If this proportion is less than a, we reject the null hypothesis that the two variables are in-
dependent, and conclude that there is a statistically significant relationship between the two
variables that make up our contingency table. (This is classed as a conditional testbecause
it is conditioned on the marginal totals actually obtained, instead of all possible marginal
totals that could have arisen given the total sample size.) I will not present a formula for
Fisher’s Exact Test because it is almost always obtained using statistical software. (SPSS
produces this statistic for all 2 3 2 tables.)
Fisher’s Exact Test has been controversial since the day he proposed it. One of the
problems concerns the fact that it is a conditional test (conditional on the fixed margin-
als). Some have argued that if you repeated the experiment exactly you would likely find
different marginal totals, and have asked why those additional tables should not be in-
cluded in the calculation. Making the test unconditional on the marginals would compli-
cate the calculations, though not excessively. This may sound like an easy debate to
resolve, but if you read the extensive literature surrounding fixed and random marginals,
you will find that it is not only a difficult debate to follow, but you will probably come
away thoroughly confused. (An excellent discussion of some of the issues can be found in
Agresti (2002), pp. 95–96.)
Fisher’s Exact Test also leads to controversy because of the issue of one-tailed versus
two-tailed tests, and what outcomes would constitute a “more extreme” result in the op-
posite tail. Instead of going into how to determine what is a more extreme outcome, I
will avoid that complication by simply telling you to decide in advance whether you
want a one- or a two-tailed test, (I strongly recommend two-tailed tests) and then report
the values given by standard statistical software. Virtually all common statistical soft-
ware prints out Fisher’s Exact Test results along with Pearson’s chi-square and related
test statistics. The test does not produce a chi-square statistic, but it does produce a p
value. In our example the pvalue is extremely small (.007), just as it was for the stan-
dard chi-square test.Fisher’s Exact Test versus Pearson’s Chi Square
We now have at least two statistical tests for 2 3 2 contingency tables, and will soon have
a third—which one should we use? Probably the most common solution is to go with
Pearson’s chi-square; perhaps because “that is what we have always done.” In fact, in pre-
vious editions of this book I recommended against Fisher’s Exact Test, primarily because
of the conditional nature of it. However in recent years there has been an important growth
of interest in permutation and randomization tests, of which Fisher’s Exact Test is an ex-
ample. (This approach is discussed extensively in Chapter 18.) I am extremely impressed
with the logic and simplicity of such tests, and have come to side with Fisher’s Exact Test.
In most cases the conclusion you will draw will be the same for the two approaches, though
this is not always the case. When we come to tables larger than 2 3 2, Fisher’s approach
does not apply, without modification, and there we almost always use the Pearson
Chi-Square. (But see Howell & Gordon, 1976.)6.4 An Additional Example—A 4 3 2 Design
Sexual abuse is a serious problem in our society and it is important to understand the fac-
tors behind it. Jankowski, Leitenberg, Henning, and Coffey (2002) examined the relation-
ship between childhood sexual abuse and later sexual abuse as an adult. They
cross-tabulated the number of childhood abuse categories (in increasing order of severity)
reported by 934 undergraduate women and their reports of adult sexual abuse. The results
are shown in Table 6.5.148 Chapter 6 Categorical Data and Chi-Square
conditional test
fixed and random
marginals