Lookup Table for the mvnconv() Function

Format

The data frame contains the following columns:

Details

Assume $$\begin{bmatrix} t_i \\ t_j \end{bmatrix} \sim \mbox{MVN} \left(\begin{bmatrix} 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 & \rho_{ij} \\ \rho_{ij} & 1 \end{bmatrix} \right)$$ is the joint distribution for test statistics $t_i$ and $t_j$. For one-sided tests, let $p_i = 1 - \Phi(t_i)$ and $p_j = 1 - \Phi(t_j)$ where $\Phi(\cdot)$ denotes the cumulative distribution function of a standard normal distribution. For two-sided tests, let $p_i = 2(1 - \Phi(|t_i|))$ and $p_j = 2(1 - \Phi(|t_j|))$. 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 $\mbox{Cov}[p_i, p_j]$.

Columns m2lp_1 and m2lp_2 contain the values for $\mbox{Cov}[-2 \ln(p_i), -2 \ln(p_j)]$.

Columns chisq1_1 and chisq1_2 contain the values for $\mbox{Cov}[F^{-1}(1 - p_i, 1), F^{-1}(1 - p_j, 1)]$, where $F^{-1}(\cdot,1)$ 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 $\mbox{Cov}[\Phi^{-1}(1 - p_i), \Phi^{-1}(1 - p_j)]$, where $\Phi^{-1}(\cdot)$ 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.