Chapter 8. Comparing Two Groups

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Ralf Bryan
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1 Chapter 8 Comparing Two Groups 1

2 Tests comparing two groups Two independent samples Two-sample t-test(normal populations) Wilcoxon rank-sum test (non-parametric) Two related samples Paired t-test (normal population) Sign Test or Wilcoxon signed rank test (non-parametric) 2

test (non-parametric) Two related samples Paired t-test

3 Two independent samples Subjects are randomly assigned to a control or treatment group (where a drug is administered). There response times to a stimulus are then measured. 3

4 Data Response time in millisecond Control Treatment

5 Two-sample t-test: SAS Test H 0 : µ t µ c = 0 against H 0 : µ t µ c 0 / Example 8.1 Two sample test / data ex8 1; infile F:\ST2137\lecdata\ex8 1.txt firstobs=2; input group $ time; proc ttest data=ex8 1; title t-test Example ; class group; var time; run; / proc ttest uses a class statement to identify the two groups of subjects. / 5

txt firstobs=2; input group $ time; proc ttest data=ex8 1; title t-test Example ;

6 Two-sample t-test: SAS output 6

7 Two-sample t-test: SAS output 7

8 Two-sample t-test: R >ex8.1=read.table( F:/ST2137/lecdata/ex8 1.txt,header=T) >attach(ex8.1) >cont=time[group== c ] >treat=time[group== t ] >var.test(cont,treat)#test if the variables are equal F test to compare two variances data: cont and treat F= , num df=4, denom df=4, p-value= alternative hypothesis: true ratio of variances is not equal to 1 95 percent confidence interval: sample estimates: ratio of variances

test(cont,treat)#test if the variables are equal F test to compare two variances data: cont and treat F=12.

9 Two-sample t-test: R >t.test(cont,treat,mu=0,var.equal=false) Welch Two Sample T-Test data: cont and treat t= df=4.641, p-value= alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: sample estimates: mean of x mean of y

10 Two-sample t-test: SPSS Analyze Compare Means Independent Sample T tests... Move time to the Test Variables panel and group to the Grouping Variable panel. Click Define Groups... Input the values for the two groups Continue OK 10

11 Two-sample t-test: SPSS 11

12 Two-sample t-test: SPSS 12

13 Two independent samples: Nonparametric tests Assumptions of two-sample t-test not met The data are not normally distributed and the sample size is small 13

14 Two independent samples: Nonparametric tests E.g. The following numbers in a psychology experiment that measured the response to a stimulus It may be due to a threshold effect. The response is either 0 (the stimulus is not detected), or, once the stimulus is detected, the average response is about 6. 14

stimulus 0 6 0 5 7 6 9 4 8 0 7 0 5 6 6 0 0 It may be due to a threshold effect.

15 Two independent samples: Nonparametric tests The data values may only represent ordered categories. E.g. Scales such as 1=very mild, 2=mild, 3=moderate, 4=strong, 5=severe reflect the strength of a response We cannot say that a score of 4(strong) is worth twice the score of 2 (mild). We need a nonparametric test to analyze differences in central tendencies for ordinal data. For very small samples, nonparametric tests are often more appropriate since assumptions concerning distributions are difficult to determine. 15

Scales such as 1=very mild, 2=mild, 3=moderate, 4=strong, 5=severe reflect the strength of a response We cannot say that a score

16 Wilcoxon rank sum test (Mann-Whitney U-test) Consider the following experiment Group A: No treatment. Group B: Treated with a drug to prevent tumor formation Both groups are exposed to a chemical that encourages tumor growth The masses (in grams) of tumors in Groups A and B are A: B: Mass: Group:B B B A A B A A A Rank:

encourages tumor growth The masses (in grams) of tumors in Groups A and B are A: 3.1 2.2 1.7 2.7 2.5 B: 0.

17 Wilcoxon rank sum test (Mann-Whitney U-test) Sum ranks of Group A= =33 Sum ranks of Group B= =12 If there were smaller tumors in Group B, we would expect the B s to be at the lower rank ordering and therefore have a smaller sum of ranks then the A s. 17

5+3+6=12 If there were smaller tumors in Group B, we would expect

18 Wilcoxon rank sum test: SAS data ex8 2; infile F:\ST2137\lecdata\ex8 2.txt firstobs=2; input group $ mass; proc npar1way data=ex8 2 wilcoxon; title Nonparametric Test to Compare Tumor Masses ; class group; var mass; exact wilcoxon; run; 18

19 Wilcoxon rank sum test: SAS 19

20 Wilcoxon rank sum test: SAS 20

21 Wilcoxon rank sum test: R >ex8.2=read.table( F:/ST2137/lecdata/ex8 2.txt,header=T) >attach(ex8.2) >gp.a=mass[group== A ] >gp.b=mass[group== B ] > >wilcox.test(gp.a,gp.b) Wilcoxon rank sum test with continuity correction data: gp.a,gp.b W=18, p=value= alternative hypothesis: true mu is not equal to 0 Warning message: cannot compute exact p-value with ties in:wilcox.test.default(gp.a,gp.b) 21

22 Wilcoxon rank sum test: SPSS We create a numeric variable Groupno to represent the groups Transform Recode into different variable... Move group to String variable panel Complete Output Variable window Change Click Old and New values and input the values Continue OK 22

23 Wilcoxon rank sum test: SPSS 23

24 Wilcoxon rank sum test: SPSS 24

25 Wilcoxon rank sum test: SPSS Analyze Nonparametric Tests Legacy Dialogs 2 Independent samples... Move mass to the Test Variable List and Groupno to the Grouping Variable Click Define Groups... Input the values for the two groups Choose Mann-Whitney U in the Test-Type 25

26 Wilcoxon rank sum test: SPSS 26

27 Wilcoxon rank sum test: SPSS 27

28 Paired t-test (Related Samples) There are many situations where each subject receives both treatments Each subject could have been measured in the absence of drug and after receiving the drug The response time for the control and treatment groups would no longer be independent. 2-sample t-test cannot be used since the groups are no longer independent. A paired t-test can be used if the differences between before and after treatments follow a normal distribution. 28

29 Paired t-test: SAS data ex8 3; infile F:\ST2137\lecdata\ex8 3.txt firstobs=2; input subject ctime ttime; proc ttest data=ex8 3; title A Paired t-test ; paired ctime*ttime; run; 29

30 Paired t-test: SAS output 30

31 Paired t-test: R >ex8.3=read.table( F:/ST2137/lecdata/ex8 3.txt,header=T) >attach(ex8.3) >t.test(control,treatment,mu=0,paired=true) Paired t-test data:control and treatment t= ,df=5,p-value= alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: sample estimates: mean of the differences

32 Paired t-test: SPSS Analyze Compare Means Paired Sample T test... Highlight the two variables control and treatment simultaneously Move these 2 variables to Paired Variables panel OK 32

33 Paired t-test: SPSS output 33

34 2 Related Samples: Nonparametric tests We apply the one sample nonparametric tests to the difference of the paired observations. One sample nonparametric tests: Sign test Wilcoxon Signed rank test Example 8.4 Consider an experiment that each subject tries each of the two drugs The time span to pain relief is measured Subject drug A drug B

35 Sign test Consider example 8.4 The differences in time span between Drug A and Drug B are Subject Difference Sign Number of positive signs=6 and number of negative signs=2 If there was no difference in the two drugs, we expect the number of positive signs(i.e. A<B) should be more or less the same as the number of negative signs( i.e. B<A) 35

36 Wilcoxon signed rank test The differences in time span between Drug A and Drug B are Subject Difference Rank of absolute Sign Sum of positive ranks: =31 Sum of negative ranks: =5 If there was no difference in the two drugs, we would expect the sum of the ranks of positive signs should be more or less the same as the ranks of negative signs. 36

37 2 Related Samples Nonparametric tests: SAS data ex8 4; infile F:\ST2137\lecdata\ex8 4.txt firstobs=2; input subject drug A drug B; diff=drug A-drug B; proc univariate data=ex8 4; title Nonparametric Test for 2 related samples ; var diff; run; 37

38 2 Related Samples Nonparametric tests: SAS output 38

39 2 Related Samples Nonparametric tests: SAS output 39

40 2 Related Samples Nonparametric tests: R >ex8.4=read.table( F:/ST2137/lecdata/ex8 4.txt,header=T) >attach(ex8.4) >diff=drug A-drug B >ncount=sum(sign(diff[diff>0]))#get the number of positive signs >binom.test(ncount,length(diff),0.5)#binom.test(obs x,n,h 0:p=0.5) Exact binomial test data: ncount and length(diff) number of successes = 6, number of trials = 8, p-value = alternative hypothesis: true probability of success is not equal to percent confidence interval: sample estimates: probability of success

41 2 Related Samples Nonparametric tests: R >wilcox.test(diff) Wilcoxon signed rank test with continuity correction data: diff V = 31, p-value = alternative hypothesis: true location is not equal to 0 Warning message: In wilcox.test.default(diff) : cannot compute exact p-value with ties 41

42 2 Related Samples Nonparametric tests: SPSS Analyze Nonparametric Tests Legacy Dialogs 2 Related Samples... Highlight the two variables drug A and drug B simultaneously Move these 2 variables to Test Pair(s) List panel OK Check Wilcoxon and Sign OK 42

43 2 Related Samples Nonparametric tests: SAS output 43

44 2 Related Samples Nonparametric tests: SPSS output 44

45 2 Related Samples Nonparametric tests: SPSS output 45

NONPARAMETRIC STATISTICS 1. depend on assumptions about the underlying distribution of the data (or on the Central Limit Theorem)

NONPARAMETRIC STATISTICS 1. depend on assumptions about the underlying distribution of the data (or on the Central Limit Theorem) NONPARAMETRIC STATISTICS 1 PREVIOUSLY parametric statistics in estimation and hypothesis testing... construction of confidence intervals computing of p-values classical significance testing depend on assumptions