Predictive Success for Event-Driven Time-to-Event Trials

From Current Events to a Final Decision

An event-driven trial continues until a prespecified number of events has accrued. At an interim analysis, the observed events provide current evidence about the treatment effect, but they do not determine the final result: future events will add information and may move the posterior distribution. Predictive success is the probability, given the data observed so far, that the final posterior will satisfy a prespecified success rule.

For a time-to-event endpoint, it is common to express the treatment effect as the log hazard ratio $\theta=\log(HR)$. A hazard ratio below 1 favors treatment under the convention that the treatment hazard is in the numerator. A decision rule might require a high final posterior probability that the hazard ratio is below a clinically specified margin. Predictive success averages the indicator of that final decision over possible future event data; it is not the current posterior probability of benefit.

Definition: Let $D_C$ denote the current trial data and $D_F$ the future data expected before the final analysis. If the final go rule is $P(\theta<\delta\mid D_C,D_F)\geq\gamma$, where $\delta$ is the log hazard-ratio margin and $\gamma$ is the required posterior probability, then predictive success is $$ P\!\left\{P(\theta<\delta\mid D_C,D_F)\geq\gamma\mid D_C\right\}. $$ The outer probability averages over future data under the model conditional on current data. The inner probability is the posterior probability used by the final decision rule.

For example, if the final rule is $P(HR<0.80\mid D_C,D_F)\geq0.95$, then $\delta=\log(0.80)$ and $\gamma=0.95$. The prediction asks how often the eventual posterior probability will reach 0.95, not how likely the true hazard ratio is to satisfy the margin according to the current posterior.

A Normal Information Model

A useful approximation treats the current posterior for the log hazard ratio as normal and represents future event information by a normal likelihood. Write $\theta\mid D_C\sim N(m,v)$, where $m$ is the current posterior mean and $v$ its variance. Let $I_F$ be the anticipated information from future events. The corresponding future estimate $Z_F$, conditional on $\theta$, is modeled as $Z_F\mid\theta\sim N(\theta,1/I_F)$. This is a summary-data approximation to updating with the future events, not a claim that each event contributes an identical independent observation in every survival model.

Assume the future likelihood is conditionally independent of the current data given $\theta$, and use the same normal likelihood approximation in the final posterior update. Then the final posterior variance and mean, conditional on a possible future estimate $Z_F=z$, are

v_F=\left(\frac{1}{v}+I_F\right)^{-1}, \qquad m_F(z)=v_F\left(\frac{m}{v}+I_Fz\right).

Under this normal posterior, the final posterior probability below the margin is $P(\theta<\delta\mid D_C,Z_F=z)=\Phi\!\left((\delta-m_F(z))/\sqrt{v_F}\right)$, where $\Phi$ is the standard normal cumulative distribution function. Thus the final go rule is equivalent to a bound on the final posterior mean:

m_F(z)\leq \delta-z_\gamma\sqrt{v_F}, \qquad z_\gamma=\Phi^{-1}(\gamma).

The future estimate itself has the posterior predictive distribution $Z_F\mid D_C\sim N(m,v+1/I_F)$. To see why, write $Z_F=\theta+\varepsilon_F$, where $\theta\mid D_C\sim N(m,v)$ and $\varepsilon_F\sim N(0,1/I_F)$ is independent of $\theta$. Their sum is normal with mean $m$ and variance $v+1/I_F$. Therefore, predictive success is the predictive probability that $Z_F$ falls on the side of its threshold that makes the final posterior mean sufficiently favorable.

Key result: In the normal information model, calculate the future-estimate cutoff $z^*$ by solving $m_F(z^*)=\delta-z_\gamma\sqrt{v_F}$. Since $m_F(z)$ increases with $z$, predictive success is $$ P(Z_F\leq z^*\mid D_C) = \Phi\!\left(\frac{z^*-m}{\sqrt{v+1/I_F}}\right). $$ This form applies to a rule requiring $\theta<\delta$; for a rule requiring $\theta>\delta$, the inequality and tail direction are reversed.

How Event Counts Supply Information

In an event-driven design, the anticipated number of additional events is often used to approximate $I_F$. With proportional hazards, randomized treatment, roughly balanced treatment allocation within the risk sets, and a standard large-sample log-rank or Cox-model approximation, the information for a log hazard ratio is approximately $D r(1-r)$. Here $D$ is the number of events contributing information and $r$ is the treatment allocation fraction. Under equal allocation, this becomes $I\approx D/4$, corresponding to an approximate standard error $2/\sqrt{D}$.

This is an information approximation, not a universal identity. The allocation fraction among people at risk can change over time, and covariate adjustment, censoring, stratification, or departures from proportional hazards can change the information supplied by a given event count. When a design has a reliable planned information calculation, use it in place of the simple $D/4$ approximation. Under the simple approximation, if $D_C$ events have occurred and the final target is $D_T$, then $I_C\approx D_C/4$ and $I_F\approx(D_T-D_C)/4$. The current posterior should already reflect the observed data and prior; do not add current event information a second time to that posterior.

Worked Example: Predictive Success With 100 Future Events

Suppose a hypothetical trial has current posterior $\theta\mid D_C\sim N(-0.30,0.04)$. The final analysis is expected to add 100 approximately balanced events, so use $I_F=100/4=25$. The final rule requires $P(HR<0.80\mid\text{final data})\geq0.95$. The normal information model is assumed to be an adequate approximation, the event target is treated as fixed, and the future likelihood is conditionally independent of current data given $\theta$.

The margin is $\delta=\log(0.80)=-0.22314$. The final posterior variance and standard deviation are $v_F=(1/0.04+25)^{-1}=1/50=0.02000$ and $\sqrt{v_F}=0.14142$. For $\gamma=0.95$, $z_\gamma=1.64485$, so the final posterior mean must satisfy $m_F\leq-0.22314-(1.64485)(0.14142)=-0.45576$.

For a future estimate $z$, the final posterior mean is $m_F(z)=\{(-0.30)/0.04+25z\}/50=(-0.30+z)/2$. Therefore success requires $(-0.30+z)/2\leq-0.45576$, or $z\leq-0.61152$. The predictive standard deviation of $Z_F$ is $\sqrt{0.04+1/25}=\sqrt{0.08}=0.28284$. Hence

P(\text{final success}\mid D_C) = \Phi\!\left(\frac{-0.61152-(-0.30)}{0.28284}\right) = \Phi(-1.101) \approx 0.135.

Under these assumptions, the probability of meeting the final 0.95 posterior criterion is about 13.5%. This is not a 13.5% posterior probability that the treatment is beneficial; it is a predictive probability of a future decision-rule outcome.

Worked Example: Updating From a Current Event Summary

Consider a second hypothetical balanced trial. A normal prior for $\theta$ is $N(0,0.25)$. The current event data are summarized by a log hazard-ratio estimate $y_C=-0.35$ with approximate likelihood $y_C\mid\theta\sim N(\theta,0.10)$. Suppose 60 additional balanced events are anticipated, and the final rule again requires posterior probability at least 0.95 that $HR<0.80$. We assume the likelihood summary and event-information approximation are suitable for this illustration.

The current posterior precision is $1/0.25+1/0.10=4+10=14$, giving $v=1/14=0.07143$ and $m=\{0(4)+(-0.35)(10)\}/14=-0.25$. The 60 future events provide approximate information $I_F=60/4=15$. The final posterior has $v_F=1/(14+15)=1/29=0.03448$ and standard deviation $0.18570$.

The required final posterior mean is at most $-0.22314-(1.64485)(0.18570)=-0.52859$. For future estimate $z$, $m_F(z)=\{14(-0.25)+15z\}/29=(-3.5+15z)/29$. Solving $(-3.5+15z)/29\leq-0.52859$ gives $z\leq-0.78860$. Under the current posterior, $Z_F$ has mean $-0.25$ and variance $0.07143+1/15=0.13810$, so its standard deviation is $0.37161$. The predictive probability is

\Phi\!\left(\frac{-0.78860-(-0.25)}{0.37161}\right) = \Phi(-1.449) \approx 0.074.

The current posterior probability that $HR<0.80$ is only $\Phi\{(-0.22314+0.25)/\sqrt{0.07143}\}=\Phi(0.1005)\approx0.540$. The predictive probability of eventual success is lower still, about 7.4%, because the final rule requires the posterior probability to reach 0.95. The two quantities answer different questions.

Worked Example: A Different Final Posterior Threshold

Suppose a hypothetical trial currently has $\theta\mid D_C\sim N(-0.40,0.04)$, and expects 40 more approximately balanced events. Let the final rule require $P(HR<0.80\mid\text{final data})\geq0.90$. Assume the normal information model and a fixed remaining event target. The future information is $I_F=40/4=10$, so $v_F=1/(25+10)=1/35=0.02857$, with standard deviation $0.16903$.

Here $z_{0.90}=1.28155$, and the final posterior mean must be at most $-0.22314-(1.28155)(0.16903)=-0.43976$. Since $m_F(z)=\{25(-0.40)+10z\}/35=(-10+10z)/35$, the condition is $z\leq-0.53918$. The predictive standard deviation of the future estimate is $\sqrt{0.04+1/10}=\sqrt{0.14}=0.37417$. Therefore,

P(\text{final success}\mid D_C) = \Phi\!\left(\frac{-0.53918-(-0.40)}{0.37417}\right) = \Phi(-0.372) \approx 0.355.

The current posterior probability of $HR<0.80$ is $\Phi\{(-0.22314+0.40)/0.20\}=\Phi(0.8843)\approx0.812$. Despite this favorable current probability, the chance of crossing the more demanding final 0.90 criterion is about 35.5%. A posterior probability below 0.90 now is not itself a prediction that the final rule will fail; future evidence can change the posterior in either direction.

Assumptions, Diagnostics, and Alternatives

The calculation is only as credible as its model for future information and outcomes. Important assumptions include a suitable proportional-hazards interpretation of $\theta$, a reasonable normal approximation for the log hazard-ratio likelihood, and a future event process consistent with the planned information. The model also assumes that the specified final decision rule, analysis population, endpoint definition, and prior will be used at the final analysis. In a randomized trial, censoring and loss to follow-up should be handled in a way compatible with the survival likelihood; informative censoring can invalidate the approximation.

Check whether the observed event accrual, treatment allocation among those at risk, censoring, and estimated log hazard-ratio uncertainty are broadly consistent with the assumptions behind the information calculation. Revisit the approximation if the event target is unlikely to be reached on the assumed schedule, risk-set allocation differs materially from the planned balance, or the data show evidence of changing treatment effects over time. In particular, a single proportional-hazards parameter may not represent a time-varying effect well. A fixed final event count does not guarantee fixed information if the risk sets or analysis model change.

The full alternative is to model future event times, censoring, and trial operations directly, then average the final decision over those possible futures. That can represent features the normal summary model omits, but requires more modeling and computation. Predictive success also differs from frequentist power: power averages a test's rejection probability under specified fixed parameter values, whereas predictive success averages a Bayesian final posterior decision under the current posterior predictive distribution. A plug-in calculation that substitutes the current posterior mean as if it were the known true effect omits uncertainty about $\theta$ and is not the predictive probability defined here.

Common Pitfalls

Confusing the inner and outer probabilities. The final posterior probability $P(\theta<\delta\mid D_C,D_F)$ is evaluated after future data are observed; predictive success is the current probability that this final quantity meets its threshold.
Counting current information twice. If $m$ and $v$ already summarize the current posterior, combine them with future information $I_F$, not with the total information from all events.
Treating every event as exactly $1/4$ information. $D/4$ is an approximation for balanced risk sets in a standard large-sample setting, not a universal survival-analysis identity.
Using the wrong direction. For a rule favoring $HR<$ a margin, lower future log-hazard-ratio estimates favor success; the predictive event is a lower-tail event.
Calling predictive success a probability of clinical benefit. It is the chance of satisfying a specified future posterior rule, conditional on the model and current data.

Key takeaway: Start with the exact final posterior decision rule, model how future event information updates the current posterior, and calculate the predictive probability that the resulting final posterior meets that rule. State the event-information approximation and its assumptions alongside the result.

Check Your Understanding

Answer each question, then compare with the concise solutions below.

In one sentence, distinguish current posterior probability $P(\theta<\delta\mid D_C)$ from predictive success.
Under equal allocation and the stated information approximation, what future information is supplied by 48 additional events?
If $v=0.05$ and $I_F=20$, calculate the final posterior variance under the normal information model.
Why does uncertainty about $\theta$ contribute to the predictive variance of $Z_F$?
Name one reason the approximation $I\approx D/4$ might be unreliable.

Solutions. (1) The current posterior probability concerns the effect given data already observed; predictive success is the probability that the final posterior probability will meet its specified threshold after future data arrive. (2) $I_F\approx48/4=12$. (3) $v_F=(1/0.05+20)^{-1}=(20+20)^{-1}=0.025$. (4) The future estimate depends both on the unknown $\theta$, distributed as $N(m,v)$ given current data, and on future sampling error with variance $1/I_F$; together they give predictive variance $v+1/I_F$. (5) Examples include materially unbalanced treatment allocation within risk sets, changing risk-set composition, or a time-varying treatment effect that conflicts with the proportional-hazards model.