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Wang-Tsiatis spending function

Source: R/spending_functions.R
spending_wt.Rd

Computes the implied cumulative alpha spending from the Wang-Tsiatis family of group sequential boundaries. The Wang-Tsiatis boundaries at analysis \(k\) with information fraction \(t_k\) are defined as: $$c_k = C \cdot t_k^{\Delta - 0.5},$$ where \(\Delta\) is the shape parameter and \(C\) is a constant calibrated so that the overall Type I error equals \(\alpha\).

Special cases:

  • \(\Delta = 0.5\): Pocock boundaries (equal Z-scale boundaries across analyses).

  • \(\Delta = 0\): O'Brien-Fleming boundaries (very conservative at early analyses).

  • \(0 < \Delta < 0.5\): intermediate between O'Brien-Fleming and Pocock.

Unlike the Lan-DeMets approximations (spending_of(), spending_pocock()), this function computes the exact boundaries from the Wang-Tsiatis family and derives the implied spending. It is computationally more expensive because it requires root-finding and multivariate normal integration at each call.

Usage

spending_wt(alpha, info_frac, delta = 0.5, maxpts = 25000, abseps = 1e-06)

Arguments

alpha

A numeric scalar of the total significance level.

info_frac

A numeric vector of information fractions at each analysis. Must be non-negative, with at most one value \(\geq 1\). The last value must be \(\geq 1\) (i.e., the final analysis must be included), because the Wang-Tsiatis constant \(C\) is calibrated over the full set of analyses.

delta

A numeric scalar for the shape parameter \(\Delta\). The default is 0.5 (Pocock). Use 0 for O'Brien-Fleming.

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 vector the same length as info_frac of cumulative alpha spent at each information fraction.

References

Wang, S. K., and Tsiatis, A. A. (1987). Approximately optimal one-parameter boundaries for group sequential trials. Biometrics, 43(1), 193-199.

See also

spending_of() and spending_pocock() for the Lan-DeMets approximations, gs_boundaries() for computing boundaries from spending functions, graph_test_shortcut_gsd() for the graphical procedure.

Examples

# Exact O'Brien-Fleming (delta = 0)
spending_wt(0.025, c(0.5, 1), delta = 0)
#> [1] 0.002582893 0.025000000

# Exact Pocock (delta = 0.5)
spending_wt(0.025, c(0.5, 1), delta = 0.5)
#> [1] 0.01469289 0.02500000

# Intermediate (delta = 0.25)
spending_wt(0.025, c(1 / 3, 2 / 3, 1), delta = 0.25)
#> [1] 0.003062016 0.012374443 0.024991263

# Compare with Lan-DeMets approximations
spending_of(0.025, c(1 / 3, 2 / 3, 1)) # Lan-DeMets OBF approximation
#> [1] 0.0001035057 0.0060483891 0.0250000000
spending_wt(0.025, c(1 / 3, 2 / 3, 1), 0) # Exact OBF
#> [1] 0.0002594522 0.0071641731 0.0250000000

# Use in graph_test_shortcut_gsd (wrap to fix delta)
# \donttest{
g <- graph_create(c(0.5, 0.5), rbind(c(0, 1), c(1, 0)))
p <- rbind(H1 = c(0.024, 0.01), H2 = c(0.015, 0.005))
graph_test_shortcut_gsd(
  graph = g, p = p, alpha = 0.025,
  info_frac = c(0.5, 1),
  spending_fn = function(a, t) spending_wt(a, t, delta = 0.25)
)
#> 
#> Test parameters ($inputs) ------------------------------------------------------
#>   Initial graph
#> 
#>   --- Hypothesis weights ---
#>   H1: 0.5
#>   H2: 0.5
#> 
#>   --- Transition weights ---
#>      H1 H2
#>   H1  0  1
#>   H2  1  0
#> 
#>   Alpha = 0.025
#> 
#>   Information fractions
#>    Analysis_1 Analysis_2
#> H1        0.5          1
#> H2        0.5          1
#> 
#>   P-values
#>    Analysis_1 Analysis_2
#> H1   0.024000   0.010000
#> H2   0.015000   0.005000
#> 
#>   Spending functions
#>     H1: spending_wt(a, t, delta = 0.25)
#>     H2: spending_wt(a, t, delta = 0.25)
#> 
#>   Look back = FALSE
#> 
#> Test summary ($outputs) --------------------------------------------------------
#>   Hypothesis   Adj.p* Reject Tested.at First.Rej.at Last.Rej.at Look.back
#>           H1 1.00000+  FALSE         2           --          --     FALSE
#>           H2 1.00000+  FALSE         2           --          --     FALSE
#>   (*) Adjusted p-values account for both the group sequential design and the
#>       graphical multiple comparison procedure. Based on repeated p-values when
#>       look_back = FALSE, and sequential p-values when look_back = TRUE.
#> 
#>   Final updated graph after removing rejected hypotheses
#> 
#>   --- Hypothesis weights ---
#>   H1: 0.5
#>   H2: 0.5
#> 
#>   --- Transition weights ---
#>      H1 H2
#>   H1  0  1
#>   H2  1  0
#> 
# }