Conditional Power with Several Remaining Looks

Why Several Future Looks Require a Joint Calculation

After an interim analysis, a trial may still have several planned opportunities to stop for efficacy. Conditional power asks for the probability of meeting a specified success criterion, given the data available now and an assumed treatment effect for the future. With more than one remaining look, the criterion may be crossed at an early future analysis, at a later one, or at either. The events are dependent: a favorable future increment affects more than one cumulative test statistic.

The calculation therefore needs the joint conditional distribution of the remaining statistics. Adding their separate crossing probabilities generally counts outcomes more than once, while multiplying their non-crossing probabilities incorrectly assumes independence. The canonical normal model provides a direct way to obtain the joint distribution and evaluate the probability of crossing any prespecified efficacy boundary.

Conditional Distribution of the Remaining Statistics

Assume a one-sided efficacy test with known variance and planned information values $I_1<\cdots<I_K$. At look $j$, the canonical statistic is $Z_j=U_j/\sqrt{I_j}$, where $U_j$ is the cumulative score. Under a fixed assumed standardized effect $\theta$, the score has independent increments: an increment from information $I_a$ to $I_b$ has mean $\theta(I_b-I_a)$ and variance $I_b-I_a$. Thus $U_j$ has mean $\theta I_j$ and variance $I_j$. These assumptions are the normal, known-variance canonical model; the results below do not account for uncertainty from estimating the variance.

Definition: Suppose the trial has reached look $r$, with observed canonical statistic $Z_r=z_r$, and let $j,k>r$ be future looks. Under the fixed assumed effect $\theta$, $$ E(Z_j\mid Z_r=z_r)=\frac{\sqrt{I_r}\,z_r+\theta(I_j-I_r)}{\sqrt{I_j}}, $$ and $$ \operatorname{Cov}(Z_j,Z_k\mid Z_r=z_r) =\frac{I_{\min(j,k)}-I_r}{\sqrt{I_j I_k}}. $$ In particular, the conditional variance at look $j$ is $(I_j-I_r)/I_j$.

The conditional mean combines the observed interim score, $\sqrt{I_r}z_r$, with the expected future score under $\theta$. The covariance comes from the future score increments shared by the two cumulative statistics. The interim score is fixed once we condition on the interim data, so it contributes no conditional variance.

Let $c_j$ be the efficacy boundary for look $j$. The conditional power for crossing at least one of the remaining upper boundaries is $$ P\left(\bigcup_{j=r+1}^{K}\{Z_j\ge c_j\}\mid Z_r=z_r\right) =1-P(Z_{r+1}<c_{r+1},\ldots,Z_K<c_K\mid Z_r=z_r). $$ The probability on the right is a multivariate normal probability with the conditional means and covariance just given. This complement formulation handles overlapping crossing events automatically.

Why the Future Statistics Are Dependent

The dependence is easiest to see from the increments. Given the interim data, write the score at a future look $j$ as the observed interim score plus the sum of future increments through that look. A later score includes increments already present in each earlier future score. Although disjoint increments are independent, the cumulative statistics are not.

U_j=U_r+(U_j-U_r), \qquad \operatorname{Cov}(U_j-U_r,U_k-U_r\mid U_r) =I_{\min(j,k)}-I_r.

Dividing each score by its own standardizing factor, $\sqrt{I_j}$ or $\sqrt{I_k}$, gives the conditional covariance formula. This also shows why conditioning matters: the shared portion before look $r$ is no longer random, while shared increments after look $r$ remain random.

For a vector of future looks, arrange the conditional means in a vector $m$ and the conditional covariances in a matrix $\Sigma$. Standardize each statistic using its conditional standard deviation. The resulting correlation matrix describes how the remaining standardized statistics move together. The probability of no crossing is the multivariate normal probability below the corresponding standardized boundaries; conditional power is one minus that probability.

Worked Example: Finding the Conditional Joint Distribution

Suppose information is measured in units with $I_r=40$, and the interim result is $z_r=1.20$. The remaining looks are at $I_1=60$, $I_2=80$, and $I_3=100$. For this calculation, assume $\theta=0.30$. Find the conditional means, variances, and correlation between the first and third future statistics.

At $I_1=60$, $$ m_1=\frac{\sqrt{40}(1.20)+0.30(60-40)}{\sqrt{60}} =\frac{7.589+6}{7.746}\approx1.754, \qquad v_1=1-\frac{40}{60}=0.3333. $$

At $I_2=80$, $$ m_2=\frac{\sqrt{40}(1.20)+0.30(80-40)}{\sqrt{80}} =\frac{7.589+12}{8.944}\approx2.190, \qquad v_2=1-\frac{40}{80}=0.5000. $$

At $I_3=100$, $$ m_3=\frac{\sqrt{40}(1.20)+0.30(100-40)}{\sqrt{100}} =\frac{7.589+18}{10}\approx2.559, \qquad v_3=1-\frac{40}{100}=0.6000. $$

The conditional covariance between the first and third future statistics is $$ \operatorname{Cov}(Z_1,Z_3\mid Z_r) =\frac{60-40}{\sqrt{60(100)}}\approx0.2582. $$ Therefore their conditional correlation is $$ \rho_{13}=\frac{0.2582}{\sqrt{(0.3333)(0.6000)}}\approx0.5774. $$ The positive correlation reflects the future increments included in both statistics. These means and covariances, not three separate marginal distributions alone, are the inputs needed for a probability of crossing at any look.

Calculating a Crossing Probability

A small example makes the role of dependence explicit. For a bivariate standard normal pair with correlation $\rho$, the probability that both components are positive is $1/4+\arcsin(\rho)/(2\pi)$. This identity follows from the bivariate normal symmetry and the angular proportion of the positive-positive region; it is useful here because it gives an exact check on the joint calculation. In applications, boundaries are generally positive and the multivariate probability is evaluated using a normal probability routine or numerical integration.

Worked Example: Two Remaining Looks and an Exact Joint Probability

Suppose $I_r=0.25$, $Z_r=0$, and the assumed effect is $\theta=0$. The two remaining information values are $I_1=0.50$ and $I_2=1.00$. For a transparent calculation of the dependence, take the specified upper boundaries to be zero at both looks.

The conditional means are both zero. The conditional variances are $$ v_1=1-\frac{0.25}{0.50}=0.50, \qquad v_2=1-\frac{0.25}{1.00}=0.75. $$ The covariance is $$ \operatorname{Cov}(Z_1,Z_2\mid Z_r) =\frac{0.50-0.25}{\sqrt{0.50(1.00)}}=0.3536, $$ so the correlation is $0.3536/\sqrt{0.50(0.75)}=0.5774=1/\sqrt{3}$.

Because both conditional means and both boundaries are zero, the probability of crossing at either look equals the probability that at least one component of a centered bivariate normal pair is positive. By complementing the event that both are nonpositive and using symmetry, $$ P(Z_1>0\text{ or }Z_2>0\mid Z_r) =1-\left(\frac14+\frac{\arcsin(1/\sqrt{3})}{2\pi}\right) \approx1-(0.25+0.09796)=0.6520. $$ This is a mathematical illustration, not a realistic efficacy design: clinical efficacy boundaries are normally positive. It verifies that the probability of any crossing is a joint probability, not the sum of marginal crossing probabilities.

Worked Example: Three Remaining Looks and the Any-Crossing Event

Continue with $I_r=0.25$, $Z_r=0$, and $\theta=0$, but now plan three future looks at information values $0.50$, $0.75$, and $1.00$. Again use zero boundaries to obtain an exact joint calculation.

The conditional variances at the three looks are $0.50$, $2/3$, and $0.75$. Applying the covariance formula and standardizing gives the pairwise correlations $$ \rho_{12}=\sqrt{\frac{0.50-0.25}{0.75-0.25}}=\frac{1}{\sqrt{2}}, \quad \rho_{13}=\sqrt{\frac{0.50-0.25}{1.00-0.25}}=\frac{1}{\sqrt{3}}, \quad \rho_{23}=\sqrt{\frac{0.75-0.25}{1.00-0.25}}=\sqrt{\frac{2}{3}}. $$ For a centered trivariate normal vector, the probability that all three components are positive is $$ \frac18+\frac{\arcsin(\rho_{12})+\arcsin(\rho_{13})+\arcsin(\rho_{23})}{4\pi}. $$ Here the three inverse sines sum to $3\pi/4$, so this probability is $1/8+3/16=5/16=0.3125$. By symmetry, the probability all three are nonpositive is also $0.3125$. Thus the conditional probability of crossing at least one zero boundary is $$ 1-\frac{5}{16}=\frac{11}{16}=0.6875. $$ The calculation includes all three looks jointly. It does not require counting each boundary-crossing event separately and correcting for every overlap.

Using Realistic Boundaries and Checking the Calculation

With positive efficacy boundaries, there is usually no simple orthant identity. For each future look, calculate its conditional mean and standard deviation, then form the standardized boundary $a_j=(c_j-m_j)/\sqrt{v_j}$. The probability of no crossing is the multivariate normal probability that each standardized residual is below its corresponding $a_j$, using the conditional correlation matrix. A reliable implementation should use the whole vector of thresholds and the full correlation matrix.

Fix the conditioning information and effect assumption.
Record the current information $I_r$, observed statistic $z_r$, and the assumed future effect $\theta$. Conditional power is always tied to this effect assumption.

Build the conditional normal distribution.
Compute each future mean and every covariance from the information values. Check that variances are positive and that the covariance matrix is symmetric and positive semidefinite.

Specify the success event.
Use the actual prespecified boundary at each remaining look. For upper efficacy boundaries, the event is crossing at least one boundary; for a two-sided rule, include the relevant lower and upper crossing regions.

Evaluate jointly and check.
Compute one minus the probability of no crossing. Compare the result with marginal probabilities and bounds: it must be at least as large as each individual crossing probability and no greater than their sum or one.

A common pitfall is to calculate the marginal chance of crossing at each future look and then treat those chances as independent. For example, multiplying the marginal non-crossing probabilities is not justified: the future canonical statistics share score increments. The correction is to use their conditional covariance matrix and evaluate the joint non-crossing probability. Another tempting shortcut is to update the assumed effect after seeing the interim estimate without saying so. That changes the conditional-power scenario; it is valid only when explicitly intended, not as an unnoticed substitution for a prespecified design effect.

For a two-sided success criterion, the event must be defined with care: at each look it may include crossing either the upper or lower boundary, and the conditional probability is the probability of the union of all such regions. The same joint-distribution principle applies, but a single upper-tail complement is no longer sufficient. The earlier material on two-sided success criteria describes that success event; here the essential additional step is accounting for dependence across all remaining looks.

Diagnostics, Alternatives, and Limitations

Several checks can reveal an incorrect calculation. Conditional variances must lie between zero and one in this canonical model, and correlations must be consistent with the information ordering. A covariance matrix that is not positive semidefinite usually signals a formula, indexing, or rounding error. The calculated any-crossing probability must be no smaller than the largest marginal crossing probability. If numerical integration reports a value outside these bounds, check the ordering of thresholds, the conditional means, and whether the software expects correlations or covariances.

The joint calculation is more exact than summing marginal probabilities, but numerical evaluation can become demanding with many remaining looks. Simulation from the specified conditional multivariate normal distribution is an alternative, especially for complex success regions; its Monte Carlo error should be reported or controlled. A simple sum of marginal probabilities is only an upper bound (capped at one), not the conditional power, and can be quite loose when looks are strongly correlated.

The result is conditional on the canonical normal model, the information schedule, the specified stopping boundaries, and the chosen effect $\theta$. It does not automatically account for variance estimation, uncertainty about the effect, or changes to the design after the interim analysis. Those features require additional modeling rather than an informal adjustment to this calculation.

Key takeaway: Conditional power across several remaining looks is the probability of a union of boundary-crossing events under the conditional joint distribution. Calculate the future means and covariance matrix from independent score increments, then evaluate the joint success region; do not assume that looks are independent.

Check Your Understanding

Answer each question using the conditional canonical model described above.

At interim information $I_r$, why does the conditional covariance between two future statistics depend only on information accumulated after $I_r$?
There are two remaining looks with marginal crossing probabilities 0.08 and 0.10. What can be said about the probability of crossing at either look without knowing their joint distribution?
For three future looks, what probability should be subtracted from one to obtain the chance of crossing at least one upper efficacy boundary?
Why is multiplying the marginal non-crossing probabilities generally incorrect?
If a computed conditional covariance matrix is not positive semidefinite, name two inputs or steps that should be checked.

Solutions. (1) Conditioning fixes the score accumulated through $I_r$, leaving only future shared increments random; their shared information is $I_{\min(j,k)}-I_r$. (2) The any-crossing probability is at least 0.10 and at most 0.18, the sum of the marginal probabilities; its exact value needs their joint distribution. (3) Subtract the joint probability that all three future statistics remain below their respective boundaries. (4) The future statistics share increments and are dependent. (5) Check the information indices and covariance formula, and check that rounding has not materially altered a matrix that should be positive semidefinite.