Lookup table for the mvnconv function.
mvnlookupThe data frame contains the following columns:
| rhos | numeric |
correlations among the test statistics |
| m2lp_1 | numeric |
Cov[-2 (p_i), -2 (p_j)]Cov[-2 ln(p_i), -2 ln(p_j)] (for one-sided tests) |
| m2lp_2 | numeric |
Cov[-2 (p_i), -2 (p_j)]Cov[-2 ln(p_i), -2 ln(p_j)] (for two-sided tests) |
| z_1 | numeric |
Cov[^-1(1 - p_i), ^-1(1 - p_j)]Cov[Phi^-1(1 - p_i), Phi^-1(1 - p_j)] (for one-sided tests) |
| z_2 | numeric |
Cov[^-1(1 - p_i), ^-1(1 - p_j)]Cov[Phi^-1(1 - p_i), Phi^-1(1 - p_j)] (for two-sided tests) |
| chisq1_1 | numeric |
Cov[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 |
Cov[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 |
Cov[p_i, p_j]Cov[p_i, p_j] (for one-sided tests) |
| p_2 | numeric |
Cov[p_i, p_j]Cov[p_i, p_j] (for two-sided tests) |
Assume [arrayc t_i \\ t_j array] MVN ([arrayc 0 \\ 0 array], [arraycc 1 & _ij \\ _ij & 1 array] )bmatrix t_i \\\ t_j bmatrix MVN (bmatrix 0 \\\ 0 bmatrix, bmatrix 1 & _ij \\\ _ij & 1 bmatrix )[t_i, t_j]' ~ MVN([0,0]', [1, rho_ij | rho_ij, 1]) is the joint distribution for test statistics t_i and t_j. For one-sided tests, let p_i = 1 - (t_i)p_i = 1 - Phi(t_i) and p_j = 1 - (t_j)p_j = 1 - Phi(t_j) where ()Phi(.) denotes the cumulative distribution function of a standard normal distribution. For two-sided tests, let p_i = 2(1 - (|t_i|))p_i = 2(1 - Phi(|t_i|)) and p_j = 2(1 - (|t_j|))p_j = 2(1 - Phi(|t_j|)). These are simply the one- and two-sided p-values corresponding to t_i and t_j.
Columns p_1 and p_2 contain the values for Cov[p_i, p_j]Cov[p_i, p_j].
Columns m2lp_1 and m2lp_2 contain the values for Cov[-2 (p_i), -2 (p_j)]Cov[-2 ln(p_i), -2 ln(p_j)].
Columns chisq1_1 and chisq1_2 contain the values for Cov[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 F^-1(,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 Cov[^-1(1 - p_i), ^-1(1 - p_j)]Cov[Phi^-1(1 - p_i), Phi^-1(1 - p_j)], where ^-1()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.