exact.pval2s {nnspat}

p-value correction to the two-sided version of exact NNCT test

Description

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)).

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 (2010)) for more details.

Usage

exact.pval2s(ptable, pval, type = "inc", double = FALSE)

Arguments

ptable

Probability of the observed 2 \times 2 NNCT under the null hypothesis using the hypergeometric distribution for Fisher's exact test.

pval

Table inclusive p-value for Fisher's exact test on the NNCT.

type

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.

double

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.

Value

A modified p-value based on the correction specified in type.

Author(s)

Elvan Ceyhan

References

Agresti A (1992). “A Survey of Exact Inference for Contingency Tables.” Statistical Science, 7(1), 131-153.

Ceyhan E (2010). “Exact Inference for Testing Spatial Patterns by Nearest Neighbor Contingency Tables.” Journal of Probability and Statistical Science, 8(1), 45-68.

See Also

exact.pval1s and tocher.cor

Examples

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")