Compute the first chi-squared stopping boundary for a group sequential design

Determines the first analysis stopping boundary based on a chi-squared global test statistic. This is used when the comparison type is a global chi-squared test (testing whether any regime differs from the others) rather than individual regime comparisons against a fixed control.

Usage

get_bounds_chi(
  alpha = 0.05,
  inf_frac = c(0.5, 1),
  spend_fn = "OF",
  corr = diag(x = 1, nrow = 1, ncol = 1),
  mu = NULL,
  B = 1000001,
  seed = 1
)

Arguments

alpha: A numeric value specifying the overall type I error rate to control. Default is 0.05.
inf_frac: A numeric vector of information fractions indicating when analyses are conducted. Values should be between 0 and 1. The length determines the number of planned analyses \(S\). Default is c(0.5, 1).
spend_fn: A character string specifying the alpha spending function. Currently used to adjust the iota scaling factors for each analysis boundary. For "OF" (O'Brien-Fleming), set iota to the information fractions. For "Pocock", use iota = rep(1, S). Default is "OF".
corr: A correlation matrix of the expected correlation between the Z-statistics of regimes against a fixed value. Should have dimension \(L \times L\) by \(L \times L\), where \(L\) is the number of regimes. For inf_frac with length greater than one, the correlation structure across multiple analyses is computed.
mu: An optional numeric vector of means for the multivariate normal distribution for all regimes at a single analysis used in the Monte Carlo simulation. Default is NULL, which uses a zero vector (null hypothesis).
B: A positive integer specifying the number of Monte Carlo samples. Default is 1000001.
seed: An integer seed for reproducibility of the Monte Carlo simulation. Default is 1.

Value

A list with the following components:

bound: A numeric vector of chi-squared stopping boundaries, one per analysis.
spending: A numeric vector of observed cumulative alpha spent at each analysis.
typeI: The overall achieved type I error rate across all analyses.
convergence: A logical value; always TRUE for Monte Carlo based computation.
iters: The number of Monte Carlo samples used (B).
dfchi: The degrees of freedom of the chi-squared statistic, equal to the rank of \(C \Sigma C'\).
alpha: The input type I error rate.
inf_frac: The input information fractions.
spend_fn: The input spending function name.
corr: The input correlation matrix.
mu: The input mean vector.
B: The input number of Monte Carlo samples.
seed: The input random seed.

Details

The function uses Monte Carlo simulation to estimate the boundary that controls the familywise type I error rate at a specified level. A contrast matrix \(C\) is constructed to form \(L-1\) linearly independent comparisons among \(L\) regimes, and the chi-squared statistic is computed as \((CZ)' (C \Sigma C')^{-1} (CZ)\).