ST4241 Design and Analysis of Clinical Trials Lecture 9: N. Lecture 9: Non-parametric procedures for CRBD

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1 ST21 Design and Analysis of Clinical Trials Lecture 9: Non-parametric procedures for CRBD Department of Statistics & Applied Probability 8:00-10:00 am, Friday, September 9, 2016

2 Outline Nonparametric tests for matched pair designs

3 Sign test Procedure of sign test Compute the difference within each pair, d i = X i1 X i2. Count n +, the number of positive d i s and n, the number of negative d i s. The sign test statistic is defined as n max = max{n +, n }. Let n = n + + n. Under the null hypothesis of no treatment effects, n max follows a binomial distribution Bio(n, 1/2). The p-value of the test (two-sided) is computed as p = 2 2 n n i=n max ( ) n. i

4 Approximated sign test If n is large (> 10), the distribution of n max can be approaximated by a standard normal distribution. By Centrol Limit Theorem, n max n 2 n = n + n n N(0, 1). Making the correction of continuity by replacing n max with n max 1/2, the above statistic becomes Z = n + n 1. n The significance of the difference is claimed at level α if Z z α/2, the upper α/2 quantile of N(0, 1).

5 Wilcoxon s signed rank test The sign test ignores the magnitude of the differences. The Wilcoxon signed rank test takes into account both the signs and the magnitude. The procedure of the Wilcoxon signed rank test is as follows: Rank the absolute values of the differences, 0 differences (if any) are ignored. Denote by R + the sum of the ranks for the pairs with positive difference. Define the test statistic Z = R + n (n +1), n (n +1)(2n +1)f where f is the factor for adjusting ties given before. Under the null hypothesis of no effect difference, the above statistic follows a standard normal distribution. The significance of difference is claimed at level α if Z > z α/2. 2

6 Remark The Wilcoxon signed rank test is in general more powerful than the sign test because of the following fact. The numerator of the Wilcoxon signed rank statistic R + n (n + 1) = R + 2n +(n + 1) + (n + n )(n + 1) = R + n + n (n + 1) + (n + n )(n + 1) n 2 = n + R+ n + (n + R+ + n R ) + (n + n )(n + 1) n = n + (n R+ n R ) + (n + n )(n + 1) n = n +n n ( R + R ) + (n + n )(n + 1).

7 Sign test for Antidepression Trial Pair Imi Pla d Pair Imi Pla d n + = 8, n = 18, Z = ( )/ 26 = < z = 1.96

8 Wilcoxon Signed rank test for Antidepression Trial Pair d d Rank Pair d d Rank

9 Wilcoxon Signed rank test for Antidepression Trial (cont.) From the rank table, it is computed that R + = 88.5, f = and the Wilcoxon signed rank statistic Z = R + n (n +1) n (n +1)(2n +1)f 2 = Since Z > 1.96, the test is significant at level α = The example illustrates that the Wilcoxon signed rank test is more powerful than the sign test.

10 Friedman test Friedman s test is for the CRBD data with g > 2. Procedure of Friedman s test: Drop the blocks with equal measurements, if any. Rank the g measurements within each of the remaining blocks. Compute the average ranks R j, j = 1,..., g, over the blocks. Compute the test statistic W = 12n g(g + 1) g j=1 ( R j g where n is the number of blocks in which at least two measurements are unequal. Claim the significance of differences at level α if W χ 2 g 1,α, the upper α qauntile of the χ 2 -distribution with df g 1. ) 2,

11 Plasma clotting trial revisited The original data of the trial is as follows: Measurement Subject

12 Friedman s test for Plasma clotting trial Obtain the ranks within each block: Rank Subject R j

13 Friedman s test for Plasma clotting trial (cont.) The Friedman s test statistic is computed as W = [( )2 + (2 2.5) 2 +( ) 2 + ( ) 2 ] = The p-value is given by P(χ 2 3 > 1.96) = The treatments are significantly different at any level α > A Remark The ranking for Friedman s test is different from that for Kruskal-Wallis test. The Kruskal-Wallis test ranks all the observations together. The Friedman s test ranks the observations within blocks.