Statistical Analysis for Education and Psychology Researchers

Interpretation of Computer Output

SAS automatically performs a two-way test if a variable representing the difference
between the two repeated observations is analyzed as the response variable, here
difference=best − poor. In Figure 7.6 the row labelled Sgn Rank gives the value of the
expected signed rank statistic, here 16.5. This Wilcoxon Signed Rank statistic is
equivalent to the observed−expected value of T+ which is the numerator of equation 7.3.
The associated probability, Pr≥|S|, is also output in this example p=0.0234. Since this p-
value is less than 0.05, for a two-tailed test, we can conclude that there is a difference
between teachers sharing behaviour in best and poorest relationship situations. If the p-
value associated with the Wilcoxon Signed Rank test had been larger than 0.05, we could
not reject the null hypothesis and would have concluded that the median of the population
differences was not significantly different from zero.

Large sample procedure approximate test

With an effective sample size of 8 you would not normally use the large sample
approximation but it is used here to illustrate the procedure.
Similar to the Wilcoxon M-W test procedure with large sample approximation, Z is
given by the general formulae, (observed-expected)/standard error where observed test
statistic is T+, (sum of the positive ranked differences), the expected value is=(n(n+1))/4,

and the standard error is
Here Z=(34.5−18)/7.141=2.311. The associated p-value is 0.021 (see Table of Z scores
in Appendix A4). This gives substantively the same answer as the small sample test.
SAS does not automatically output the observed test statistic T+ or its standard error
when the Wilcoxon signed ranks test is performed with PROC UNIVARIATE. The SAS
programme Wilcoxsr, see Figure 10, Appendix A3, outputs the rank scores, the sum, T+,
standard error, expected T and a Z score with associated probability (2-tailed). Output
from this SAS programme using data from the worked example is shown:
Wilcoxon signed ranks test
OBS first value second
value

absolute

difference

difference ranked

differences

1 4 2 2 2 5.0

2 3 1 2 2 5.0

3 5 3 2 2 5.0

4 3 1 2 2 5.0

5 5 1 4 4 8.0

6 4 3 1 1 1.5

7 3 4 1 −1 1.5

8 4 2 2 2 5.0

Summary Statistics

Number of

subjects

observed value

(T)

expected

value

SE z score p-value (2-tailed

test)

Inferences involving rank data 231