# 9.6 - Alpha Spending Function approach

A few drawbacks to the group sequential approach to interim statistical testing include the strict requirements that (1) the number of scheduled analyses, R, must be determined prior to the onset of the trial, and (2) there is equal spacing between scheduled analyses with respect to patient accrual. The alpha spending function approach was developed to overcome these drawbacks: (DeMets DL, Lan KK, 1994, Interim analysis: The alpha spending function approach, *Statistics in Medicine* 13: 1341-1352.)

Let τ denote the information fraction available during the course of a clinical trial. For example, in a clinical trial with a target sample size, *N*, in which treatment group means will be compared, the information fraction at an interim analysis is τ = *n*/*N*, where n is the sample size at the time of the interim analysis. If your target sample size is 500 and you have taken measurements on 400 patients then τ = .8

If the clinical trial involves a time-to-event endpoint, then the information fraction is τ = *d*/*D*, where *D* is the target number of events for the entire trial and *d* is the events that have occurred at the time of the interim analysis.

The alpha spending function, α(τ), is an increasing function. At the beginning of trial: t = 0 and α(t) = 0; at the end** **of trial: t = 1 and α(t) = α, the desired overall significance level. In other words, every time an analysis is performed, part of the overal alpha is "spent". For the *r*^{th} interim analysis, where the information fraction is τ_{r}, 0 ≤ τ_{r} ≤ 1, α(τ_{r}) determines the probability of any of the first r analyses leading to rejection of the null hypothesis when the null hypothesis is true. Obtaining the critical values consecutively requires numerically integrating the distribution function. A program is available in this module, along with the Demets-Lan paper.

As a simple example, suppose investigators are planning a trial in which patients are examined every two weeks over a 12-week period. The investigators would like to incorporate an interim analysis when one-half of the subjects have completed at least one-half of the trial. This corresponds to τ = 0.25.

A simple spending function that is a compromise between the Pocock and O'Brien-Fleming functions, is α(τ) = τα, 0 ≤ τ ≤ 1. This leads to a significance level of 0.012 at the interim analysis and a significance level of 0.04 at the final analysis (calculations not shown). Many variations of spending functions have been devised.

Regardless of whether a sequential, group sequential, or alpha spending function approach is invoked, the estimates of a treatment effect will be biased when a trial is terminated at an early stage. The earlier the decision, the larger the bias. Intuitively, if target sample size is 200 and the trial terminates after 25 patients because of a significant different between treatment groups, you recognize the potential for a lot of bias in this situation. Are 25 patients a representative sample from the population?