Statistical Analysis for Education and Psychology Researchers

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Interpretation of Computer Output

The SAS programme Runs. produced the following output:

(^) One-tailed test Two-tailed test
OBS Z p-value p=value
1 0.46526 0.3209 0.6418
The associated probability of obtaining a |Z| value (absolute Z value) of 0.46526 when the
null hypothesis is true is p=0.6418 (the p-value is doubled, 0.3209×2, for a two-tailed
test). The null hypothesis of randomness cannot be rejected at the 5 per cent level and the
researcher can therefore conclude that the sequence of classroom observations is random.
7.4 Wilcoxon Mann-Whitney Test (also called Wilcoxon’s rank sum
test)
When to Use
This is a test of the difference between two independent random samples which is used to
determine whether two samples could have reasonably come from the same population.
The Wilcoxon M-W test is sensitive to differences in the location of the central tendency
of distributions. If two distributions have similar shape and dispersion it is effectively a
test of the difference in medians between the two groups. It is often described as a
nonparametric analogue to the independent t-test but unlike the t-test it does not test
specifically for differences between means.
When assumptions of an underlying normal distribution are not satisfied, or data are
already in the form of ranks, the Wilcoxon M-W test is a useful and powerful alternative
to the independent t-test. As it is based on rank scores, in practice, the procedure can be
used with ordinal, interval and ratio levels of measurement. The test is particularly useful
when distributions are heavy tailed, that is the distribution contains many values that are
distant from the mean (see kurtosis in Chapter 3, section 3.4). The test is more powerful
than the t-test for heavy tailed distributions, for both relatively small sample sizes and in
the asymptotic limit (large sample sizes).
The two samples (groups) need not be the same size and the test has an exact sampling
distribution for the test statistic SR (sum of rank scores) which rapidly approaches the
normal distribution as sample sizes increase (i.e., when there are about twenty scores in
the larger of the two samples). Many statistical texts provide tables of critical values of SR
for different combinations of sample sizes (for the two groups being compared).
However, a normal approximation based on the standard error of the test statistic SR is
adequate for most occasions and with smaller samples,<20 in any of the groups, a
continuity correction can be applied to the calculated Z score. A few tied scores will
have little effect on the test statistic but if there are a number of tied scores, and in
Statistical analysis for education and psychology researchers 218