases with continuous and ranked data using Monte Carlo techniques. The ranked data violates the normality assumption of the parametric Hotelling's T2.; The simulation results demonstrated that the parametric Hotelling's T2 was conservative with respect to Type I error rates for the test of equality of two population centroids. For example, when n1 = n2 = 50 and a nominal alpha level of 0.05 with correlation coefficient of 0.2, the Type I error rate was 0.0175. However, when the correlation coefficient was large (e.g. 0.7) the Type I error rates for parametric Hotelling's T 2 test results were inflated. Further, Type I error inflations became progressively worse when large correlations were coupled with increased sample sizes. For example, when the correlation coefficient was 0.7 the Type I error were: 0.071 with n1 = n 2 = 10; 0.0854 with n1 = n 2 = 3 0; and 0.1107 with n1 = n 2 = 50 when the distribution was normal. In short, the parametric Hotelling's T2 was robust only for Cauchy and Exponential distributions with correlation coefficient of 0.7 and n1= n2 = 10. For these two cases, the Rank Transform had power advantages over the parametric Hotelling's T2.; When the sample sizes for the two-sample case were not equal, the parametric Hotelling's T2 test failed as a test due to severe Type I error inflations. For example, when n 1 = 10, n2 = 30 and correlation coefficient of 0.2 the test was rejecting 100% of the time across all distributions.; The Rank Transform Hotelling's T2 demonstrated robust Type I error rates for small sample sizes, but conservative Type I error rates for large sample sizes. For example, for n 1 = n2 = 10, with correlation coefficient of 0.7 and nominal alpha level 0.05 the Type I error rate for Rank Transform T2 was 0.0526, but 0.039 when sample sizes were n1 = n2 = 50. For n1 = n2 = 30 the corresponding Type I error rate was 0.0457.; The other purpose of this dissertation was to propose a method for simulating continuous and ranked bivariate data, with a pre-specified correlation. The empirical results of the proposed method demonstrated that the estimates where in close agreement with their associated population parameters. In summary, the absolute errors of r, the sample estimate, were within 10-2. |
