Statistical Inference and Null HypothesisInferences made in the Fisher’s exact test are about population proportions. Stated in
general terms the null hypothesis is that the proportions in the two independent samples
are not related (statistically independent) to the dependent binary variable. In the example
of upper limb injuries and handedness the null hypothesis would be that age when the
injury occurred is independent of handedness.
Test AssumptionsData should be discrete (counts) and may be nominal or ordinal provided members of
each independent sample can be classified into one of two mutually exclusive groups.
The test should be used when the underlying distribution is hypergeometric. This
implies that both row and column marginals are fixed. For example, assume that in total
ten male subjects with upper limb injuries are randomly assigned to two groups (A and
B). Each subject is then classified as right- or left-handed, and the following 2×2 table is
obtained:
Ten subjects with upper limb injuries randomly assigned to two groups
(^) Lefthanded Right-handed
Group A 2 3 5 (Fixed row total)
Group B 4 1 5 (Fixed row total)
(Fixed column totals) 6 4
Provided there is no association between handedness and group membership then each
group can be treated as a random sample from the population with upper limb injuries
described by the column marginal totals (6,4). This population is described by the fixed
column marginal totals (6,4). Whereas a different randomization is likely to have
produced different cell frequencies, the column marginal totals would remain as before
(6,4), provided there was no difference between Group A and Group B. In this case both
marginal totals are fixed and the distribution of cell frequency counts is described by the
hypergeometric distribution. Inferences are made with respect to the target population of
subjects (males with upper limb injuries).
Two alternative sampling strategies that can give rise to a 2×2 contingency table are
simple random sampling and stratified random sampling. If ten male subjects with upper
limb injuries were sampled at random from the population (simple random sampling),
and each subject was asked two questions: Are you left-or right-handed? Did your injury
occur ≤6-years-old or >6-years-old? The only fixed marginal total is the overall total, and
the row and column marginal totals will be random. The following data may be obtained:
Ten subjects with upper limb injuries randomly assigned to two groups
(^) Left-handed Right-handed
≤6-years-old 3 3
6-years-old 1 3
10 (Fixed total)
In this design, inferences can be drawn with respect to the target population of males with
upper limb injuries. The null hypothesis would be no association between the row and
Inferences involving binomial and nominal count data 179