Statistical Methods in Medical Research 4ed

526 Empirical methods for categorical data
the marginal totals, the test involves evaluation of the probability that P rixi is
at least as large as the observed value (Agresti, 1990, §4.8.2). The exact method
involves a large amount of calculation and is only feasible with appropriate
statistical software, such as StatXact.
The exact test may be extended to cover the testing of a trend when there is a
stratifying variable to take into account (§15.7). The x 2 test for this situation is
(15.21). An exact test is based on the probability of obtaining a value of P
h Th at
least as large as that observed and, from (15.4), this is equivalent to basing the
test on the sum of P rixi over the h strata. Conditioning is on the marginal totals
in all the strata.
Example 15.9
In Table 15.8 there are several low counts and so there might be some doubt as to the
accuracy of the analysis shown in Example 15.8. Using StatXact the test of trend is as
follows.
The sum of P rixi over the three strata is 35. The exact probability that this sum would
be greater than or equal to 35 if there were no effect of length of exposure, and subject to
the marginal totals in each of the three age groups, is 0 0050 from StatXact. The
corresponding one-sided probability using the x2 …1† of 7 42 in Example 15.8 is 0 0032.
However, this is based on a test without a continuity correction and is, therefore,
equivalent to a mid-P test (see §4.4), whereas the exact value is the usual P value.
If the test statistic in Example 15.8 had been corrected for continuity, then its value
would have been 7 242 =8 45 ˆ 6 52, giving a one-sided P value of 0 0054very similar to
the exact value of 0 0050.
Conversely, since from StatXact the probability that P rixi over the three strata is
1
exactly 35 is 0 0033, the exact one-sided mid-P value is 0 0050 2 …0 0033† ˆ0 0034, very
near to the value from the uncorrected x2 of 0 0032.
The tests have been compared on the basis of their one-sided values since, with the
exact test, there can be ambiguity on how to obtain a two-sided value (see discussion in
Example 4.13). The two-sided levels using the x2 are, of course, simply double the onesided
levels, giving 0 0065 and 0 011 for the uncorrected and corrected values. One option
of obtaining a two-sided exact level, which we advocated, is to double the one-sided level,
giving 0 0067 and 0 010 for the mid-P and P values, respectively.
So, in this example, even though the frequencies are not large, the tests based on x2 statistics proved very satisfactory approximations to the exact tests.
Combination of 2 2 tables
This is an extension of the exact test in a single 2 2 table (§4.5). The one-tailed
significance level is the probability that P rAi is equal to or greater than its
observed value, where for any value of P rAi the probability is calculated by
considering all the possible combinations of tables, with the same marginal
totals, over the strata that produce this total (Mehta et al., 1985; Hirji et al.,
1988; Agresti, 1990, §7.4.4).