errorSpent: Error Spending

In lrstat: Power and Sample Size Calculation for Non-Proportional Hazards and Beyond

Error Spending

Description

Obtains the error spent at given spending times for the specified error spending function.

Usage

errorSpent(t, error, sf = "sfOF", sfpar = NA)

Arguments

t	A vector of spending times, typically equal to information fractions.
error	The total error to spend.
sf	The spending function. One of the following: "sfOF" for O'Brien-Fleming type spending function, "sfP" for Pocock type spending function, "sfKD" for Kim & DeMets spending function, and "sfHSD" for Hwang, Shi & DeCani spending function. Defaults to "sfOF".
sfpar	The parameter for the spending function. Corresponds to rho for "sfKD" and gamma for "sfHSD".

Details

This function implements a variety of error spending functions commonly used in group sequential designs, assuming one-sided hypothesis testing.

O'Brien-Fleming-Type Spending Function

This spending function allocates very little alpha early on and more alpha later in the trial. It is defined as:

\alpha(t) = 2 - 2\Phi\left(\frac{z_{\alpha/2}}{\sqrt{t}}\right),

where \Phi is the standard normal cumulative distribution function, z_{\alpha/2} is the critical value from the standard normal distribution, and t \in [0, 1] denotes the information fraction.

Pocock-Type Spending Function

This function spends alpha more evenly throughout the study:

\alpha(t) = \alpha \log(1 + (e - 1)t),

where e is Euler's number (approximately 2.718).

Kim and DeMets Power-Type Spending Function

This family of spending functions is defined as:

\alpha(t) = \alpha t^{\rho}, \quad \rho > 0.

• When \rho = 1, the function mimics Pocock-type boundaries.

• When \rho = 3, it approximates O’Brien-Fleming-type boundaries.

Hwang, Shih, and DeCani Spending Function

This flexible family of functions is given by:

\alpha(t) = \begin{cases} \alpha \frac{1 - e^{-\gamma t}}{1 - e^{-\gamma}}, & \text{if } \gamma \ne 0 \\ \alpha t, & \text{if } \gamma = 0. \end{cases}

• When \gamma = -4, the spending function resembles O’Brien-Fleming boundaries.

• When \gamma = 1, it resembles Pocock boundaries.

Value

A vector of errors spent up to the interim look.

Author(s)

Kaifeng Lu, [e-mail]

Examples

errorSpent(t = 0.5, error = 0.025, sf = "sfOF")

errorSpent(t = c(0.5, 0.75, 1), error = 0.025, sf = "sfHSD", sfpar = -4)

lrstat documentation built on June 10, 2025, 1:07 a.m.