In using Fisher's exact test on the \(2 \times 2\) nearest neighbor contingency tables (NNCTs) a correction may be needed for the \(p\)-value. For the one-sided alternatives, the probabilities of more extreme tables are summed up, including or excluding the probability of the table itself (or some middle way).
There is additional complexity in \(p\)-values for the two-sided alternatives. A recommended method is adding up probabilities of the same size and smaller than the probability associated with the current table. Alternatively, one can double the one-sided \(p\)-value (see (agresti:1992;textualnnspat).
Let the probability of the contingency table itself be \(p_t=f(n_{11}|n_1,n_2,c_1;\theta_0)\) where \(\theta_0=(n_1-1)(n_2-1)/(n_1 n_2)\) which is the odds ratio under RL or CSR independence and \(f\) is the probability mass function of the hypergeometric distribution.
**Type (I):** For double the one-sided \(p\)-value, we propose the following four variants:
[(i)] twice the minimum of \(p_{inc}\) for the one-sided tests, which is table-inclusive version for this type of two-sided test, and denoted as \(p^I_{inc}\),
[(ii)] twice the minimum of \(p_{inc}\) minus twice the table probability \(p_t\), which is table-exclusive version of this type of two-sided test, and denoted as \(p^I_{exc}\),
[(iii)] table-exclusive version of this type of two-sided test plus \(p_t\), which is mid-\(p\)-value for this test, and denoted as \(p^I_{midd}\),
[(iv)]Tocher corrected version (see tocher.cor for details).
**Type (II):** For summing the \(p\)-values of more extreme ---than that of the table--- cases in both directions, the following variants are obtained. The \(p\)-value is \(p=\sum_S f(t|n_1,n_2,c_1;\theta=1)\) with
[(i)] \(S=\{t:\,f(t|n_1,n_2,c_1;\theta=1) \leq p_t\}\), which is called table-inclusive version, \(p^{II}_{inc}\),
[(ii)] the probability of the observed table is included twice, once for each side; that is \(p=p^{II}_{inc}+p_t\), which is called twice-table-inclusive version, \(p^{II}_{tinc}\),
[(iii)] table-inclusive minus \(p_t\), which is referred as table-exclusive version, \(p^{II}_{exc}\),
[(iv)] table-exclusive plus one-half the \(p_t\), which is called mid-\(p\) version, \(p^{II}_{mid}\) and,
[(v)]Tocher corrected version, \(p^{II}_{Toc}\), is obtained as before.
See (ceyhan:exact-NNCT;textualnnspat) for more details.
exact.pval2s(ptable, pval, type = "inc", double = FALSE)A modified \(p\)-value based on the correction specified in type.
Probability of the observed \(2 \times 2\) NNCT under the null hypothesis using the hypergeometric distribution for Fisher's exact test.
Table inclusive \(p\)-value for Fisher's exact test on the NNCT.
The type of the \(p\)-value correction for the two-sided exact test on the NNCT, default="inc".
Takes on values "inc", "exc", "mid", "tocher" (or equivalently 1-4, respectively) for table inclusive,
table-exclusive, mid-\(p\)-value, and Tocher corrected \(p\)-value, respectively.
A logical argument (default is FALSE) to determine whether type I or II correction should be
applied to the two-sided \(p\)-value. If TRUE type I correction (for doubling the minimum of the one-sided \(p\)-value)
is applied, otherwise, type II correction (using the probabilities for the more extreme tables) is applied.
Elvan Ceyhan
exact.pval1s and tocher.cor
ct<-matrix(sample(20:40,4),ncol=2)
ptab<-prob.nnct(ct)
pv<-.23
exact.pval2s(ptab,pv)
exact.pval2s(ptab,pv,type="exc")
exact.pval2s(ptab,pv,type="mid")
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