mvnlookup: Lookup Table for the mvnconv() Function
In ozancinar/poolR: Methods for Pooling P-Values from (Dependent) Tests
mvnlookup R Documentation
Lookup Table for the mvnconv() Function
Description
Lookup table for the mvnconv function.\loadmathjax
Usage
mvnlookup
Format
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)
Details
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.
ozancinar/poolR documentation built on Oct. 1, 2024, 12:28 a.m.
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| mvnlookup | R Documentation |
Lookup Table for the mvnconv() Function
Description
Lookup table for the mvnconv function.\loadmathjax
Usage
mvnlookupFormat
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) |
Details
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.
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