Preface to First Edition - lib

STATISTICAL TESTS 51assumed to have a normal distribution with mean µ and the null hypothesishere is that the mean difference is zero, i.e., H 0 : µ = 0. The paired t-statisticist =¯ds/ √ nwhere ¯d is the mean difference between the paired measurements and s is itsstandard deviation. Under the null hypothesis, t follows a t-distribution withn − 1 degrees of freedom. A 100(1 − α)% confidence interval for µ can beconstructed bywhere P(t ≤ t α,n−1 ) = 1 − α/2.¯d ± t α,n−1 s/ √ n3.2.2 Non-parametric Analogues of Independent Samples and Paired t-TestsOne of the assumptions of both forms of t-test described above is that the datahave a normal distribution, i.e., are unimodal and symmetric. When departuresfrom those assumptions are extreme enough to give cause for concern,then it might be advisable to use the non-parametric analogues of the t-tests,namely the Wilcoxon Mann-Whitney rank sum test and the Wilcoxon signedrank test. In essence, both procedures throw away the original measurementsand only retain the rankings of the observations.For two independent groups, the Wilcoxon Mann-Whitney rank sum testapplies the t-statistic to the joint ranks of all measurements in both groupsinstead of the original measurements. The null hypothesis to be tested is thatthe two populations being compared have identical distributions. For two normallydistributed populations with common variance, this would be equivalentto the hypothesis that the means of the two populations are the same. Thealternative hypothesis is that the population distributions differ in location,i.e., the median.The test is based on the joint ranking of the observations from the twosamples (as if they were from a single sample). The test statistic is the sum ofthe ranks of one sample (the lower of the two rank sums is generally used). Aversion of this test applicable in the presence of ties is discussed in Chapter 4.For small samples, p-values for the test statistic can be assigned relativelysimply. A large sample approximation is available that is suitable when thetwo sample sizes are greater and there are no ties. In R, the large sampleapproximation is used by default when the sample size in one group exceeds50 observations.In the paired situation, we first calculate the differences d i = y 1i − y 2i betweeneach pair of observations. To compute the Wilcoxon signed-rank statistic,we rank the absolute differences |d i |. The statistic is defined as the sumof the ranks associated with positive difference d i > 0. Zero differences arediscarded, and the sample size n is altered accordingly. Again, p-values for© 2010 by Taylor and Francis Group, LLC