Statistical Analysis for Education and Psychology Researchers

Statistical Inference and Null Hypothesis

Statistical inferences are about counts or relative frequencies with respect to defined
characteristics in two populations. The number of observations that fall into a particular
category in one group are compared with the proportion of observations that fall into the
same category from the other group. Groups may refer to two independent measurements
from one population (sample) of subjects, or to two independent populations from which
two samples have been randomly selected. For example, in a study of the impact of an
MEd in-service programme, one measure of interest might be confidence following
completion of the in-service degree programme (professional confidence increased/not
increased) another measure could be gender (male/female). The one-sample χ^2 test of
independence would be used with this design to detect any association between
confidence and gender.
Groups may also refer to two independent populations, for, example in a study of
parents of children who have special educational needs two populations of parents (two
independent samples) may be investigated; parents who were contacted via voluntary
organisations, and parents who were contacted by the psychological services. With this
sampling design a two-sample χ^2 test of homogeneity of proportions would be
appropriate.
Whereas the precise form of the research hypothesis differs depending upon the
sampling design, fortunately, the null hypothesis is the same for both the χ^2 tests of
independence and homogeneity. For the one-sample χ^2 test of independence (random row
and column marginal totals) the parameters being estimated are the proportions of each of
the four outcomes (frequency distributions in each cell of the 2×2 table) in the population
from which the sample was drawn. The research hypothesis is that the row and column
variables interact, that is they are not independent, and the observed proportions in the
four cells will therefore differ depending upon the particular row and column
classification. Similarly, for the two-sample χ^2 test of homogeneity of proportions (fixed
column or row marginal totals) the parameters being estimated are the proportions in the
four outcomes in the population which the sample proportions are intended to estimate.
The research hypothesis is that the distribution of proportions (for one categorical
variable) is different in the two populations (the other categorical variable with fixed
marginal totals). A more general way of stating this research hypothesis is to say that
there is a relationship between the two categorical variables. The null hypothesis for both
one-sample and two-sample χ^2 tests is that there is no interaction (relationship) between
column and row variables. If the null hypothesis were true, the four cell proportions
would be equal, and there would be no significant differences between observed cell
frequencies and expected cell frequencies (under the null hypothesis of no interaction).
The χ^2 distribution is completely determined by a single parameter, the degrees of
freedom (df). Whenever we evaluate the χ^2 statistic we need to consider the appropriate
df. Degrees of freedom are determined by the number of rows and columns in a
contingency table specifically, df=(number of rows−1)×(number of columns−1) and is
hence always 1 in a 2×2 table. The χ^2 test and associated df provides a probability for the
difference between observed and expected frequencies. When the observed and expected
frequencies are identical the χ^2 statistic will be zero. Any deviation from this will always
be positive, the larger the χ^2 value, the greater the statistical significance (departure from
the null hypothesis).

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