Statistical Analysis for Education and Psychology Researchers

Summary Statistics for COND by Rank

Controlling for Subject

Cochran—Mantel—Haenszel Statistics (Based on Table Scores)

Statistic Alternative Hypothesis DF Value Prob

1 Nonzero Correlation 1 1.144 0.285

2 Row Mean Scores 3 7.983 0.046

3 General Association 12 13.100 0.362

Total Sample=24

Figure 7.10: CMH statistic from

PROC FREQ based on data shown in

Table 7.7

In this example, the ANOVA CMH statistic is equivalent to Friedman’s Chi-square with
k−1, i.e., 3 degrees of freedom. The appropriate statistic and alternative hypothesis for the
repeated measures design is statistic 2 in the output, the row means (medians) differ.
Looking at the CMH test statistic this is 7.983, identical to the value in the worked
example, adjusted for ties. This Chi-square statistic is evaluated with 3 df. The associated
probability of obtaining a test statistic as large as this is given in the third column of the
output and is 0.046. SAS outputs the actual pro-bability rather than a set probability level.
We can conclude that there is a significant difference, at the 5 per cent level, among the
four experimental conditions.
Sometimes it is inconvenient to consult a Chi-square table for critical values, on these
occasions the SAS function, PROBCHI can be used instead. For example, to determine
the probability associated with a Chi-square value of 7.983 with 3 df the following SAS
code can be entered:

data a;

p=1−probchi(7.983,3);

put p=;

run;

The probability value returned in the SAS LOG is:

P=0.0463643541

NOTE: The data set WORK.A has 1 observations and 1

variables.

Pairwise Multiple-comparisons for post hoc Analysis

Similar to the Kruskal-Wallis test when an obtained H statistic is significant, an
investigator may wish to determine which of the conditions (repeated measures) differ
when the Friedman test statistic is found to be statistically significant. A post hoc
pairwise multiple-comparison procedure, adjusting for the number of pairwise
comparisons should be used. Computational details and worked examples are described
by Siegel and Castellan (1988). The SAS programme Friedx automatically performs

Inferences involving rank data 245