E509A: Principle of Biostatistics. (Week 11(2): Introduction to non-parametric. methods ) GY Zou.
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1 E509A: Principle of Biostatistics (Week 11(2): Introduction to non-parametric methods ) GY Zou gzou@robarts.ca
2 Sign test for two dependent samples Ex 12.1 subj baseline post diff sign Is 7 + significant?
3 Under H 0, each subject could get a + with p =0.5. We thus can use binomial distribution to obtain P -value. Let x denote the number of + signs, then Pr(X = x) = ( n x) 0.5 x (1 0.5) n x = ( 10 x ) 0.5 n x Pr By definition P = = which is one-sided. Two-sided P = = mid-p = =.1133, two-sided mid-p = = Sign test disregards a lot of information.
4 The Wilcoxon signed-rank test (for two dependent samples) The null hypothesis is the median of the differences is 0, i.e., H 0 : M d =0. subj diff sign rank rank sign Wilcoxon signed-rank test statistic is the sum of the positive ranks, denoted by T.
5 Anybody can propose a test, the difficulty is to figure out the property of the test. For T, if no positive rank, T =0; if all positive, T = n = n(n +1)/2, wheren is the number of observations with nonzero difference The mean for T becomes n(n +1)/4 and variance under H 0 is a large sample test var 0 (T )= n(n + 1)(2n +1) 24 Z = T n(n+1) 4 n(n+1)(2n+1) 24 N(0, 1), under H 0
6 Ex 12.2 Z = T n(n+1) 4 n(n+1)(2n+1) = (9+1) 4 9(9+1)(2 9+1) =2.13 which yields a P -value of (1-sided). Two-sided p-value is =
7 Wilcoxon-Mann-Whitney (WMW) test for two independent samples I use WMW is because this test has been proposed at least 7 times (Kruskal 1957 J Am Stat Assoc 52: ). Idea:Supposewehaven 1 observations from group 1 and denoted as x 1,x 2,,x n1 ). We also have n 2 observations from group 2, denoted as y 1,y 2,,y n2. x 1 x 2 x n1 Total y 1 y 2. y n2 Total U In each cell, if x i <y j we put 1, if x i >y j, we put 0, if x i = y j we put 0.5. There should be n 1 n 2 comparisons. Once we are done, we sum them up to get U statistic (commonly referred to as Wilcoxon-Mann-Whitney U statistic)
8 Check to see if U<n 1 n 2 U then use U to proceed, otherwise use n 1 n 2 U as U The distribution of U under H 0 U ranges from 0 to n 1 n 2,meann 1 n 2 /2; Under H 0,thevarianceofU can be shown to be Z = n 1 n 2 (n 1 + n 2 +1) 12 U n 1n 2 2 n1 n 2 (n 1 +n 2 +1) which is asymptotically distributed as N(0, 1). This looks different from your book, because 12 S = U + n 1(n 1 +1) 2 where n 1 is the sample size for U used for the test.
9 Ex Total Total U =6 Z = U n 1n 2 2 n1 n 2 (n 1 +n 2 +1) = (4+4+1) = P -value is 0.28 (1-sided). Two-sided p-value is = Some prefer 0.5 continuity correction so that 12 Z = U n 1n n1 n 2 (n 1 +n 2 +1) 12 = (4+4+1) 12 = 0.433
10 data a; input group response cards; ; proc npar1way wilcoxon data=a; class group; var response; run;
11 The NPAR1WAY Procedure Wilcoxon Scores (Rank Sums) for Variable response Classified by Variable group Sum of Expected Std Dev Mean group N Scores Under H0 Under H0 Score Wilcoxon Two-Sample Test Statistic Normal Approximation Z One-Sided Pr > Z Two-Sided Pr > Z t Approximation One-Sided Pr > Z Two-Sided Pr > Z Z includes a continuity correction of 0.5. Kruskal-Wallis Test Chi-Square DF 1 Pr > Chi-Square
12 napr1way for Wilcoxon-Mann-Whitney data roc; input disease out count; cards; ; proc npar1way wilcoxon data=roc; class disease; var out; freq count; run;
13 The NPAR1WAY Procedure Wilcoxon Scores (Rank Sums) for Variable out Classified by Variable disease Sum of Expected Std Dev Mean disease N Scores Under H0 Under H0 Score Average scores were used for ties. Wilcoxon Two-Sample Test Statistic Normal Approximation Z One-Sided Pr < Z Two-Sided Pr > Z t Approximation One-Sided Pr < Z Two-Sided Pr > Z Z includes a continuity correction of 0.5. Kruskal-Wallis Test Chi-Square DF 1 Pr > Chi-Square
14 Misconception of WMW test Hart A Mann-Whitney test is not just a test of medians: differences in spread can be important. BMJ 323: If no distribution assumption is made, then the null hypothesis is the probability that a member of the first population drawn at random will exceed a member of the second population drawn at random is 50% Blind date in City of Toronto: the probability of the man taller than the women. If we assume two population distributions have same shape, then WMW is testing the equality of medians. Otherwise, it is not testing the equality of medians.
15 Hart also note As Altman states, one form of the test statistic is an estimate of the probability that one variable is less than the other, although this statistic is not output by many statistical packages. Here I present a simple way using SAS proc freq with measures option. This is because Pr(X <Y) canalsobeexpressedintermsofandsommers d (Somers, 1962, American Sociological Review 27: ) as Pr(X <Y)=(d +1)/2. SAS gives d and its standard error, from which we can obtain estimate for Pr(X <Y) and its confidence interval because [ ] var Pr(X var( d) <Y) = ŝ.e.( ŝ.e.( d) Pr(X <Y)) = 4 2
16 Example. A clinical trial (Aurlien et al 1998 Bone Marrow Transplant 21: ) involving 35 patients with malignant lymphoma was conducted to estimate the effect in response between Hodgkin s disease patients and non-hodgkin s patients with respect to time (days) to neutrophil recovery. Trt 1 (n 1 =25): 8, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 15 Trt2 (n 2 =10): 10, 10, 11, 11, 11, 12, 13, 16, 17, 24 Denoting X and Y as the responses obtained from non-hodgkin s and Hodgkin s patients, respectively, we are interested in Pr(X <Y).
17 data a; input group response cards; ; proc freq; tables group* response/ measures cl; test measures; run;
18 The FREQ Procedure Statistics for Table of group by response Statistic Value ASE Gamma Kendall s Tau-b Stuart s Tau-c Somers D C R Somers D R C Pearson Correlation Spearman Correlation Lambda Asymmetric C R Lambda Asymmetric R C Lambda Symmetric Uncertainty Coefficient C R Uncertainty Coefficient R C Uncertainty Coefficient Symmetric Sample Size = 35
19 PROC FREQ output provides d =0.18 with standard error , this gives Pr(X <Y)= =0.59 with standard error ŝ.e.( Pr(X <Y)) = 2 95% CI for Pr(X <Y) is then which include = ± = (0.38, 0.80) This confidence interval works only if point estimate is not far away from Otherwise the limits could be outside the range (0,1). Newcombe (2006, Statistics in Medicine, 25(4): ) was able to write two papers on the topic.
20 data a; input group response cards; ; proc freq; tables group* response/ measures cl; test measures; run;
21 Somers D C R Somers D C R ASE \% Lower Conf Limit \% Upper Conf Limit Test of H0: Somers D C R = 0 ASE under H Z One-sided Pr > Z Two-sided Pr > Z Somers D R C Somers D R C ASE \% Lower Conf Limit \% Upper Conf Limit Test of H0: Somers D R C = 0 ASE under H Z One-sided Pr > Z Two-sided Pr > Z Somers d: C R, response given row.
22 Effect: x ȳ versus x ȳ 2 s2 s 2 (Cohen s effect size) Let x i and y j be normal observations for two independent groups, respectively. Pr(X <Y) is given by Pr(X x ȳ <Y)=Φ( ) 2 s 2 where Φ is the Standard Normal Distribution, e.g. Φ(0) = 0.5, Φ(0.3) = Pr(X <Y) denote the probability of a randomly chosen observation from one group is less than a randomly chosen observation from the other group. Cohen (1977 Statistical Power Analysis for the Behavioral Sciences. San Diego, CA: Academic Press, Section 2.2.1) attempted to provide an intuitively compelling and meaningful interpretation for the effect size by using percent nonoverlap index which he denoted as U =Φ( x ȳ 3 ). s 2 What U 3 really represents is the proportion of individual scores in one group that are less than the average of scores in the other group.
23 Area (A) under the receiver operating characteristic (ROC) curve The parameter Pr(X <Y) we discussed here is actually the area under the receiver operating characteristic (ROC) as shown by Bamber (1975, JMath Psychol 12: ). ROC plots were developed in the 1950s for evaluating radar signal detection. Hanley and McNeil (1982 Radiology 143: 29 36) is a classic. Such plot is obtained by calculating the sensitivity and specificity for every distinct observed data value and plotting sensitivity against 1-specificity. The area under the ROC curve is usually regarded as a global measure of diagnostic accuracy. A one-page article by Altman and Bland (1994 BMJ 309: 188) may be a good starting point in this field.
24 Test value D + D Pr(T + D + ) Pr(T D ) < 1=(T + ) 50/50 0/ =(T ), > 1=(T + ) 48/50 28/ ,2=(T ), > 2=(T + ) 44/50 42/ < 4=(T ), 4, 5=(T + ) 34/50 47/ < 5=(T ), 5=(T + ) 20/50 49/ > 5=(T + ) 0/50 50/50 Total 50 50
25 True positive False positive Sensitivity=Pr(T + D + ) 1-specificity=1 Pr(T D )
26
27
28 SAS proc freq to calculate ÂUC =0.91 and its standard error options nocenter ls=64; data roc; input disease out count; cards; ; proc freq; tables disease*out/norow nocol nopercent measures cl; weight count; run;
29 test results D D Diagnostic accuracy means given disease status, what is the probability of a test results. In this case, it is Somers D C R
30 Statistics for Table of disease by out Statistic Value ASE Gamma Kendall s Tau-b Stuart s Tau-c Somers D C R Somers D R C Pearson Correlation Spearman Correlation Lambda Asymmetric C R Lambda Asymmetric R C Lambda Symmetric Uncertainty Coefficient C R Uncertainty Coefficient R C Uncertainty Coefficient Symmetric Sample Size = 100
31 In diagnostic research, the area under the ROC curve is close to 1, the simple CI method may produce upper limit that is greater than 1. To avoid this, one may take a logit transformation logit(a) =ln The 95% CI for logit(a) is given by (l, u) = logit(â) ± Z ŝ.e.(â) Â(1 Â) A 1 A CI for A is then e l 1+e l, e u 1+e u
32 Ex: Â =0.91 and ŝ.e.(â) = (l, u) =log ± (1.91) =( , ) e e , e =(0.83, 0.95) 1+e
33 As an example for using continuous data as diagnostic tool, consider data presented by Altman and Bland (1994 BMJ 309: 188) Values of an index of mixed epidermal cell lymphocyte reactions in bone-marrow transplant recipients who did or did not develop graft-versus-host disease. Without GVHD: With GvHD:
34 data a; do i = 1 to 20; group=1; input output; end; do i =1 to 17; group =2; input output; end; cards; ; proc freq ; tables group*response / measures CL; run;
35 Statistics for Table of group by response 95\% Statistic Value ASE Confidence Limits Gamma Kendall s Tau-b Stuart s Tau-c Somers D C R Somers D R C Pearson Correlation Spearman Correlation Lambda Asymmetric C R Lambda Asymmetric R C Lambda Symmetric Uncertainty Coefficient C R Uncertainty Coefficient R C Uncertainty Coefficient Symmetric Sample Size = 37
36 The AUC is estimated as with 95% Interval given by ( , ), i.e., (.6563,.9349) Criterion for interpretation of area under ROC curve AUC Interpretation 0.50 to 0.75 fair 0.75 to 0.92 good 0.92 to 0.97 very good 0.97 to 1.00 excellent
37 Non-parametric for k>2 independent samples (p. 558) Non-parametric ANOVA, Kruskal-wallis test Assume there are k populations to be compared and that a sample of n j observations is available from pop j, j =1, 2,,k; The null hypothesis is that all populations have the same prob distribution; All obs ranked without regard to group membership and then the sums of ranks of the observations in each group are calculated. Denote these rank sums as R 1,R 2,,R k ; The degree to which the R j s differ is given by KW = 12 N(N +1) k j=1 R 2 j n j 3(N +1) where N is total sample size. Under H 0, KW distributed as χ 2 k 1.
38 Ex (Int J Cancer 1980) Number of Glucocorticoid Receptor (GR) sites per Leukocyte Cell (N)ormal: 3500, 3500, 3500, 4000,4000,4000,4300,4500,4500,4900,5200,6000,6750,8000 (H)airy-cell leukemia; 5710, 6110,8060,880,11400; (C)hronic Lymphatic; 2390, 3330, 3580, 3880, 4280, 5120; Chronic (M)yelocytic: 6320, 6860, 11400, (A)cute: 3230, 3880, 7640, 7890, 8280, 16200, 18250, 29900
39 data leukaemia; input group$ ngrs; cards; N 3500 N 3500 N A A A ; proc boxplot; plot ngrs*group; run;
40
41 proc npar1way data=leukaemia wilcoxon; class group; var ngrs; run;
42 The NPAR1WAY Procedure Wilcoxon Scores (Rank Sums) for Variable ngrs Classified by Variable group Sum of Expected Std Dev Mean group N Scores Under H0 Under H0 Score N H C M A Average scores were used for ties. Kruskal-Wallis Test Chi-Square DF 4 Pr > Chi-Square
43 Spearman (Rank) correlation (p. 560) x cigar y exc = R x R y r S = COV (R x,r y ) var(rx )var(r y ) = = 0.453
44 CI for ρ S : CI for.5ln 1+ρ S 1 ρ S : (l.u) =.5ln 1+( 0.453) 1 (.453) ± 1.96/ 12 3 CI for ρ S : e 2l 1 e 2l +1 = e2 ( ) 1 e 2 ( ) +1 = e 2u 1 e 2u +1 = e2 ( ) 1 e 2 ( ) +1 =0.163
45 data spearman; input cigar exc cards; ; proc corr SPEARMAN FISHER; run;
46 The SAS System 12:19 Wednesday, November 22, The CORR Procedure 2 Variables: cigar exc Simple Statistics Variable N Mean Std Dev Median Minimum Maximum cigar exc Spearman Correlation Coefficients, N = 12 Prob > r under H0: Rho=0 cigar exc cigar exc Spearman Correlation Statistics (Fisher s z Transformation) With Sample Bias Correlation Variable Variable N Correlation Fisher s z Adjustment Estimate cigar exc Spearman Correlation Statistics (Fisher s z Transformation) With p Value for Variable Variable 95\% Confidence Limits H0:Rho=0 cigar exc
47 data spearman; input cigar exc cards; ; proc corr SPEARMAN FISHER (BIASADJ=no); run;
48 The CORR Procedure 2 Variables: cigar exc Simple Statistics Variable N Mean Std Dev Median Minimum Maximum cigar exc Spearman Correlation Coefficients, N = 12 Prob > r under H0: Rho=0 cigar exc cigar exc Spearman Correlation Statistics (Fisher s z Transformation) With Sample p Value for Variable Variable N Correlation Fisher s z 95\% Confidence Limits H0:Rho=0 cigar exc
49 Sample size (Noether GE. Sample size determination for some common nonparametric statistics. J Am Stat Assoc 1987;82:6457). No reference list. Wilcon-Mann-Whitney test for 2-independent samples n = (Z 1 α/2 + Z 1 β ) 2 6(p 0.50) 2 where n is size of each group and p =Pr(X<Y). 1st paragraph of Statistical Analysis section: Estimates of sample size were based on the number of new enhancing lesions observed during the first 12 weeks after the first infusion in a previous clinical trial of natalizumab. Using methods based on the Wilcoxon-Mann-Whitney statistic (Noether, 1987) appropriate for a two-group comparison at a two-sided level of significance of 5 percent, we calculated that approximately 73 patients were needed in each group for the study to have 80 percent power. (NEJM 348 (1): JAN ) p = Z 1 α/2 + Z 1 β 6 n =
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