The FREQ Procedure

# Example 8: Computing Friedman's Chi-Square Statistic

Procedure features:
TABLES statement, multiple requests
TABLES statement options:
 CMH2 NOPRINT SCORES= SCOROUT

This example

• computes the first two Cochran-Mantel-Haenszel statistics

• uses rank scores to compute the Cochran-Mantel-Haenszel statistics

• suppresses the display of contingency tables for each stratum.

Friedman's test is a nonparametric test for treatment differences in a randomized complete block design. Each block of the design may be a subject or a homogeneous group of subjects. If blocks are groups of subjects, the number of subjects in each block must equal the number of treatments. Treatments are randomly assigned to subjects within each block. If there is one subject per block, then the subjects are repeatedly measured once they are under each treatment. The order of treatments is randomized for each subject.

In this setting, Friedman's test is identical to the ANOVA (row means scores) CMH statistic when the analysis uses rank scores (SCORES=RANK). The three-way table uses subject (or subject group) as the stratifying variable, treatment as the row variable, and response as the column variable. PROC FREQ handles ties by assigning midranks to tied response values. If there are multiple subjects per treatment in each block, the ANOVA CMH statistic is a generalization of Friedman's test.

 ```options nodate pageno=1 linesize=80; data hypnosis; length Emotion \$ 10; input Subject Emotion \$ SkinResponse @@; datalines; 1 fear 23.1 1 joy 22.7 1 sadness 22.5 1 calmness 22.6 2 fear 57.6 2 joy 53.2 2 sadness 53.7 2 calmness 53.1 3 fear 10.5 3 joy 9.7 3 sadness 10.8 3 calmness 8.3 4 fear 23.6 4 joy 19.6 4 sadness 21.1 4 calmness 21.6 5 fear 11.9 5 joy 13.8 5 sadness 13.7 5 calmness 13.3 6 fear 54.6 6 joy 47.1 6 sadness 39.2 6 calmness 37.0 7 fear 21.0 7 joy 13.6 7 sadness 13.7 7 calmness 14.8 8 fear 20.3 8 joy 23.6 8 sadness 16.3 8 calmness 14.8 ;```
 ```proc freq data=hypnosis; tables subject*emotion*skinresponse emotion*skinresponse /cmh2 scores=rank noprint;```
 ``` title1 'Examining the Effect of Hypnosis on Skin Potential'; run;```

 PROC FREQ computes Cochran-Mantel-Haenszel statistics across strata controlling for Subject. Because CMH statistics are based on rank scores, the Row Mean Scores Differ statistic is identical to Friedman's chi-square (Q=6.45). The p-value of .09 indicates that differences in skin potential response for different emotions are significant at the 10% level but not at the 5% level. When you do not stratify by subject, the Row Mean Scores Differ CMH statistic is identical to a Kruskal-Wallis test and is not significant (p=.904). Thus, adjusting for subject is critical to reducing the background variation due to subject differences.