Calculate the sequential p-value for a single hypothesis

A sequential p-value is the minimum significance level at which a group sequential boundary would be crossed at any analysis up to and including the current one. It is computed using the group sequential boundaries derived from the spending function and the joint distribution of test statistics across analyses.

Sequential p-values are used in graphical multiple comparison procedures for group sequential designs. They allow the separation of the group sequential testing (handled by the spending function and boundaries) from the multiplicity adjustment (handled by the graph). See Maurer and Bretz (2013) for details.

Usage

sequential_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 at information fraction t. 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 sequential p-value.

Details

For a hypothesis tested at analyses $k = 1, \ldots, K$ with p-values $p^{(k)}$ and information fractions $t^{(k)}$, the sequential p-value is the minimum $\tilde{p}$ such that for some analysis $k$, the observed p-value $p^{(k)}$ crosses the group sequential boundary $c_k(\tilde{p})$ derived from the spending function: $$\tilde{p} = \min\{\alpha : p^{(k)} \le c_k(\alpha) \text{ for some } k\},$$ where $c_k(\alpha)$ is the nominal p-value boundary at analysis $k$ when the total significance level is $\alpha$.

The boundary $c_k(\alpha)$ is computed from the spending function $f(\alpha, t)$ using the joint distribution of test statistics. Specifically, the Z-scale boundary $b_k$ satisfies $$P(Z_1 < b_1, \ldots, Z_k < b_k) = 1 - f(\alpha, t_k),$$ and $c_k = 1 - \Phi(b_k)$. Note that $c_k \neq f(\alpha, t_k) - f(\alpha, t_{k-1})$ for $k > 1$ due to the correlation between test statistics across analyses.

The sequential p-value is found using stats::uniroot() on the function $g(\alpha) = \max_k (z_k - b_k(\alpha))$, where $z_k = \Phi^{-1}(1 - p^{(k)})$ is the observed Z-statistic and $b_k(\alpha)$ is the Z-scale boundary. This function is monotonically increasing in $\alpha$ since boundaries become less stringent as $\alpha$ increases.

References

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

Liu, Q., and Anderson, K. M. (2008). On adaptive extensions of group sequential trials for clinical investigations. Journal of the American Statistical Association, 103(484), 1621-1630.

Examples

# A hypothesis tested at two analyses (interim at 50% and final at 100%)
sequential_p(
  p = c(0.024, 0.01),
  info_frac = c(0.5, 1),
  spending_fn = spending_of
)
#> [1] 0.01009409

# Sequential p-value with Pocock spending
sequential_p(
  p = c(0.024, 0.01),
  info_frac = c(0.5, 1),
  spending_fn = spending_pocock
)
#> [1] 0.01850038

# Sequential p-value updates as more analyses are conducted
# After analysis 1 only
sequential_p(
  p = 0.05,
  info_frac = 0.3,
  spending_fn = spending_of
)
#> [1] 0.2830392

# After analyses 1 and 2
sequential_p(
  p = c(0.05, 0.02),
  info_frac = c(0.3, 0.7),
  spending_fn = spending_of
)
#> [1] 0.05185881

# After all three analyses
sequential_p(
  p = c(0.05, 0.02, 0.01),
  info_frac = c(0.3, 0.7, 1),
  spending_fn = spending_of
)
#> [1] 0.01071832