Power Calculation Under an Alternative Drift

Why the Alternative Changes the Calculation

A group sequential boundary is crossed by a sequence of correlated statistics, not by a single final statistic. Under the null hypothesis \(H_0\), a Gaussian transition model describes how the statistic moves between planned looks. To calculate power for a specified treatment effect, the transition variances and dependence structure remain the same under the canonical model, but the transition means change. Leaving the null transition means unchanged would calculate null crossing probabilities, not power.

This tutorial uses a fixed treatment effect \(\delta\), measured in the units of the treatment-effect parameter, and cumulative information \(I_k\) at look \(k\). The direction of the effect is chosen so that a positive \(\delta\) favors efficacy, which is assessed by crossing an upper boundary. Information is assumed to be known at each look.

This model assumes a stable treatment effect across the information range, a correctly specified information scale, and Gaussian score increments with independent increments over disjoint intervals. The resulting joint distribution of the \(Z_k\) values is not independent: earlier observations contribute to later statistics. The recursion accounts for that dependence through the conditional transition distribution.

Deriving the Alternative-Drift Transition

Let \(I_{k-1}<I_k\), with \(Z_{k-1}=z\). The score increment \(S_k-S_{k-1}\) is independent of the previous score and has mean \(\delta(I_k-I_{k-1})\) and variance \(I_k-I_{k-1}\). Since \(S_{k-1}=z\sqrt{I_{k-1}}\), adding the increment and standardizing by \(\sqrt{I_k}\) gives the transition.

The first term in the conditional mean carries forward the previous statistic. The second is the mean contribution from information newly added at look \(k\). The conditional variance is the variance of that new score increment after standardization. Under \(H_0\), set \(\delta=0\); under the alternative, retain the specified treatment effect in every transition.

If information is expressed as fractions \(t_k=I_k/I_K\) of final information, it is convenient to write \(\theta=\delta\sqrt{I_K}\). Then the mean of \(Z_k\) is \(\theta\sqrt{t_k}\), and the conditional transition becomes

Here \(\theta\) is the effect measured in standard-error units at final information; it is not the treatment effect in its original measurement units. The formula also shows why simply adding \(\delta\sqrt{I_k-I_{k-1}}\) to a standardized statistic is incorrect: the new score contribution must be divided by \(\sqrt{I_k}\), the denominator for the statistic at the current look.

Recursing the Probability of Efficacy Crossing

Let \(u_k\) be the upper efficacy boundary at look \(k\). If there is also a lower stopping boundary \(l_k\), the continuation region is \(C_k=(l_k,u_k)\); with an upper efficacy boundary only, take \(C_k=(-\infty,u_k)\). The survivor density \(f_k(z)\) is an unnormalized density: its integral is the probability of having reached look \(k\) without having crossed a stopping boundary. It is not a density conditional on survival.

At the first look, the process has its alternative marginal distribution, restricted to the continuation region. At each later look, the survivor density is propagated through the alternative transition and restricted again to the new continuation region. Probability entering the upper stopping region is counted as efficacy crossing. With a competing lower stopping boundary, probability entering that lower region is not counted as efficacy crossing.

Here \(\phi(x;m,v)\) is the normal density with mean \(m\) and variance \(v\), \(m_k(z)\) and \(v_k\) are the conditional mean and variance in the transition formula, and \(\mathbf{1}\{\cdot\}\) is an indicator. The upper-crossing mass at the first look is the probability under the alternative that \(Z_1\ge u_1\). At later looks, the crossing mass is obtained by integrating the propagated density over \(y\ge u_k\). The cumulative power is the sum of these upper-crossing masses over the planned looks. If there is no competing lower stopping boundary and the final look stops everyone remaining, power can equivalently be calculated as one minus the final survivor mass.

Worked Calculations

Numerical Checks, Alternatives, and Limitations

In practice, the integrals in the recursion are evaluated numerically, for example by quadrature or a discretized state-space method. The earlier material on Gaussian transition integration and survivor-density recursion supplies the computational framework; the essential change here is to use the alternative conditional mean. At each look, verify that the transition variance is nonnegative, that the mean contains the increment-specific drift, and that the survivor mass does not exceed the mass entering the look.

• Check the information scale. Confirm that \(I_k\) is cumulative and increasing. The transition uses \(I_k-I_{k-1}\), not \(I_k\), for the new score increment.
• Check the drift direction. A beneficial effect should move the mean toward the efficacy boundary under the chosen sign convention. A negative effect may reduce upper-boundary power.
• Check probability accounting. Upper-crossing mass plus lower-crossing mass plus continuing survivor mass should account for the probability that has reached each look, up to numerical integration error.
• Check numerical stability. Refine the grid or quadrature and confirm that cumulative power changes negligibly at the precision being reported. A result outside \([0,1]\), or a materially changing result under modest refinement, signals a computational or implementation problem.

A common mistake is to keep the null transition mean, \(\sqrt{I_{k-1}/I_k}\,z\), and change only the final marginal distribution or boundary. That omits the alternative drift from each new score increment, so the recursive path probabilities are wrong. The correction is to add \(\delta(I_k-I_{k-1})/\sqrt{I_k}\) to every conditional mean, then integrate only over paths that have not already stopped.

This recursive approach respects the sequential stopping rule and can accommodate unequal information increments and look-specific boundaries. A fixed-time calculation using only the final marginal distribution is simpler, but it does not in general give power for a group sequential design. Direct multivariate-normal probability calculations can provide an alternative way to compute crossing probabilities, while simulation can be useful as a rough independent check; neither removes the need to specify the alternative and stopping regions correctly. The recursion also has limitations: it relies on the canonical Gaussian model and specified information times, and its numerical accuracy depends on the integration method. If the treatment effect varies over time, information is misspecified, or the test statistic has materially non-Gaussian behavior, the resulting power may not describe the trial well.