Statistical Analysis for Education and Psychology Researchers

If the two age groups are treated as separate populations, and the achieved samples are
treated as random samples from these populations, then the appropriate underlying
distribution is product multinomial and inferences are made with respect to the
populations, conditional on the observed marginal totals. A two-tailed test is appropriate
given the exploratory nature of the study. The investigators reported that among the 577
injured men, the proportion of left-handers was significantly higher when the age the
accident occurred was ≤6 years (28 per cent) [7/25× 100] than when it occurred later
(10.7 per cent) [59/552×100], Fisher’s exact test: p<0.02.

Worked Example

Assume the following data table was obtained when a random sample of males with
upper limb injuries was selected.
Age when injury occurred
Handedness: ≤6-years-old >6-years-old (Random row totals)
Left-handed 3(A) 1(B) 4(A+B)
Right-handed 3(C) 3(D) 6(C+D)
(Random column totals) 6(A+C) 4(B+D) 10 (TOTAL N FIXED)

Figure 6.4: Ten subjects with upper

limb injuries selected at random

Cells in the table are labelled A to D and marginal totals are (A+B), (C+D), (A +C) and
(B+D).
In this example the overall total, N, is fixed and the row and column marginal totals
are random. Whereas the sampling design suggests that the multinomial distribution is
appropriate, if we make the assumption that inferences are conditional on the observed
marginal totals, then the hypergeometric distribution can be used. The exact probability,
under the null hypothesis (assuming random assignment), of observing this frequency
distribution in the four cells, or a more extreme distribution, can be evaluated using the
following equation:

Probability

of observed

frequency

distribution

for Fisher’s

test—6.5

where N is the overall total, 10, and A to D are the cell frequencies and marginal totals as
illustrated in Figure 6.4.
Using data presented in Figure 6.4, the null hypothesis is that the proportions of left-
handed males with upper limb injuries is the same in both age cohorts (≤6 years, and >6
years), that is H 0 : P 1 =P 2 where P 1 is the probability that a male from the age cohort ≤ 6
years will be left-handed (cell A) and P 2 is the probability that a male from the age cohort

6 years will be left-handed (cell B). There are three possible alternative hypotheses: H 1 :

Inferences involving binomial and nominal count data 181