mvnlookup: Lookup Table for the mvnconv() Function
In poolr: Methods for Pooling P-Values from (Dependent) Tests

Lookup table for the mvnconv function.\loadmathjax

mvnlookup

The data frame contains the following columns:

rhos	`numeric`	correlations among the test statistics
m2lp_1	`numeric`	\mjeqn\mboxCov[-2 \ln(p_i), -2 \ln(p_j)]Cov[-2 ln(p_i), -2 ln(p_j)] (for one-sided tests)
m2lp_2	`numeric`	\mjeqn\mboxCov[-2 \ln(p_i), -2 \ln(p_j)]Cov[-2 ln(p_i), -2 ln(p_j)] (for two-sided tests)
z_1	`numeric`	\mjeqn\mboxCov[\Phi^-1(1 - p_i), \Phi^-1(1 - p_j)]Cov[Phi^-1(1 - p_i), Phi^-1(1 - p_j)] (for one-sided tests)
z_2	`numeric`	\mjeqn\mboxCov[\Phi^-1(1 - p_i), \Phi^-1(1 - p_j)]Cov[Phi^-1(1 - p_i), Phi^-1(1 - p_j)] (for two-sided tests)
chisq1_1	`numeric`	\mjeqn\mboxCov[F^-1(1 - p_i, 1), F^-1(1 - p_j, 1)]Cov[F^-1(1 - p_i, 1), F^-1(1 - p_j, 1)] (for one-sided tests)
chisq1_2	`numeric`	\mjeqn\mboxCov[F^-1(1 - p_i, 1), F^-1(1 - p_j, 1)]Cov[F^-1(1 - p_i, 1), F^-1(1 - p_j, 1)] (for two-sided tests)
p_1	`numeric`	\mjeqn\mboxCov[p_i, p_j]Cov[p_i, p_j] (for one-sided tests)
p_2	`numeric`	\mjeqn\mboxCov[p_i, p_j]Cov[p_i, p_j] (for two-sided tests)

Assume \mjtdeqn\left[\beginarrayc t_i \\ t_j \endarray\right] \sim \mboxMVN \left(\left[\beginarrayc 0 \\ 0 \endarray\right], \left[\beginarraycc 1 & \rho_ij \\ \rho_ij & 1 \endarray\right] \right)\beginbmatrix t_i \\\ t_j \endbmatrix \sim \mboxMVN \left(\beginbmatrix 0 \\\ 0 \endbmatrix, \beginbmatrix 1 & \rho_ij \\\ \rho_ij & 1 \endbmatrix \right)[t_i, t_j]' ~ MVN([0,0]', [1, rho_ij | rho_ij, 1]) is the joint distribution for test statistics \mjseqnt_i and \mjseqnt_j. For one-sided tests, let \mjeqnp_i = 1 - \Phi(t_i)p_i = 1 - Phi(t_i) and \mjeqnp_j = 1 - \Phi(t_j)p_j = 1 - Phi(t_j) where \mjeqn\Phi(\cdot)Phi(.) denotes the cumulative distribution function of a standard normal distribution. For two-sided tests, let \mjeqnp_i = 2(1 - \Phi(|t_i|))p_i = 2(1 - Phi(|t_i|)) and \mjeqnp_j = 2(1 - \Phi(|t_j|))p_j = 2(1 - Phi(|t_j|)). These are simply the one- and two-sided \mjseqnp-values corresponding to \mjseqnt_i and \mjseqnt_j.

Columns p_1 and p_2 contain the values for \mjeqn\mboxCov[p_i, p_j]Cov[p_i, p_j].

Columns m2lp_1 and m2lp_2 contain the values for \mjeqn\mboxCov[-2 \ln(p_i), -2 \ln(p_j)]Cov[-2 ln(p_i), -2 ln(p_j)].

Columns chisq1_1 and chisq1_2 contain the values for \mjeqn\mboxCov[F^-1(1 - p_i, 1), F^-1(1 - p_j, 1)]Cov[F^-1(1 - p_i, 1), F^-1(1 - p_j, 1)], where \mjeqnF^-1(\cdot,1)F^-1(.,1) denotes the inverse of the cumulative distribution function of a chi-square distribution with one degree of freedom.

Columns z_1 and z_2 contain the values for \mjeqn\mboxCov[\Phi^-1(1 - p_i), \Phi^-1(1 - p_j)]Cov[Phi^-1(1 - p_i), Phi^-1(1 - p_j)], where \mjeqn\Phi^-1(\cdot)Phi^-1(.) denotes the inverse of the cumulative distribution function of a standard normal distribution.

Computation of these covariances required numerical integration. The values in this table were precomputed.

poolr documentation built on Jan. 27, 2022, 1:12 a.m.