Compute group sequential boundaries from an alpha spending function

Given a significance level, information fractions, and a spending function, compute the group sequential boundaries at each analysis. The boundaries are computed on the Z-scale using the recursive relationship between cumulative spending and the joint distribution of test statistics. The null hypothesis is rejected at analysis \(k\) if the test statistic \(Z_k \ge b_k\).

At analysis \(k\), the Z-scale boundary \(b_k\) satisfies $$P(Z_1 < b_1, \ldots, Z_k < b_k) = 1 - f(\alpha, t_k),$$ where \(f(\alpha, t_k)\) is the cumulative spending at information fraction \(t_k\), and \((Z_1, \ldots, Z_k)\) follows the canonical joint distribution with mean zero and correlations given by gs_corr().

Usage

gs_boundaries(alpha, info_frac, spending_fn, maxpts = 25000, abseps = 1e-06)

Arguments

alpha

A numeric scalar of the significance level to be spent across analyses.

info_frac

A numeric vector of information fractions at each analysis. Must be monotonically non-decreasing with values in (0, 1].

spending_fn

A spending function that takes two arguments: alpha (significance level) and info_frac (information fraction), and returns the cumulative alpha spent. See spending_of().

maxpts

An integer scalar for the maximum number of function values for mvtnorm::GenzBretz. The default is 25000.

abseps

A numeric scalar for the absolute error tolerance for mvtnorm::GenzBretz. The default is 1e-6.

Value

A list with elements:

• bounds_z - A numeric vector of Z-scale boundaries at each analysis.

• bounds_nominal - A numeric vector of nominal p-value boundaries at each analysis, i.e., \(c_k = 1 - \Phi(b_k)\).

See also

spending_of(), spending_pocock(), spending_hsd(), spending_linear() for spending functions, sequential_p() for sequential p-values, graph_test_shortcut_gsd() for graphical MCPs with group sequential designs, gs_corr() for the correlation matrix.