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
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
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
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.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.