Obtains the error spent at given spending times for the specified error spending function.
errorSpent(t, error, sf = "sfOF", sfpar = NA)A vector of errors spent up to the interim look.
A vector of spending times, typically equal to information fractions.
The total error to spend.
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".
The parameter for the spending function. Corresponds to rho for "sfKD" and gamma for "sfHSD".
Kaifeng Lu, kaifenglu@gmail.com
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.
errorSpent(t = 0.5, error = 0.025, sf = "sfOF")
errorSpent(t = c(0.5, 0.75, 1), error = 0.025, sf = "sfHSD", sfpar = -4)
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