Examples

Example 52.1. 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 52.1.1 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 52.1.1: Wilcoxon Two-Sample Test

  The NPAR1WAY Procedure   Wilcoxon Scores (Rank Sums) for Variable Response   Classified by Variable Treatment   Sum of      Expected       Std Dev          Mean   Treatment       N        Scores      Under H0      Under H0         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 52.1.2 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 52.1.2: Median Two-Sample Test

  Median Scores (Number of Points Above Median) for Variable Response   Classified by Variable Treatment   Sum of      Expected       Std Dev          Mean   Treatment       N        Scores      Under H0      Under H0         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 52.1.3 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.

Output 52.1.3: Empirical Distribution Function Statistics

  Kolmogorov-Smirnov Test for Variable Response   Classified by Variable Treatment   EDF at    Deviation from Mean   Treatment       N       Maximum        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   Summed Deviation   Treatment          N           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   Deviation   Treatment        N      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  

 

Example 52.2. The Exact Wilcoxon Two-Sample Test

Researchers conducted an experiment to compare the effects of two stimulants. Thirteen randomly selected subjects received the first stimulant and six randomly selected subjects received the second stimulant. The reaction times (in minutes) were measured while the subjects were under the influence of the stimulants.

The following SAS statements create the data set React , which contains the observed reaction times for each stimulant. The variable Stim represents Stimulant 1 or 2. The variable Time contains the reaction times observed for subjects under the stimulant.

  data React;   input Stim Time @@;   datalines;   1 1.94   1 1.94   1 2.92   1 2.92   1 2.92   1 2.92   1 3.27   1 3.27   1 3.27   1 3.27   1 3.70   1 3.70   1 3.74   2 3.27   2 3.27   2 3.27   2 3.70   2 3.70   2 3.74   ;  

PROC NPAR1WAY tests the null hypothesis that there is no difference between the effects of the two stimulants against an alternative hypothesis that stimulant 1 has smaller reaction times than stimulant 2. The WILCOXON option specifies that Wilcoxon scores are to be used. The CLASS statement specifies that the variable Stim determines the classes. The VAR statement identifies Time as the response variable. The EXACT option requests the exact p -values. Since the sample size is small, the normal approximation may not be completely accurate, and it is appropriate to compute the exact test.

  proc npar1way wilcoxon data=React;   class Stim;   var Time;   exact;   run;  

Output 52.2.1 displays the results of the Wilcoxon two-sample test. The Wilcoxon statistic equals 79.50. Since this value is greater than 60.0, the expected value under the null hypothesis, PROC NPAR1WAY displays the right-sided p -values. The one-sided exact p -value equals 0.0527, which is not significant at the 0.05 level. The normal approximation yields a one-sided p -value of 0.0421, which is significant at the 0.05 level. Thus, the normal approximation may result in rejection of the null hypothesis, while the exact test may result in acceptance.

Output 52.2.1: Wilcoxon Two-Sample Test

  The NPAR1WAY Procedure   Wilcoxon Scores (Rank Sums) for Variable Time   Classified by Variable Stim   Sum of      Expected       Std Dev          Mean   Stim       N        Scores      Under H0      Under H0         Score   --------------------------------------------------------------------   1         13        110.50         130.0     11.004784         8.500   2          6         79.50          60.0     11.004784        13.250   Average scores were used for ties.   Wilcoxon Two-Sample Test   Statistic (S)               79.5000   Normal Approximation   Z                            1.7265   One-Sided Pr >  Z            0.0421   Two-Sided Pr > Z           0.0843   t Approximation   One-Sided Pr >  Z            0.0507   Two-Sided Pr > Z           0.1014   Exact Test   One-Sided Pr >=  S            0.0527   Two-Sided Pr >= S - Mean   0.1054   Z includes a continuity correction of 0.5.   Kruskal-Wallis Test   Chi-Square         3.1398   DF                      1   Pr > Chi-Square    0.0764  

 

Example 52.3. The Exact Savage Multisample Test

A researcher conducting a laboratory experiment randomly assigned 15 mice to receive one of three drugs. The survival time (in days) was then recorded.

The following SAS statements create the data set Mice , which contains the observed survival times for all the mice. The variable Trt denotes the treatment received by a mouse. The variable Days contains the number of days the mouse survived.

  data Mice;   input Trt $ Days @@;   datalines;   1 1 1 1 1 3 1 3 1 4   2 3 2 4 2 4 2 4 2 15   3 4 3 4 3 10 3 10 3 26   ;  

PROC NPAR1WAY tests the null hypothesis that there is no difference in the survival times among the three drugs against an alternative hypothesis of difference among the drugs. The SAVAGE option specifies that Savage scores are to be used. The variable Trt is the CLASS variable, and the VAR statement specifies that the variable Days is the response variable. The EXACT statement requests the exact test.

  proc npar1way savage data=Mice;   class Trt;   var Days;   exact;   run;  

Output 52.3.1 shows the results of the Savage test. The exact p -value is 0.0445, which is significant at the 0.05 level. However, the p -value based on the chi-square approximation is 0.0638, which results in nonrejection of the null hypothesis at the 0.05 level.

Output 52.3.1: Savage Multisample Test

  The NPAR1WAY Procedure   Savage Scores (Exponential) for Variable Days   Classified by Variable Trt   Sum of      Expected       Std Dev          Mean   Trt       N        Scores      Under H0      Under H0         Score   -------------------------------------------------------------------   1         5   3.367980           0.0      1.634555   0.673596   2         5      0.095618           0.0      1.634555      0.019124   3         5      3.272362           0.0      1.634555      0.654472   Average scores were used for ties.   Savage One-Way Analysis   Chi-Square                     5.5047   DF                                  2   Asymptotic Pr >  Chi-Square    0.0638   Exact      Pr >= Chi-Square    0.0445