Statistical Analysis for Education and Psychology Researchers

column variables and the distribution of frequencies is described by the multinomial
distribution.
The third sampling design is stratified random sampling. If two groups of 5 male
subjects are sampled at random from each of two populations, male subjects with upper
limb injuries that occurred when they were ≤6-years-old and a similar group but with
injuries occurring when subjects were >6-years-old, and these subjects were asked, ‘Are
you right- or left-handed?’ The following contingency table may be obtained:
Random sample of five subjects with upper limb injuries selected from two populations
≤6-years-old >6-years-old
Left-handed 1 0
Right-handed 4 5
(Fixed column totals) 5 5

In this table, the column marginal totals are fixed, the row marginals are random and the
underlying frequency distribution is described by the product multinomial distribution.
The null hypothesis in this design is that the proportions of left-handers (or we could say
right-handers) is the same in both age cohort populations (≤6 years, and >6 years).
Inferences are made with respect to the target populations and the notion of independence
does not make sense in this sampling design.

What should be done if we have a small overall sample or small expected

cell values in a 2×2 table, or we want to consider a question about

statistical independence between row and column variables but have used

the wrong sampling design?

We can still use Fisher’s exact test, but any inferences are made conditional on the
observed marginal totals.

Example from the Literature

In a study of handedness plasticity, part of which was designed to investigate handedness
(left or right) and upper limb injury, Dellatolas et al. (1993) surveyed 9591 men aged 17–
27 who were conscripted into the French army. Of this sample 577 reported that they had
previously sustained an upper limb injury, 25 were ≤ 6-years-old when the injury
occurred and 552 sustained their injuries when they were >6-years-old. One question
addressed by the investigators was whether there was any difference between two age
groups in the proportions of left-handed males who had sustained upper limb injuries.
Data was set out in a 2×2 contingency table as follows:
577 subjects with upper limb injuries
Age when injury occurred
Handedness: ≤6-years-old >6-years-old (Random row totals)
Left-handed 7 59 66
Right-handed 18 493 511
(Fixed column totals) 25 552 577

Statistical analysis for education and psychology researchers 180