Time-to-Event Go Rules Using Hazard-Ratio Margins

Why a Hazard-Ratio Go Rule Needs a Margin

In a time-to-event trial, treatment benefit is often summarized by comparing the event hazard in the treatment group with that in the control group. A decision rule must say how much posterior evidence is enough to proceed, and what level of benefit counts as success. These are separate choices: a threshold for posterior probability controls the strength of evidence required, while a hazard-ratio margin defines the treatment effect the trial is trying to establish.

The direction of benefit depends on the event. In what follows, the event is an adverse outcome, such as death or disease progression, so a lower hazard in the treatment group is favorable. The treatment-to-control convention is used throughout; reversing that ratio would reverse the relevant inequality.

Definition: Let $\theta$ denote the treatment-to-control hazard ratio (HR), and let $D$ denote the observed trial data. For a prespecified margin $M$ and posterior probability threshold $\gamma$, the posterior go rule is to declare a go if $$ P(\theta < M \mid D) \geq \gamma. $$ The probability is calculated under the prespecified model and prior, conditional on the observed data. The margin $M$ defines the effect of interest; $\gamma$ defines the required posterior evidence.

For superiority against no treatment effect, the usual margin is $M=1$: an HR below 1 favors treatment. For a clinically meaningful benefit, the margin is smaller than 1. For example, $M=0.80$ requires evidence that the treatment hazard is at least 20% lower than the control hazard, under the hazard-ratio interpretation. A rule for $M=0.80$ is more demanding than a rule for $M=1$, because every HR below 0.80 is also below 1, but not every HR below 1 is below 0.80.

The margin and probability threshold should be set before examining the trial results. A threshold such as $\gamma=0.90$ or $0.95$ is a decision choice, not a universal statistical constant. The appropriate choice depends on the consequences of an incorrect go decision, the development context, and the broader decision framework.

Model and Calculation

A common working model is a proportional-hazards model. For participant $i$, let $x_i=1$ for treatment and $x_i=0$ for control. The model specifies

h_i(t \mid x_i)=h_0(t)\exp(x_i\beta), \qquad \theta=\exp(\beta),

where $h_i(t \mid x_i)$ is the instantaneous event hazard at time $t$, $h_0(t)$ is the baseline hazard, and $\beta$ is the treatment log hazard ratio. The proportional-hazards assumption means that the treatment-to-control hazard ratio is constant over time. A Bayesian analysis specifies a prior for the model parameters and updates it using the trial data to obtain a posterior distribution for $\beta$, and therefore for $\theta$.

This model-based interpretation relies on an appropriate event definition and follow-up plan, adequate handling of censoring, and a credible proportional-hazards approximation. For example, independent censoring conditional on the modelled information means that, among participants with the same modelled information, censoring does not depend on their unobserved event time. Randomization supports a causal treatment comparison, but does not by itself guarantee proportional hazards or eliminate problems caused by informative censoring.

Because the logarithm is increasing, $\theta<M$ is equivalent to $\beta<\log(M)$. If the posterior for $\beta$ is approximately normal with location $a$ and scale $b$, then

P(\theta<M\mid D) =P(\beta<\log(M)\mid D) \approx \Phi\!\left(\frac{\log(M)-a}{b}\right),

where $\Phi$ is the standard normal cumulative distribution function. If posterior simulation is used instead, the probability is estimated by the fraction of posterior draws for which $\exp(\beta)<M$. In either approach, the comparison is with the margin on the HR scale, or equivalently with its logarithm on the log-HR scale.

Justification: The go probability is the posterior expectation of an indicator: $$ P(\theta<M\mid D) =E\!\left[\mathbf{1}(\theta<M)\mid D\right]. $$ If posterior draws are $\theta^{(1)},\ldots,\theta^{(S)}$, replacing this expectation by its sample average gives $$ \widehat{P}(\theta<M\mid D) =\frac{1}{S}\sum_{s=1}^{S}\mathbf{1}\!\left(\theta^{(s)}<M\right). $$ Thus the rule is directly the posterior mass below the prespecified margin, not a comparison of the posterior mean HR with that margin.

Worked Decisions

Worked Example: Superiority Does Not Guarantee a Meaningful Margin

Hypothetical trial. A randomized trial compares a treatment with control for an adverse event. A Bayesian proportional-hazards analysis of the trial data gives the approximate posterior $\beta\mid D\sim N(-0.25,0.12^2)$. The prespecified evidence threshold is $\gamma=0.95$. Calculate the posterior probability of superiority and the probability of an HR below 0.80.

For superiority, $M=1$, so $\log(M)=0$. Therefore, $$ P(\theta<1\mid D)\approx\Phi\!\left(\frac{0-(-0.25)}{0.12}\right) =\Phi(2.083)\approx 0.981. $$ This exceeds 0.95, so the superiority go rule is met.

For the clinically meaningful margin $M=0.80$, $\log(0.80)\approx -0.2231$. Thus $$ P(\theta<0.80\mid D)\approx\Phi\!\left(\frac{-0.2231-(-0.25)}{0.12}\right) =\Phi(0.224)\approx 0.589. $$ This does not reach 0.95. The trial’s posterior evidence supports a lower treatment hazard than control, but it does not provide strong enough evidence under this rule that the reduction reaches the 20% margin. The same posterior can therefore pass a superiority rule and fail a more demanding clinical-benefit rule.

Worked Example: Applying a Clinically Meaningful HR Margin

Hypothetical trial. In a second randomized trial with the same adverse event, the fitted model gives $\beta\mid D\sim N(-0.40,0.12^2)$. The trial team prespecified $M=0.80$ and $\gamma=0.90$. Determine whether the rule is met.

The log margin is $\log(0.80)\approx -0.2231$. Hence $$ P(\theta<0.80\mid D)\approx \Phi\!\left(\frac{-0.2231-(-0.40)}{0.12}\right) =\Phi(1.474)\approx 0.930. $$ Since $0.930\geq 0.90$, this rule gives a go decision. The interpretation is that, under the specified posterior model, the probability that the treatment-to-control HR is below 0.80 is about 93.0%. It is not a 93% probability that the treatment works in some unrestricted sense, nor does it state that every patient’s hazard is reduced by 20%.

For comparison, the probability of any superiority is $$ P(\theta<1\mid D)\approx \Phi\!\left(\frac{0-(-0.40)}{0.12}\right) =\Phi(3.333)\approx 0.9996. $$ That larger probability is not a substitute for the prespecified clinically meaningful-margin calculation. It answers a different question.

Worked Example: Estimating the Go Probability From Posterior Draws

Hypothetical trial. A third analysis retains 10,000 approximately independent posterior draws of the treatment HR after model fitting and diagnostics. Of these, 8,600 are below 1 and 5,750 are below 0.80. The rule uses $M=0.80$ and $\gamma=0.90$.

The estimated posterior probability for the clinical margin is $$ \widehat{P}(\theta<0.80\mid D)=\frac{5750}{10000}=0.575. $$ The superiority estimate is $8600/10000=0.860$, but the go rule concerns the 0.80 margin. Since $0.575<0.90$, the estimated posterior probability does not meet the rule. The result does not establish that the HR is at least 0.80; rather, the required posterior probability below 0.80 has not been reached.

If the draws were independent, the approximate Monte Carlo standard error for the 0.575 estimate would be $$ \sqrt{\frac{0.575(1-0.575)}{10000}}\approx 0.00494. $$ Here the difference from the threshold is much larger than this Monte Carlo error. With MCMC draws, however, autocorrelation reduces the effective information, so effective sample size and Monte Carlo error should be assessed rather than treating the raw draw count as independent observations.

Interpretation, Diagnostics, and Alternatives

An HR margin concerns a relative instantaneous hazard among participants who remain at risk, under the model’s assumptions. It does not directly specify a difference in the proportion of participants experiencing the event by a particular time. When proportional hazards is doubtful—for example, if treatment effects emerge late or change direction—the fitted constant HR can obscure clinically important time patterns. Diagnostics should therefore include examination of whether treatment effects appear to vary over time, alongside checks of model fit, censoring assumptions, and the posterior computation.

The go probability also depends on the prior and likelihood, not only on the observed event counts or the point estimate. A posterior concentrated near the margin may yield a probability close to the decision threshold. In such a case, Monte Carlo error can affect the reported probability, and the posterior calculation should be precise enough to support the decision. For simulated posteriors, inspect mixing and effective sample size and quantify Monte Carlo error for the tail probability being used.

A point-estimate rule would compare a single estimated HR with the margin, but would not express uncertainty about the HR. A credible interval provides a range of posterior values and can be informative, but the go rule is explicitly about posterior probability mass below the chosen margin. A frequentist confidence interval or log-rank test uses a different inferential framework and does not directly report the posterior probability in this rule. Finally, absolute survival probabilities at a specified time or restricted mean survival time can answer clinically useful questions on an absolute time or risk scale. They are complementary summaries, especially when proportional hazards is questionable; they do not automatically equal a hazard-ratio criterion.

Common pitfall: “The posterior mean HR is 0.78, which is below the 0.80 margin, so the go rule is met.” This is incorrect: the mean alone does not determine the posterior probability below the margin. The correction is to calculate $P(\theta<0.80\mid D)$ and compare that probability with the prespecified $\gamma$. A posterior spread wide enough to place substantial mass above 0.80 may leave the rule unmet even when the posterior mean is below it.

A well-defined decision therefore states the adverse event, treatment-to-control HR convention, model and prior, margin $M$, posterior threshold $\gamma$, and the resulting posterior probability. It should also report enough context to make the assumptions and clinical meaning of the HR assessable. A go decision is the consequence of that prespecified rule; it is not a claim that the model assumptions are certainly true or that the treatment will benefit every individual.

Check Your Understanding

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

A trial has $\beta\mid D\sim N(-0.18,0.08^2)$. Calculate the posterior probability of an HR below 0.90. Does it meet a rule requiring probability at least 0.90?
Why can a rule using $M=0.80$ fail even when the rule using $M=1$ passes?
For posterior HR draws, 7,200 of 10,000 draws are below 0.75. Estimate the posterior probability below 0.75 and decide whether a threshold of 0.70 is met.
Name one consequence of nonproportional hazards for a go rule based on a single HR.

Solutions. (1) $\log(0.90)\approx-0.1054$, so the standardized value is $(-0.1054+0.18)/0.08=0.933$. The probability is approximately $\Phi(0.933)=0.825$, below 0.90, so the rule is not met. (2) The event $\theta<0.80$ is more restrictive than $\theta<1$; posterior mass between 0.80 and 1 contributes to superiority probability but not to the probability of meeting the 0.80 margin. (3) The estimate is $7200/10000=0.720$, which exceeds 0.70, so the rule is met, subject to adequate posterior computation and Monte Carlo precision. (4) A constant HR may average over changing effects and conceal when, or in which direction, the treatment effect occurs, making the single-margin interpretation misleading.

Time-to-Event Go Rules Using Hazard-Ratio Margins

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