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Alpha spending functions for group sequential designs

Source: R/spending_functions.R
spending_functions.Rd

Alpha spending functions determine how the total significance level (alpha) is allocated across interim and final analyses in a group sequential design. Given the total alpha and the information fraction(s) at one or more analyses, a spending function returns the cumulative alpha spent at each information fraction.

Four commonly used spending functions are provided:

  • spending_of() for the Lan-DeMets O'Brien-Fleming approximation,

  • spending_pocock() for the Lan-DeMets Pocock approximation,

  • spending_hsd() for the Hwang-Shih-DeCani family,

  • spending_linear() for linear (uniform) spending.

Usage

spending_of(alpha, info_frac)

spending_pocock(alpha, info_frac)

spending_hsd(alpha, info_frac, gamma = -4)

spending_linear(alpha, info_frac)

Arguments

alpha

A numeric scalar of the total significance level to be spent. Must be between 0 and 1.

info_frac

A numeric scalar or vector of information fractions. Values must be non-negative. When info_frac = 0, the spending is 0. When info_frac >= 1, the spending is capped at alpha.

gamma

A numeric scalar for the gamma parameter of the Hwang-Shih-DeCani spending function. Common choices are gamma = -4 (approximates O'Brien-Fleming), gamma = 1 (approximates Pocock), and gamma = 0 (linear spending). The default is gamma = -4.

Value

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

Details

All spending functions satisfy the following properties:

  • \(f(\alpha, 0) = 0\),

  • \(f(\alpha, 1) = \alpha\),

  • \(f(\alpha, t)\) is non-decreasing in \(t\).

The cumulative alpha spent at analysis \(k\) is \(f(\alpha, t_k)\), and the incremental spending is $$\Delta\alpha_k = f(\alpha, t_k) - f(\alpha, t_{k-1}).$$

Note that the incremental spending is not the nominal significance level (boundary) at analysis \(k\). The boundary must be derived from the spending using the joint distribution of test statistics across analyses. See sequential_p() and graph_test_shortcut_gsd() for details.

Spending function formulas

  • O'Brien-Fleming (spending_of): $$f(\alpha, t) = 2\left(1 - \Phi\left(\frac{\Phi^{-1}(1 - \alpha/2)} {\sqrt{t}}\right)\right).$$ This is the Lan-DeMets approximation to O'Brien-Fleming boundaries. It is very conservative at early analyses and spends most of the alpha at the final analysis.

  • Pocock (spending_pocock): $$f(\alpha, t) = \alpha \cdot \ln(1 + (e - 1) \cdot t).$$ This spends alpha more evenly across analyses compared to O'Brien-Fleming.

  • Hwang-Shih-DeCani (spending_hsd): $$f(\alpha, t) = \alpha \cdot \frac{1 - e^{-\gamma t}}{1 - e^{-\gamma}}, \quad \gamma \neq 0,$$ $$f(\alpha, t) = \alpha \cdot t, \quad \gamma = 0.$$ With gamma = -4, it approximates O'Brien-Fleming; with gamma = 1, it approximates Pocock.

  • Linear (spending_linear): $$f(\alpha, t) = \alpha \cdot t.$$

References

Lan, K. K. G., and DeMets, D. L. (1983). Discrete sequential boundaries for clinical trials. Biometrika, 70(3), 659-663.

Hwang, I. K., Shih, W. J., and De Cani, J. S. (1990). Group sequential designs using a family of type I error probability spending functions. Statistics in Medicine, 9(12), 1439-1445.

Examples

# O'Brien-Fleming spending at 50% information
spending_of(0.025, 0.5)
#> [1] 0.001525323

# Cumulative spending across analyses (vectorized)
spending_of(0.025, c(0, 0.5, 1))
#> [1] 0.000000000 0.001525323 0.025000000

# Compare spending functions at information fractions (1/3, 2/3, 1)
spending_of(0.025, c(1 / 3, 2 / 3, 1))
#> [1] 0.0001035057 0.0060483891 0.0250000000
spending_pocock(0.025, c(1 / 3, 2 / 3, 1))
#> [1] 0.01132081 0.01908456 0.02500000
spending_hsd(0.025, c(1 / 3, 2 / 3, 1), gamma = -4)
#> [1] 0.001303062 0.006246445 0.025000000
spending_linear(0.025, c(1 / 3, 2 / 3, 1))
#> [1] 0.008333333 0.016666667 0.025000000

# User-defined spending function: piecewise combination.
# Use O'Brien-Fleming for the first half of alpha (conservative at
# early analyses), and Pocock for the second half (more aggressive).
# This can be useful when a hypothesis starts with a small weight
# (OBF spending) and later receives additional weight via graph
# propagation (Pocock spending for the increment).
spending_piecewise <- function(alpha, info_frac, threshold = 0.0125) {
  spending_of(pmin(alpha, threshold), info_frac) +
    spending_pocock(pmax(alpha - threshold, 0), info_frac)
}
spending_piecewise(0.025, c(1 / 3, 2 / 3, 1))
#> [1] 0.005675579 0.011762668 0.025000000
# Compare: alpha = 0.0125 uses only OBF
spending_piecewise(0.0125, c(1 / 3, 2 / 3, 1))
#> [1] 1.517362e-05 2.220386e-03 1.250000e-02
spending_of(0.0125, c(1 / 3, 2 / 3, 1))
#> [1] 1.517362e-05 2.220386e-03 1.250000e-02
# Pocock spending at 50% information
spending_pocock(0.025, 0.5)
#> [1] 0.01550286
# Hwang-Shih-DeCani spending at 50% information
spending_hsd(0.025, 0.5, gamma = -4)
#> [1] 0.002980073
spending_hsd(0.025, 0.5, gamma = 1)
#> [1] 0.01556148
spending_hsd(0.025, 0.5, gamma = 0)
#> [1] 0.0125
# Linear spending at 50% information
spending_linear(0.025, 0.5)
#> [1] 0.0125