taken as reasonable. An analysis of non-response by gender would strengthen the
independence of observations assumption. (Close scrutiny of the published data shows
that the total of males and females does not add up to 127. The sum of females should be
79 and not 49. As a check in a Chi-square analysis the sum of the cell frequency counts
should equal the total sample size, that is no observation is double counted or missed
out.)
Two-sample χ^2 test of homogeneity of proportionsIn a study of parents of children with special educational needs by Riddell, Brown and
Duffield (1994), parents of twenty-two children attending private schools and the parents
of 131 children attending state schools were sampled. The investigators examined, as part
of the overall purpose of the study, whether there was any association between sampling
method of contacting parents to elicit study information (via voluntary organizations or
psychological services) and type of school attended (state or private). Data as presented
by the investigators is shown in Figure 6.2.
(^) Voluntary
Organizations
Psychological
Services
Random Row
Total
Private 21 1 22
State 62 69 131
Fixed Column
Total
83 70 153
Figure 6.2: Method of contacting
parents by type of school child
attended
The authors used a χ^2 test of homogeneity of proportions with fixed column marginal
totals, 83 parents contacted via voluntary organizations and 70 parents contacted via
psychological services. The column variable in the 2×2 table represents two independent
populations of parents. The authors suggest that the achieved samples from these two
populations are likely to be unrepresentative of parents (non-random samples). Each
parent contacted was classified on a response variable into state or private according to
the type of school his or her child attended. The row marginal totals were therefore
random and subject to sampling error.
The research question addressed by the investigators was whether there was a
statistical relationship between method of contacting parents and type of school the child
attended. This research question could be rephrased as: ‘Of the parents whose children
attend private school, how does the proportion (or per cent) of parents contacted by
voluntary organizations compare with the proportion of parents contacted by the
psychological services?’
The null hypothesis would be that the population proportions (or percentages) of
parents contacted via voluntary organizations and psychological services whose children
attend private schools are equal. A more general form of this null hypothesis is that there
is no relationship (statistical interaction) between the row and column variables—that is
method of contact and type of school attended.
Inferences involving binomial and nominal count data 167