Statistical Analysis for Education and Psychology Researchers

From the observed frequencies, 4.55 per cent (1/22×100) of parents whose children
attended private schools were contacted by psychological services and 95.45 per cent
(21/22×100) of these parents were contacted by voluntary organizations. It is evident
from these results that there is no need for a statistical test of any significant interaction
between method of contact and type of school, however, the investigators reported a Chi-

square value of p<0.001, and went on to conclude that there was, not
unexpectedly, an association between method of contact and type of school attended by
the child.

Comment on the Analysis

All assumptions appear to be met with the possible exception that the samples may not
have been representative of the two populations of parents. This is a good example where
it is difficult to define target populations. Sampling is always a critical aspect of a study
design if inferential statistical procedures are to be used and the authors rightly draw the
readers attention to this aspect of the study.

Worked Example

The χ^2 statistic for a contingency table is a kind of standardized measure of the overall
difference between the entire set of observed and expected cell counts. The χ^2 test
compares the observed frequency counts (we already know these) in each of the cells in
the contingency table with the expected frequency counts for each of the cells (we have
to estimate expected frequencies). The expected cell counts are estimated under the
assumption that the null hypothesis is true, that is there is no association between the row
and column variables.
The expected count for any cell in a 2×2 table is estimated by the joint probability of
the appropriate row i and column j in the 2×2 table. For example, the probability, say, of
increased understanding and being a male is equal to the probability of increased
understanding multiplied by the probability of being a male, derived from the
multiplication rule for independent events (P(U and M)= P(U)P(M)). This joint
probability is the product of the marginal probabilities for the appropriate row i and
column j in a contingency table.
These marginal probabilities are not themselves observable but can be estimated by
the row and column sample proportions, that is row proportion= (row total)/sample total,
and column proportion=(column total)/sample total.
In a 2×2 table, each categorical variable is binary and the mean (expected) count for a
binary variable B(n, π), is np for sample data, where p is the joint probability ricj for a
particular cell frequency and n is the sample total. The expected frequency count is
therefore:

Statistical analysis for education and psychology researchers 168