Statistical Analysis for Education and Psychology Researchers

The investigators reported a χ^2 value of 14.00, df=1. The difference between the reported
value in the original paper and the calculated value in the worked example is attributable
to the investigators’ use of a continuity adjusted χ^2 (which was not reported in the
original paper). This adjustment was unnecessary, the sample size was>20 and none of
the expected cell frequencies were<5. The tendency for the continuity adjusted χ^2 to
provide conservative probabilities is evident here because the adjusted χ^2 is less than the
unadjusted statistic, χ 12 adj 14<χ 12 15.64 (χ 12 refers to χ^2 with 1 df). In this particular
instance, the different values of the adjusted and unadjusted χ^2 statistics do not affect the
interpretation.

Interpretation

To evaluate the statistical significance of the estimated χ^2 statistic you need to calculate
the appropriate degrees of freedom for the contingency table, here df=1, and refer to a
table of critical χ^2 values (see Table 2, Appendix A4). An alpha level is first selected,
usually p≤0.05 or p≤0.001, although depending upon the statistical table there may be
other alpha values to choose from. If we select alpha as p<0.001, the last column in Table
2 of Appendix A4, we then move down this column until we intersect with the
appropriate row for degrees of freedom.
In this example the critical χ^2 value is 10.828, rounded to 10.83, the intersection of last
column and first row in Table 1. Since the calculated value is greater than the value from
the statistical tables (what would be expected under the null hypothesis of no interaction
between row and column variables) then we can reject the null hypothesis and conclude
that the two variables, increased understanding of educational issues, and gender are
related. Generally, it is good practice to inspect and report on the differences in observed
and expected frequencies, that is to consider how the null hypothesis is untrue.
Descriptive percentages are helpful in doing this. For example, the results of the analysis
could be presented as follows:

(^) Understanding of Educational Issues
(^) An increase reported No increase reported
(^) Observed (^) Expected Observed (^) Expected
Gender:
Male 21 12 27 36
Female 10 19 69 60
Total 31 96
Notice as a check, observed and expected totals are equal. Proportionately more males
43.8 per cent (21/48×100) than females 12.7 per cent (10/79×100) reported an increase in
understanding of educational issues following the in-service MEd course.
Computer Analysis
Data from the example of the χ^2 test of independence (see Figure 6.1) is illustrated. When
data is in the form of frequency counts a simple way to enter and analyze this data in SAS
is to use the frequencies procedure PROC FREQ with the weight statement. For example,
if we want to analyze data from the study about teachers’ understanding of educational
Statistical analysis for education and psychology researchers 170