Example 47.1: Two-Sample Location Tests and EDF Statistics

The NPAR1WAY Procedure

Example 47.1: Two-Sample Location Tests and EDF Statistics

Fifty-nine female patients with rheumatoid arthritis who participated in a clinical trial were assigned to two groups, active and placebo. The response status (excellent=5, good=4, moderate=3, fair=2, poor=1) of each patient was recorded.

The following SAS statements create the data set Arthritis, which contains the observed status values for all the patients. The variable Treatment denotes the treatment received by a patient, and the variable Response contains the response status of the patient. The variable Freq contains the frequency of the observation, which is the number of patients with the Treatment and Response combination.

   data Arthritis;
      input Treatment $ Response Freq @@;
      datalines;
   Active 5 5 Active 4 11 Active 3 5 Active 2 1 Active 1 5
   Placebo 5 2 Placebo 4 4 Placebo 3 7 Placebo 2 7 Placebo 1 12
   ;

PROC NPAR1WAY tests the null hypothesis that there is no difference in the patient response status against an alternative hypothesis that the patient response status differs in the two treatment groups. The WILCOXON option requests the Wilcoxon test for difference in location, and the MEDIAN option requests the median test for difference in location. The EDF option requests empirical distribution function statistics. The variable Treatment is the CLASS variable, and the VAR statement specifies that the variable Response is the response variable.

   proc npar1way wilcoxon median edf data=Arthritis;
      class Treatment;
      var Response;
      freq Freq;
   run;

Output 47.1.1: Wilcoxon Two-Sample Test

The NPAR1WAY Procedure

Wilcoxon Scores (Rank Sums) for Variable Response Classified by Variable Treatment
Treatment	N	Sum of Scores	Expected Under H0	Std Dev Under H0	Mean Score
Active	27	999.0	810.0	63.972744	37.000000
Placebo	32	771.0	960.0	63.972744	24.093750
Average scores were used for ties.

Wilcoxon Two-Sample Test
Statistic	999.0000

Normal Approximation
Z	2.9466
One-Sided Pr > Z	0.0016
Two-Sided Pr > \|Z\|	0.0032

t Approximation
One-Sided Pr > Z	0.0023
Two-Sided Pr > \|Z\|	0.0046
Z includes a continuity correction of 0.5.

Kruskal-Wallis Test
Chi-Square	8.7284
DF	1
Pr > Chi-Square	0.0031

Output 47.1.1 shows the results of the Wilcoxon analysis. The Wilcoxon two-sample test statistic equals 999.0, which is the sum of the Wilcoxon scores for the smaller sample (Active). This sum is greater than 810.0, its expected value under the null hypothesis of no difference between the two samples Active and Placebo. The one-sided p-value is 0.0016, which shows that the patient response for the Active treatment is significantly more than for the Placebo group.

Output 47.1.2: Median Two-Sample Test

The NPAR1WAY Procedure

Median Scores (Number of Points Above Median) for Variable Response Classified by Variable Treatment
Treatment	N	Sum of Scores	Expected Under H0	Std Dev Under H0	Mean Score
Active	27	18.916667	13.271186	1.728195	0.700617
Placebo	32	10.083333	15.728814	1.728195	0.315104
Average scores were used for ties.

Median Two-Sample Test
Statistic	18.9167
Z	3.2667
One-Sided Pr > Z	0.0005
Two-Sided Pr > \|Z\|	0.0011

Median One-Way Analysis
Chi-Square	10.6713
DF	1
Pr > Chi-Square	0.0011

Output 47.1.2 shows the results of the median two-sample test. The statistic equals 18.9167, with a one-sided p-value of 0.0005. This shows that the response for the Active treatment is significantly more than for the Placebo group.

Output 47.1.3: Empirical Distribution Function Statistics

The NPAR1WAY Procedure

Kolmogorov-Smirnov Test for Variable Response Classified by Variable Treatment
Treatment	N	EDF at Maximum	Deviation from Mean at Maximum
Active	27	0.407407	-1.141653
Placebo	32	0.812500	1.048675
Total	59	0.627119
Maximum Deviation Occurred at Observation 3
Value of Response at Maximum = 3.0

Kolmogorov-Smirnov Two-Sample Test (Asymptotic)
KS	0.201818	D	0.405093
KSa	1.550191	Pr > KSa	0.0164

Cramer-von Mises Test for Variable Response Classified by Variable Treatment
Treatment	N	Summed Deviation from Mean
Active	27	0.526596
Placebo	32	0.444316

Cramer-von Mises Statistics (Asymptotic)
CM	0.016456	CMa	0.970912

Kuiper Test for Variable Response Classified by Variable Treatment
Treatment	N	Deviation from Mean
Active	27	0.000000
Placebo	32	0.405093

Kuiper Two-Sample Test (Asymptotic)
K	0.405093	Ka	1.550191	Pr > Ka	0.1409

Output 47.1.3 shows empirical distribution function statistics comparing these two samples. The asymptotic p-value for the Kolmogorov-Smirnov test is 0.0164. This indicates rejection of the null hypothesis that the distributions are identical for the two groups.

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