Statistical Analysis for Education and Psychology Researchers

One-sample χ^2 test of independence with random row and column

marginal totals

In this design a random sample is drawn from a single population of subjects but with
two measures for each subject, that is the row and column binary variables. The total
sample size, n, is fixed but the frequencies in both row and column marginal totals are
random and not known or fixed in advance. The random marginal frequencies depend
upon (contingent upon, hence the term contingency table) how each subject is classified
on both binary variables. That is, each subject would be allocated to one of the four cells
in the 2×2 table.
For example, a researcher investigating the carers’ role in supporting relatives with
dementia may be specifically interested in the relationship between length of experience
as carers (column variable) and dominant feelings about their roles as carers (row
variable). A single random sample of 100 carers was selected from a population of carers
who have relatives suffering from dementia. The carers were asked two questions: how
long have they been looking after their relatives? (responses classified into greater than or
equal to five years or less than five years), and what were their dominant feelings about
their carer role? (responses were classified into predominantly anger or guilt). The
research hypothesis was that feelings about the carer role was related to the length of
experience as a carer. The null hypothesis is that the row and column variables are
independent, that is the expected proportions (counts) in each cell of the contingency
table would be equal and would not differ from the observed counts. In more general
terms this would be stated as there is no relationship between time as a carer and
dominant feelings about the carers’ role.

Two-sample χ^2 test of homogeneity of proportions with fixed column (or

row) marginal totals

This design is used to compare the distribution of proportions in two independent
populations. In a 2×2 contingency table each variable is treated as binary. For example,
the column variable in a 2×2 table may represent two independent populations, males and
females, and the row variable (response variable) may represent examination
performance classified as pass or fail. The researcher may want to investigate whether the
proportion of candidates passing is related to gender. For example, an independent
random sample of fifty males, and a separate random sample of fifty females would be
selected. The column total for males and females in this example is fixed by the
researcher. Each male and female would be classified into a pass or fail category, the row
marginal totals are random (not fixed by the researcher) and subject to sampling error. If
the proportion of candidates who pass is represented by P, then the proportion of fails
would be 1−P (the variable is binary). Usually a count of the number of passes is given as
a percentage, and a comparison is made between the percentage of males and females
who pass. The null hypothesis would be the population proportion (or percentage) of
males and females who pass is equal, or put another way there is no difference between
males and females in the percentage who pass the exam. A more general form of this null
hypothesis is that there is no relationship between gender and examination performance.

Inferences involving binomial and nominal count data 163