gamma_k: Order k product-type approximation of multivariate normal...
In oyvind-bakke/fwerapprox: FWER Approximations

Description Usage Arguments Details Value Examples

Computes an order k Glaz–Johnson approximation to a multivariate standard normal probability with a given correlation matrix. Gives a familywise error rate level bound in multiple testing for a given local (per-hypothesis) significance level.

1	gamma_k(k, alphaloc, corr, miwasteps = 4096, genz = FALSE)

`k`	Order of the approximation.
`alphaloc`	Local significance level (se above).
`corr`	Either a list of correlations in which the first component is a vector of first-order correlations, the second is a vector of second-order correlations, up to a vector of `k` - 1-order correlations, or a correlation matrix. Typically the `corrs` or `corrmatrix` component given by `scorestatcorr`.
`miwasteps`	The `steps` parameter used for `Miwa` (`algorithms`) in `pmvnorm`.
`genz`	If `TRUE`, the `GentzBretz` algorithm (`algorithms`) is used for `pmvnorm`, otherwise the `Miwa` algorithm is used.

The function is quite slow. For second-order approximation, it is better to use gamma2.

P(|T_1| < c, …, |T_m| < c) is approximated for (T_1, …, T_m) multivariate standard normal with covariances (at least up to order k-1) given by corr, where c is qnorm(1 - alphaloc/2). If (T_1,… T_m) is a test statistic vector for m hypotheses, a local significance level of alphaloc, i.e. rejection of a null hypothesis if the p-value is less than alphaloc, will control familywise error rate at the 1 - gamma_k(alphaloc, corr) level.

# Normal model with three environmental covariates:
result <- scorestatcorr(y_normal ~ sex + activity + agecategory, xg[,1:200], 2)
al <- uniroot(function(a) gamma2(a, result$corrs[[1]]) - .95, c(1e-5, 5e-4), tol = 1e-14)$root
al3 <- uniroot(function(a) gamma_k(3, a, result$corrs) - .95, c(1e-5, 5e-4), tol = 1e-14)$root
0.05/2000 # Bonferroni
1 - 0.95^(1/2000) # Sidak
al # order 2 FWER approximation
al3 # order 3 FWER approximation