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 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 ResponseClassified by Variable Treatment Treatment N Sum ofScores ExpectedUnder H0 Std DevUnder H0 MeanScore 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 correctionof 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 VariableResponseClassified by Variable Treatment Treatment N Sum ofScores ExpectedUnder H0 Std DevUnder H0 MeanScore 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 VariableResponseClassified by Variable Treatment Treatment N EDF atMaximum Deviation fromMeanat 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 forVariable ResponseClassified 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 ResponseClassified 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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