Chapter 8 Comparing Two Groups 1
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
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
Data Response time in millisecond Control Treatment 80 100 93 103 83 104 89 99 98 102 4
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
Two-sample t-test: SAS output 6
Two-sample t-test: SAS output 7
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=12.3953, num df=4, denom df=4, p-value=0.03177 alternative hypothesis: true ratio of variances is not equal to 1 95 percent confidence interval: 1.290573 119.051498 sample estimates: ratio of variances 12.39535 8
Two-sample t-test: R >t.test(cont,treat,mu=0,var.equal=false) Welch Two Sample T-Test data: cont and treat t=-3.8302 df=4.641, p-value=0.01410 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -21.931655-4.068345 sample estimates: mean of x mean of y 88.6 101.6 9
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
Two-sample t-test: SPSS 11
Two-sample t-test: SPSS 12
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
Two independent samples: Nonparametric tests E.g. The following numbers in a psychology experiment that measured the response to a 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. The response is either 0 (the stimulus is not detected), or, once the stimulus is detected, the average response is about 6. 14
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
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: 3.1 2.2 1.7 2.7 2.5 B: 0.0 0.0 1.0 2.3 Mass: 0.0 0.0 1.0 1.7 2.2 2.3 2.5 2.7 3.1 Group:B B B A A B A A A Rank: 1.5 1.5 3 4 5 6 7 8 9 16
Wilcoxon rank sum test (Mann-Whitney U-test) Sum ranks of Group A=4+5+7+8+9=33 Sum ranks of Group B=1.5+1.5+3+6=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
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
Wilcoxon rank sum test: SAS 19
Wilcoxon rank sum test: SAS 20
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=0.06506 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
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
Wilcoxon rank sum test: SPSS 23
Wilcoxon rank sum test: SPSS 24
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
Wilcoxon rank sum test: SPSS 26
Wilcoxon rank sum test: SPSS 27
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
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
Paired t-test: SAS output 30
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=-4.3481,df=5,p-value=0.007372 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -11.66874-2.997923 sample estimates: mean of the differences -7.333333 31
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
Paired t-test: SPSS output 33
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 1 2 3 4 5 6 7 8 drug A 20 40 30 45 19 27 32 26 drug B 18 36 32 46 15 22 29 25 34
Sign test Consider example 8.4 The differences in time span between Drug A and Drug B are Subject 1 2 3 4 5 6 7 8 Difference 2 4-2 -1 4 5 3 1 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
Wilcoxon signed rank test The differences in time span between Drug A and Drug B are Subject 1 2 3 4 5 6 7 8 Difference 2 4-2 -1 4 5 3 1 Rank of absolute 3.5 6.5 3.5 1.5 6.5 8 5 1.5 Sign + + - - + + + + Sum of positive ranks: 3.5+6.5+3.5+8+5+1.5=31 Sum of negative ranks: 3.5+1.5=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
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
2 Related Samples Nonparametric tests: SAS output 38
2 Related Samples Nonparametric tests: SAS output 39
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 = 0.2891 alternative hypothesis: true probability of success is not equal to 0.5 95 percent confidence interval: 0.3491442 0.9681460 sample estimates: probability of success 0.75 40
2 Related Samples Nonparametric tests: R >wilcox.test(diff) Wilcoxon signed rank test with continuity correction data: diff V = 31, p-value = 0.07895 alternative hypothesis: true location is not equal to 0 Warning message: In wilcox.test.default(diff) : cannot compute exact p-value with ties 41
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
2 Related Samples Nonparametric tests: SAS output 43
2 Related Samples Nonparametric tests: SPSS output 44
2 Related Samples Nonparametric tests: SPSS output 45