Clinical Trials

❘❙❚■ Chapter 19 | Comparison of MeansTwo-sample Z-testIf the sample sizes n 1and n 2are large (say n 1and n 2> 50) [2–4], we can use aZ-statistic to test the null hypothesis in equation (6), as discussed earlier:XZ = 1– X 2SE(X 1– X 2)where SE(X 1– X 2)= S 21+ S 2 1/ 22( n 1n 2)According to statistical theory (central limit theorem), Z is approximatelynormally distributed with mean 0 and standard deviation 1 when n 1and n 2are large [2]. Therefore, the treatment difference will be significant at the α levelof significance if:|Z| ≥ Z α/2where Z α/2is the critical value of the standard normal distribution (see Chapter 17) [6].The corresponding 100(1 – α)% CI for μ 1– μ 2is given by:(X 1– X 2) + Z α/2× SE(X 1– X 2)Unlike the two-sample t-test, the two-sample Z-test does not require the standarddeviations to be similar (ie, σ 1= σ 2= σ), although there are still assumptions ofnormality, independence, and homogeneity [2–4].Two-sample Wilcoxon rank-sum (Mann–Whitney) testIn previous sections, statistical inferences have been primarily based on theassumption that the outcome under evaluation follows a normal distribution.In practice, this ‘normality’ might not be present. For example, consider that theoutcome is length of stay in hospital – this outcome usually has a skeweddistribution, with most people staying a short duration. We now need differentstatistical methods using nonparametric methods that do not require normalityto draw statistical conclusions. Corresponding to the one-sample t-test, pairedt-test, and two-sample t-test to address Questions 1, 2, and 3, three nonparametric(or distribution-free) methods are available [7–9]:• Wilcoxon signed rank test• Wilcoxon matched pairs signed rank-sum test• Wilcoxon rank-sum (Mann–Whitney) test210