Futility Spending With Unequal Information Fractions

Why Unequal Information Fractions Matter

Interim analyses are often scheduled at unequal intervals in calendar time. That fact alone does not tell us how much information has accumulated at each look. Enrollment may be faster or slower than expected, event accrual may vary, or measurements may arrive on a different schedule. Futility boundaries are calibrated using the statistical information available at each analysis, not the number of days since the trial began.

This distinction matters because the standardized test statistics at different looks are dependent, with dependence determined by their information fractions. When a later boundary is calibrated, its target is the probability of a new futility crossing among paths that did not cross earlier. Unequal information fractions affect both that dependence and the conditional distribution used in the calculation.

For the calculations below, assume the usual canonical group sequential model: information is correctly represented by \(I_k\), the standardized statistics have a joint normal distribution, and their increments are independent when expressed on the information scale. Under the design alternative, write \(\theta=\delta\sqrt{I_K}\), where \(\delta\) is the alternative effect on the scale used to standardize the statistic. Then \(Z_k\) has mean \(\theta\sqrt{t_k}\) and variance 1, while for \(j<k\),

These assumptions are the basis for the standard joint-normal calibration. In practice, the statistic and information measure must be appropriate for the trial; if the canonical model is a poor description, the calculated boundaries may not achieve their nominal operating characteristics.

Calibrating the New Crossing at Each Look

Let \(l_k\) be the lower futility boundary at look \(k\). A path crosses for futility at look \(k\) if \(Z_k\leq l_k\) and it has not crossed at any earlier look. Thus the beta-spending calibration is

where \(P_A\) denotes probability under the design alternative. Summing these mutually exclusive first-crossing probabilities gives \(B(t_k)\), the cumulative target through look \(k\). At the first look, the equation is simply \(P_A(Z_1\leq l_1)=B(t_1)\). At a later look, the equation must exclude paths that already crossed.

For two looks, put \(\rho=\sqrt{t_1/t_2}\), \(\mu_k=\theta\sqrt{t_k}\), and \(s=\sqrt{1-\rho^2}\). The joint normal model implies

where \(\phi\) and \(\Phi\) are the standard normal density and distribution functions. To see why, condition on \(Z_2=\mu_2+y\). The conditional distribution of \(Z_1\) has mean \(\mu_1+\rho y\) and variance \(1-\rho^2\). The factor involving \(\Phi\) is therefore the conditional probability that the first-look statistic is above its futility boundary. Integrating over values of \(Z_2\) below \(l_2\) yields the probability of a new crossing.

Unequal Calendar Spacing Is Not Unequal Beta Spending

The spending function is evaluated at the observed or planned information fractions. It is not evaluated at the fraction of calendar time elapsed. Nor does an uneven calendar schedule automatically imply a front-loaded or back-loaded beta-spending schedule. Those are distinct design choices: the calendar determines when analyses occur, while \(B(t)\) determines the cumulative crossing target as information accrues.

Marginal Probability Is Not a New-Crossing Probability

A common shortcut is to choose \(l_k\) so that \(P_A(Z_k\leq l_k)=\Delta B_k\). That is generally incorrect after the first look. The marginal probability includes paths that crossed at earlier analyses. The target \(\Delta B_k\), by contrast, applies only to surviving paths. A marginal quantile can be useful for a first look or for a look with no earlier possibility of crossing, but later boundaries require the joint crossing calculation.

Calculation Checks and Design Limitations

A numerical solution is credible only if its inputs and event definition match the design. For a planned schedule, verify that information fractions are strictly increasing and that the cumulative spending values are nondecreasing. Each increment \(\Delta B_k\) must be nonnegative. Also check that the requested new-crossing probability is attainable: it cannot exceed the probability of reaching look \(k\) without a previous futility crossing. A solver failure or a boundary that moves unexpectedly can signal an incoherent spending schedule, incorrect information fractions, or a mistake in the survival event.

For more than two looks, the event at look \(k\) requires survival of every earlier lower boundary. The joint probability can be evaluated using the canonical multivariate normal distribution or by integrating sequential conditional distributions. Using only the immediately preceding statistic is justified when the canonical independent-increment model gives the required Markov conditional structure; the full event must still exclude crossings at all earlier looks.

Compared with direct joint-probability calibration, a marginal-quantile rule is simpler but does not generally spend the specified incremental beta after the first look. Assigning equal beta increments at equally spaced calendar dates is another tempting substitute; it is valid only if those increments are actually the chosen spending schedule at the information fractions, not merely because the dates are evenly spaced. The standard beta-spending approach is flexible about timing, but it depends on a correctly specified information scale and design alternative. If the effect used for calibration is poorly chosen, or if information fractions differ materially from the planned values, actual futility probabilities can differ from the design targets.