Overall Power Across Sequential Logrank Looks

Why Overall Power Is a Joint Probability

A sequential logrank design can reject at an interim look or continue to a later look, depending on the statistics and prespecified boundaries. The probability of rejection by the end of monitoring therefore depends on the joint path of the cumulative statistics. The marginal power at each look does not, by itself, give the overall power: cumulative statistics are correlated, and some paths stop before later looks.

This tutorial uses the canonical normal approximation for planned information times. It assumes the event information at each look is specified, the large-sample approximation is adequate, and the logrank score increments have the independent-increment structure needed for the canonical model. In a proportional-hazards planning calculation, a specified alternative is represented by a constant drift. If these assumptions are poor, the resulting power can be inaccurate; the calculation is not a substitute for checking the model and design assumptions.

Definition: Let $I_1<I_2<\cdots<I_K$ be the planned information levels, and let $Z_k$ be the cumulative, standardized logrank statistic at look $k$, oriented so that larger values favor the treatment benefit being tested. Under the canonical approximation and a specified alternative with drift $\theta$, the vector $(Z_1,\ldots,Z_K)$ is multivariate normal with $$ E(Z_k)=\theta\sqrt{I_k}, \qquad \operatorname{Corr}(Z_j,Z_k)=\sqrt{\frac{I_j}{I_k}}\quad (j<k). $$ Here $\theta$ is the alternative drift per square root of information. A one-sided efficacy boundary $u_k$ is crossed at look $k$ if $Z_k\ge u_k$. Overall power is the probability that the sequential stopping rule rejects by the final look under the specified alternative.

The correlation is essential: $Z_k$ includes accumulated information from earlier events, so it is not a new independent statistic at each look. Equivalently, the unstandardized cumulative score at information $I_k$ has mean $\theta I_k$ and variance $I_k$; independent increments of that score produce the stated correlation after standardization. These are the canonical joint-distribution and independent-increment assumptions used in the earlier material on sequential logrank statistics.

Writing the Rejection Probability

First consider efficacy monitoring without futility stopping. Rejection by look $K$ occurs if at least one efficacy boundary is crossed. Its complement is the event that every look remains below its boundary:

\text{Power} =P\!\left(\bigcup_{k=1}^{K}\{Z_k\ge u_k\}\right) =1-P(Z_1<u_1,\ldots,Z_K<u_K).

The final probability is a multivariate normal probability with means $\theta\sqrt{I_k}$ and the canonical correlations. This expression automatically accounts for overlap among look-specific crossing events. Adding marginal crossing probabilities would count some paths more than once.

With a futility boundary $l_k$, where $l_k<u_k$, a path that falls below $l_k$ stops without rejection and cannot cross a later efficacy boundary. For efficacy rejection at look $j$, the path must remain in the continuation region at every earlier look and cross efficacy at $j$. These first-rejection events are disjoint, so their probabilities can be added:

\text{Power} =\sum_{j=1}^{K} P\!\left( l_i<Z_i<u_i\ \text{for }i<j,\ Z_j\ge u_j \right).

If a design has no futility boundary at a look, the corresponding lower restriction is omitted. For two-sided monitoring, the efficacy event at a look is instead the prespecified two-sided rejection region, such as $Z_k\ge u_k$ or $Z_k\le -u_k$; the same joint-probability principle applies.

Justification: Why Marginal Powers Cannot Be Added

At a single look, the marginal probability of crossing an upper boundary under the canonical model is

P(Z_k\ge u_k)=1-\Phi\!\left(u_k-\theta\sqrt{I_k}\right),

where $\Phi$ is the standard normal cumulative distribution function. This is the power at that look if the statistic is considered marginally. Across looks, however, the events overlap. For two looks, the probability of either crossing is

P(Z_1\ge u_1)+P(Z_2\ge u_2) -P(Z_1\ge u_1,\ Z_2\ge u_2).

The joint term is not generally zero. With more looks, intersections of multiple crossing events also matter. The complement formulation or a sum over disjoint first-rejection paths handles these dependencies directly.

Worked Example: Power at One Planned Look

Suppose a hypothetical one-sided logrank design has planned information $I_1=100$, alternative drift $\theta=0.30$ per square root of information, and efficacy boundary $u_1=2.00$. Assume the canonical normal approximation is appropriate and there is no other stopping rule.

The statistic has mean $\theta\sqrt{I_1}=0.30\sqrt{100}=3.00$ and variance 1. The rejection probability is

P(Z_1\ge2.00) =1-\Phi(2.00-3.00) =1-\Phi(-1.00) =\Phi(1.00) \approx0.8413.

Thus the power at this sole look, and therefore the overall power, is about 84.1% under the specified alternative. This calculation is the one-look special case; it does not require a joint probability because there is only one statistic.

Worked Example: Two Correlated Looks

Consider a hypothetical two-look design with $I_1=25$, $I_2=100$, and drift $\theta=0.20$. For a transparent illustration of dependence, set the upper boundaries equal to the alternative means: $u_1=1.00$ and $u_2=2.00$. There is no futility boundary. This boundary choice is for demonstrating the probability calculation, not a recommended design.

The means are $0.20\sqrt{25}=1.00$ and $0.20\sqrt{100}=2.00$, so $X_k=Z_k-E(Z_k)$ has mean zero at each look. The correlation is

\rho_{12}=\sqrt{\frac{25}{100}}=0.50.

No rejection means $X_1<0$ and $X_2<0$. For two standard normal variables with correlation $\rho$, the probability that both are nonpositive is $1/4+\arcsin(\rho)/(2\pi)$. Therefore,

P(\text{no rejection}) =\frac14+\frac{\arcsin(0.50)}{2\pi} =\frac14+\frac{\pi/6}{2\pi} =\frac14+\frac1{12} =\frac13.

The overall power is $1-1/3=2/3$, or approximately 66.7%. Each marginal crossing probability is 0.50, so adding them would give 1.00, not the overall power. The difference arises because some paths cross both boundaries; they are one rejection, not two.

Worked Example: Three Looks and the Joint Orthant Probability

In another hypothetical illustration, let $I_1=25$, $I_2=50$, $I_3=100$, and $\theta=0.20$. Set each upper boundary equal to its alternative mean: $u_1=1.00$, $u_2=0.20\sqrt{50}\approx1.414$, and $u_3=2.00$. There is no futility stopping. These boundaries again simplify the calculation to illustrate how the joint distribution determines overall power.

After subtracting the means, the no-rejection event is that all three centered statistics are nonpositive. Their correlations are

\rho_{12}=\sqrt{\frac{25}{50}}=\frac{1}{\sqrt2},\qquad \rho_{13}=\sqrt{\frac{25}{100}}=\frac12,\qquad \rho_{23}=\sqrt{\frac{50}{100}}=\frac{1}{\sqrt2}.

For three centered, jointly normal variables with unit variances, the probability that all are nonpositive is

\frac18+ \frac{\arcsin(\rho_{12})+\arcsin(\rho_{13})+\arcsin(\rho_{23})}{4\pi}.

To justify this identity, write each nonpositive indicator as $(1-\operatorname{sgn}(X_i))/2$ and expand the product. By central symmetry, the single-sign and three-sign expectations are zero. For a bivariate standard normal pair, representing the pair as $U$ and $\rho U+\sqrt{1-\rho^2}V$, with independent standard normal $U,V$, shows that the probability of different signs is $\arccos(\rho)/\pi$: the angle of $(U,V)$ is uniform, and the sign-disagreement sectors have total angle $2\arccos(\rho)$. Hence $E[\operatorname{sgn}(X_i)\operatorname{sgn}(X_j)]=2\arcsin(\rho_{ij})/\pi$. Substitution in the expanded product gives the displayed three-variable formula.

Here the inverse sines are $\pi/4$, $\pi/6$, and $\pi/4$. Their sum is $2\pi/3$, so

P(\text{no rejection}) =\frac18+\frac{2\pi/3}{4\pi} =\frac18+\frac16 =\frac7{24}.

Thus overall power is $1-7/24=17/24\approx0.7083$, or 70.8%. Each look has marginal crossing probability 0.50; summing those three marginal probabilities would give 1.50 and is not a valid overall-power calculation.

Computing General Designs and Checking the Result

For ordinary boundaries, the multivariate normal probability usually has no simple closed form. It can be evaluated by numerical multivariate normal integration or by integrating the conditional distribution of each look given the preceding statistics. A useful computational representation is the independent score-increment model. If $\Delta I_k=I_k-I_{k-1}$, with $I_0=0$, the score increment over that interval has mean $\theta\Delta I_k$, variance $\Delta I_k$, and is independent of earlier increments under the canonical assumptions. Cumulative scores are formed by adding these increments and dividing by $\sqrt{I_k}$ to obtain $Z_k$. In computation, paths that cross efficacy are counted as rejections; paths that cross futility are stopped and counted as non-rejections.

Several checks help detect errors:

Check the covariance structure. For $j<k$, the canonical correlation must be $\sqrt{I_j/I_k}$, between zero and one. Later cumulative statistics are not independent of earlier ones.
Check path handling. If there is futility stopping, a simulated or integrated path must not be allowed to reach later looks after it has stopped.
Check the probability scale. Overall power must lie between zero and one. With efficacy-only stopping, it must be at least as large as the marginal crossing probability at any one included look, because each such crossing is part of the overall rejection event.
Check limiting behavior. If later looks are removed, the calculation should reduce to the one-look power. With futility rules held fixed, making efficacy boundaries harder to cross cannot increase power. Changing futility boundaries changes which paths continue, so its effect on power is not generally monotone.

The principal alternative to joint integration is simulation from the canonical model: generate independent normal score increments, construct the cumulative $Z_k$ values, apply the stopping rule, and estimate power by the fraction of paths that reject. Simulation is flexible and accommodates complicated rules, but its estimate has Monte Carlo error and requires enough paths for adequate precision. Direct multivariate normal integration is often more precise for a modest number of looks, while a validated recursive computation can be efficient for many looks. None of these methods repairs a misspecified drift, incorrect information levels, or an unsuitable normal approximation.

Limitations and Common Mistakes

The canonical calculation treats information levels and the joint normal model as an adequate representation of the planned design. In an event-driven trial, realized event counts can be discrete, risk sets can change, and actual information can differ from planned information. Those features may affect boundary crossing probabilities, especially when event counts are small. The continuous-information calculation should therefore be interpreted as an approximation under its assumptions, not as an exact guarantee for every realized data path.

A common mistake is to calculate a power at each look and add the values. The mistaken reasoning is that every look offers another independent chance to reject. The correction is that the statistics are cumulative and correlated, and a sequential test rejects at most once; overall power is the probability of a union of crossing events, with stopping rules applied along the path. A related mistake is to use only the final-look power. That omits paths that crossed an efficacy boundary earlier and stopped successfully.

Key takeaway: Overall sequential logrank power is a joint path probability under the specified alternative. Use the canonical means and correlations, apply efficacy and futility boundaries in their actual order, and report the probability of rejection by any permitted look.

Check Your Understanding

Try each question, then compare your reasoning with the solutions.

For two looks at information $I_1=36$ and $I_2=144$, what is the canonical correlation between the cumulative statistics?
Under the alternative, each of three looks has marginal crossing probability 0.40. Can overall power be obtained by adding these three values? Why or why not?
For a one-look design with $I_1=64$, $\theta=0.25$, and $u_1=2.00$, calculate power using the canonical approximation.
In a design with a futility boundary at the first look, what must be done with paths that cross that boundary when calculating rejection probability at the second look?

Solutions. (1) The correlation is $\sqrt{36/144}=0.50$. (2) No. The look statistics are correlated, crossing events can overlap, and stopping rules affect which later looks are reached; overall power must be calculated as a joint path probability. (3) The mean is $0.25\sqrt{64}=2.00$, so power is $1-\Phi(2.00-2.00)=0.5000$. (4) Stop those paths at look 1 and count them as non-rejections; they cannot contribute a later rejection.