Determine sample size for a chi-squared global test in a group sequential SMART design

For specified operating characteristics, iteratively increases the sample size until the desired power is achieved using a chi-squared global test statistic. The chi-squared test assesses whether any treatment regime differs from the others, rather than comparing individual regimes against a fixed control.

Usage

get_sample_size_chi(
  variances,
  beta,
  delta,
  bounds,
  n_init = 100,
  corr = bounds$corr,
  inf_frac = bounds$inf_frac,
  n_split = NULL,
  seed = 2,
  B = 10001,
  lambda = 20,
  max_iter = 100
)

Arguments

variances: A numeric vector of length \(L\) of variances of the value estimators, i.e., \(\sqrt{N} \times \mathrm{Cov}(\hat{\theta})\). These should reflect the population variances rather than sample variance or standard errors.
beta: A numeric value between 0 and 1 specifying the type II error rate. Power is 1 - beta.
delta: A numeric vector of length \(L\) of change in the regime values under the alternative hypothesis.
bounds: A numeric vector of chi-squared stopping boundaries for analyses \(1, \ldots, S\), or the output list from get_bounds_chi.
n_init: A positive integer specifying the initial total trial sample size \(N\) to begin the search. Default is 100.
corr: A correlation matrix of dimension \(L \times L\) between regime value estimators. Defaults to the correlation from the bounds list if provided.
inf_frac: A numeric vector of information fractions indicating when analyses are conducted. Defaults to the information fractions from the bounds list if provided.
n_split: A numeric vector indicating the proportion of the sample size at the analysis times \(s = 1, \ldots, S\). If NULL (default), assumes the split is proportional to the information fractions.
seed: An integer seed for reproducibility of the Monte Carlo simulation. Default is 2.
B: A positive integer specifying the number of Monte Carlo samples for power estimation. Default is 10001.
lambda: A positive numeric value specifying the incremental sample size increases when raising the sample size to find the required power via simulation. After this is found, iteration from \(n-lambda\) to \(n+lambda\) will be done to find the correct exact sample size.
max_iter: The maximum sample sizes n to evaluate.

Value

A list with the following components:

N: A numeric vector of sample sizes at each analysis.
power: The achieved power at the final sample size.
prop_rej: A numeric vector of cumulative rejection probabilities at each analysis.
variances: The input variances.
beta: The input type II error rate.
delta: The input alternative differences.
bounds: The input stopping boundaries.
n_init: The input initial sample size.
corr: The input correlation matrix.
inf_frac: The input information fractions.
n_split: The input sample size split proportions.
seed: The input random seed.
B: The input number of Monte Carlo samples.
lambda: The input step size for the sample size search.

Details

The function uses Monte Carlo simulation to estimate power at each candidate sample size. A contrast matrix \(C\) is constructed to form \(L-1\) linearly independent comparisons among \(L\) regimes, and the chi-squared statistic is computed as \((CZ)' (C \Sigma C')^{-1} (CZ)\).