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Chapter 8
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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)
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Two independent samples
• Subjects are randomly assigned to a control or treatment group (where a drug is administered).
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Data
Response time in millisecond Control Treatment 80 100 93 103 83 104 89 99 98 102
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Two-sample t-test: SAS
Test H0 : µt − µc = 0 against H0 : µt − µc 6= 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;
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>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:
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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
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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
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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
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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.
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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.
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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
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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.
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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;
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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:
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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
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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
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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.
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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;
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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:
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Paired t-test: SPSS
• “Analyze”→ “Compare Means”→ “Paired Sample T test...”
• Highlight the two variables “control” and “treatment” simultaneously
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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
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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)
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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.
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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;
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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
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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:
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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”
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