ST4241 Design and Analysis of Clinical Trials Lecture 7: N. Lecture 7: Non-parametric tests for PDG data

Transcription

1 ST4241 Design and Analysis of Clinical Trials Lecture 7: Non-parametric tests for PDG data Department of Statistics & Applied Probability 8:00-10:00 am, Friday, September 2, 2016

2 Outline

3 Non-parametric methods Non-parametric tests are distribution-free; that is, the distribution of the test statistics does not depend on the underlying distribution of the responses. Non-parametric methods should be used if responses are non-normally distributed, especially, when the parametric tests based on the original data and on the transformed data do not coincide.

4 Kolmogorov-Smirnov test Kolmogorov-Smirnov test is for testing whether there is any difference between two groups. The test statistic is given by M = max ˆF 1 (x) ˆF 2 (x), x where ˆF (x) = 1 ni n i j=1 I {X ij x}, i = 1, 2, are the empirical distribution functions of the two groups.the difference between the two groups is judged significant at level α if n1 n 2 M > κ α. n 1 + n 2 Critical values for Kolmogorov-Smirnov test are given below: α κ α

5 Computation of Kolmogorov-Smirnov test statistic Let x 11,..., x 1n1 and x 21,..., x 2n2 be the observations of group 1 and 2 respectively. Order the two samples together in a sequence x (1), x (2),..., x (n). where n is the number of untied values and only one value of a tie is kept in the sequence. At each value x (k), compute ˆF 1 (x (k) = 1 n 1 I {X 1j x n (k) }, ˆF 2 (x (k) = 1 n 2 I {X 2j x 1 n (k) }. 2 Compute j=1 j=1 max{ ˆF 1 (x (1) ˆF 2 (x (1),..., ˆF 1 (x (n) ˆF 2 (x (n) }

6 Peptic ulcer example revisited Lysozyme levels in the gastric juice of 29 patients with peptic ulcer and of 30 normal controls. Group 1 (n 1 = 29) Group 2 (n 2 = 30)

7 Computation of ˆF i x ˆF1 (x) ˆF2 (x) x ˆF1 (x) ˆF2 (x) x ˆF1 (x) ˆF2 (x)

8 Test statistic and conclusion M = max ˆF 1 (x) ˆF 2 (x) = = x n1 n M = 0.25 n 1 + n = < κ 0.2 = The null hypothesis that the two groups have no difference cannot be rejected even at level α = 0.2.

9 Mann-Whitney-Wilcoxon test Mann-Whitney-Wilcoxon (MWW) test is for testing whether two groups have the same location. The procedure of the MWW test is as follows. Rank all the observations together. Denote the rank of individual i by R i. Average the ranks for each group. Denote the average by R 1 and R 2. The MWW test statistic is given by χmww = 12n 1n 2 ( R 1 R 2 ) 2 n 2. (n + 1) where n j is the size of group j and n = n 1 + n 2.

10 Mann-Whitney-Wilcoxon test (cont.) If several observations are tied, each is given the average of their ranks they would have received if they had not been tied. When there are ties, the denominator of MWW statistic is adjusted by a factor f defined as T i=1 f = 1 t i(t i 1)(t i + 1), n (n 1)(n + 1) where T is the total number of values at which there are ties, t i is the number of ties at the ith tied value, i.e., in the case of ties χmww = 12n 1n 2 ( R 1 R 2 ) 2 n 2. (n + 1)f Under null hypothesis that there is no difference between the groups, χmww χ 2 1. The null hypothesis is rejected at level α if χmww χ 2 1α.

11 Peptic ulcer example (cont.) Original measurements together with ranks (in parentheses). Group 1 (n 1 = 29) Group 2 (n 2 = 30) 0.2(1.5) 4.9(26) 17.6(48) 0.2(1.5) 2.5(17) 8.8(35) 0.3(3.5) 5.0(27) 18.9(49) 0.3(3.5) 2.8(18) 9.1(36) 0.4(5.5) 5.3(28) 20.7(51.5) 0.4(5.5) 3.6(20) 10.3(38) 1.1(8) 7.5(32.5) 24.0(53) 0.7(7) 4.8(24) 15.6(43) 2.0(13.5) 9.8(37) 25.4(54) 1.2(9) 4.8(24) 16.1(44) 2.1(15) 10.4(39) 40.0(56) 1.5(10.5) 5.4(29) 16.5(46) 3.3(19) 10.9(40) 42.2(57) 1.5(10.5) 5.7(30) 16.7(47) 3.8(21) 11.3(41) 50.0(58) 1.9(12) 5.8(31) 20.0(50) 4.5(22) 12.4(42) 60.0(59) 2.0(13.5) 7.5(32.5) 20.7(51.5) 4.8(24) 16.2(45) 2.4(16) 8.7(34) 33.0(55)

12 Peptic ulcer example (cont.) The two mean ranks are: The test statistic R 1 = , R2 = ( )2 χmww = There are T = 8 groups of ties: = (t 1 = 2), 0.3(t 2 = 2), 0.4(t 3 = 2), 1.5(t 4 = 2), 2.0(t 5 = 2), 4.8(t 6 = 3), 7.5(t 7 = 2), 20.7(t 8 = 2). f = , 320 = Adjusted test statistic: χmww = 2.58/ = The p-value is

13 Kruskal-Wallis test KW test is for the comparison of more than two groups. The ranking procedure is the same as in MWW test. KW test statistic is given by H = 1 S 2 [ g i=1 R 2 i n i ] N(N + 1)2, 4 where R i is the sum of the ranks in group i, N = g i=1 n i, and S 2 = 1 g n i R 2 N(N + 1)2 ij. N 1 4 i=1 j=1 When there are no ties, the statistic reduces to 12 g Ri 2 H = 3(N + 1). N(N + 1) n i i=1

14 Kruskal-Wallis test (cont.) Under null hypothesis that there is no difference among the groups, asymptotically, H χ 2 g 1. The asymptotic approximation can be used when n i 5. The null hypothesis is rejected at level α if H χ 2 g 1,α. Tensile example In an experiment to determine if the cotton weight percent in a synthetic fiber affects the tensile strength, five levels of cotton weight percent were considered. At each level, five replicates are run. The measurement of tensile strength and their ranks are given in the next slide.

15 Tensile example (cont.) Tensile strengths and ranks (in parentheses). The last row gives the total rank of each level Weight percent of cotton (2) 12 (9.5) 14 (11) 19 (20.5) 7 (2) 7 (2) 17 (14) 18 (16.5) 25 (25) 10 (5) 15 (12.5) 12 (9.5) 18 (16.5) 22 (23) 11 (7) 11 (7) 18 (16.5) 19 (20.5) 19 (20.5) 15 (12.5) 9 (4) 18 (16.5) 19 (20.5) 23 (24) 11 (7) g i=1 ni j=1 R2 ij = , S 2 = 1 25(26)2 24 [ H = g i=1 R 2 i n i = ] = (26)2 [ ] = > χ 2 4,0.01 = The null hypothesis of no difference among the five weights is rejected at a level smaller than 0.01.

16 R functions The analysis of parallel-groups design data can be implemented in R by the following functions: sample() for random assignments. t.test() for comparison of two groups. lm() for comparison of multiple groups. ks.test() for Komokorov-Smirnov test. wilcox.test() for WMM test. kruskal.test() fro KW test.