Calculate the repeated p-value for a single hypothesis at a given analysis

A repeated p-value at analysis $k$ is the minimum significance level at which the group sequential boundary at analysis $k$ would be crossed. Unlike the sequential p-value, which considers all analyses up to $k$, the repeated p-value only considers the boundary at analysis $k$ itself.

The sequential p-value equals the minimum of repeated p-values across analyses: $\tilde{p}_k = \min_{l=1}^{k} \hat{p}_l$, where $\hat{p}_l$ is the repeated p-value at analysis $l$.

Usage

repeated_p(
  p,
  info_frac,
  spending_fn,
  tol = 1e-06,
  maxpts = 25000,
  abseps = 1e-06
)

Arguments

p: A numeric vector of p-values at each analysis for a single hypothesis. The length must match the length of info_frac. All values must be non-missing and between 0 and 1.
info_frac: A numeric vector of information fractions at each analysis. Values must be in (0, 1] and monotonically non-decreasing. The length should match the length of p.
spending_fn: A spending function. Must accept two arguments: alpha (total significance level) and info_frac (information fraction), and return the cumulative alpha spent. Built-in options include spending_of(), spending_pocock(), spending_hsd(), and spending_linear().
tol: A numeric scalar for the tolerance of the root-finding algorithm. The default is 1e-6.
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 numeric scalar of the repeated p-value at the last analysis in the input vectors.

Details

For a hypothesis tested at analyses $k = 1, \ldots, K$ with p-values $p^{(k)}$ and information fractions $t^{(k)}$, the repeated p-value at analysis $K$ is the minimum $\hat{p}$ such that the observed p-value $p^{(K)}$ crosses the group sequential boundary $c_K(\hat{p})$ at that analysis: $$\hat{p}_K = \min\{\alpha : p^{(K)} \le c_K(\alpha)\}.$$

Note that computing the boundary $c_K(\alpha)$ requires knowledge of all previous information fractions $t^{(1)}, \ldots, t^{(K)}$ because the boundary at analysis $K$ depends on the cumulative spending and the joint distribution of test statistics.

The repeated p-value is found using stats::uniroot() on the function $g(\alpha) = z_K - b_K(\alpha)$, where $z_K = \Phi^{-1}(1 - p^{(K)})$ is the observed Z-statistic at analysis $K$ and $b_K(\alpha)$ is the Z-scale boundary.

References

Maurer, W., and Bretz, F. (2013). Multiple testing in group sequential trials using graphical approaches. Statistics in Biopharmaceutical Research, 5(4), 311-320.

Examples

# Repeated p-value at the second analysis (interim at 50%, final at 100%)
repeated_p(
  p = c(0.024, 0.01),
  info_frac = c(0.5, 1),
  spending_fn = spending_of
)
#> [1] 0.01009393

# Compare with sequential p-value (which is the minimum repeated p-value)
sequential_p(
  p = c(0.024, 0.01),
  info_frac = c(0.5, 1),
  spending_fn = spending_of
)
#> [1] 0.01009409

# Repeated p-values at each analysis
# Analysis 1
repeated_p(
  p = 0.05,
  info_frac = 0.3,
  spending_fn = spending_of
)
#> [1] 0.2830392

# Analysis 2
repeated_p(
  p = c(0.05, 0.02),
  info_frac = c(0.3, 0.7),
  spending_fn = spending_of
)
#> [1] 0.05185878