Reporting Time-To-Event Effects with Survival Probabilities

Why Report Both Relative and Absolute Time-to-Event Effects?

A hazard ratio summarizes a relative comparison of event rates over time, under the model used to estimate it. It does not, by itself, tell readers what proportion of patients are expected to remain alive or event-free at a clinically meaningful time. A hazard ratio of 0.75 could accompany quite different absolute outcomes in populations with different baseline risks. Reporting survival probabilities at a specified time, and preferably their difference, helps make the expected outcome more tangible.

A clear report names the event, the treatment groups, the population, and the time horizon. For example, “the probability of remaining alive without progression at 2 years among adults enrolled in the trial” is more informative than “2-year survival” if the endpoint is actually progression-free survival. The interpretation also depends on how the trial handles events such as treatment switching or discontinuation; the reported population and estimand should match the analysis.

Definition: For treatment group \(a\), the survival probability at time \(t\) is \(S_a(t)=P(T_a>t)\), where \(T_a\) is the time to the specified event under that treatment in the stated target population. The survival difference at time \(t\), defined here as treatment minus control, is \(\Delta_S(t)=S_T(t)-S_C(t)\). A positive value means a larger probability of remaining event-free in the treatment group at that time.

The word “survival” is often used even when the event is not death. State whether the outcome is overall survival, time to relapse, progression-free survival, or another endpoint. State the time unit as well: a 2-year probability is not interchangeable with a 24-month probability unless the time origin and follow-up definition agree.

What the Hazard Ratio Contributes

A hazard is an instantaneous event rate among individuals who have not yet experienced the event. Under a proportional-hazards model, the treatment hazard is a constant multiple of the control hazard at each time. The hazard ratio (HR) is that multiplier: an HR below 1 indicates a lower treatment hazard, while an HR above 1 indicates a higher treatment hazard. It is a relative rate comparison, not a ratio of survival probabilities and not a percentage-point change in the probability of surviving.

Under proportional hazards, the relationship between the hazard ratio and survival probabilities can be derived. Let \(h_C(t)\) be the control hazard and suppose \(h_T(t)=\theta h_C(t)\) for every \(t\), where \(\theta\) is the HR. The cumulative hazard is the integral of the hazard over time, and survival equals the exponential of the negative cumulative hazard. Therefore:

H_T(t)=\int_0^t h_T(u)\,du =\theta\int_0^t h_C(u)\,du =\theta H_C(t), \qquad S_T(t)=\exp\{-H_T(t)\} =\exp\{-\theta H_C(t)\} =S_C(t)^\theta.

This relation explains why an HR alone does not determine an absolute survival difference: \(S_C(t)\), which depends on baseline risk and time, is also needed. For \(0<S_C(t)<1\) and \(\theta<1\), \(S_C(t)^\theta>S_C(t)\), but the size of the difference depends on \(S_C(t)\). In a Bayesian analysis, this calculation should generally be made for each joint posterior draw of the treatment effect and baseline survival, then summarized. Substituting a posterior mean or median HR into a separate summary of control survival does not generally produce the posterior mean or median survival difference.

Worked Example: Why the Same Hazard Ratio Can Imply Different Survival Differences

Suppose, for illustration, that event times follow an exponential model in a hypothetical population. The control hazard is 0.12 events per person-year, and the treatment hazard is 0.09 events per person-year. The hazards are constant over time, so the proportional-hazards assumption holds, with HR \(=0.09/0.12=0.75\). Calculate survival and the treatment-minus-control difference at 2 and 5 years.

For a constant hazard \(\lambda\), survival is \(S(t)=\exp(-\lambda t)\). At 2 years:

S_C(2)=\exp(-0.12\times2)=\exp(-0.24)=0.7866, \qquad S_T(2)=\exp(-0.09\times2)=\exp(-0.18)=0.8353.

The 2-year survival difference is \(0.8353-0.7866=0.0487\), or 4.87 percentage points. At 5 years:

S_C(5)=\exp(-0.12\times5)=\exp(-0.60)=0.5488, \qquad S_T(5)=\exp(-0.09\times5)=\exp(-0.45)=0.6376.

The 5-year difference is \(0.6376-0.5488=0.0888\), or 8.88 percentage points. Both horizons have the same HR of 0.75, but different absolute survival differences. The calculation depends on the illustrative exponential model and its constant hazards; these figures are not data from a clinical study.

Summarizing Posterior Results at a Chosen Horizon

For a Bayesian report, give a posterior summary for the HR and an absolute survival summary at a prespecified, clinically relevant time. State the point summary used, such as posterior median or posterior mean, and provide a credible interval with its level. For the absolute outcome, report the treatment and control survival probabilities as well as their difference. The difference communicates the absolute contrast; the two probabilities show the event-free outcome in each group.

Posterior quantities should preserve their joint dependence. For posterior draw \(m\), calculate \(S_T^{(m)}(t)\), \(S_C^{(m)}(t)\), and \(\Delta_S^{(m)}(t)=S_T^{(m)}(t)-S_C^{(m)}(t)\). Summarize the resulting draws separately. The posterior interval for the difference is not obtained by subtracting the lower control limit from the upper treatment limit, or by subtracting two marginal interval endpoints in any fixed order.

Worked Example: Reporting a Posterior Hazard Ratio and Two-Year Survival

In a hypothetical randomized trial of adults with a specified disease, the event is death or disease progression. The analysis targets the trial population and reports survival, meaning the probability of remaining alive and progression-free. Suppose joint posterior draws under the fitted model give a median HR of 0.78 with a 95% credible interval of 0.60 to 1.01. At 2 years, the posterior mean control survival is 0.79 and the posterior mean treatment survival is 0.83. The posterior mean difference is 0.04, with a 95% credible interval of −0.01 to 0.09.

The point survival difference is calculated as \(0.83-0.79=0.04\), or 4 percentage points. A suitable report is: “In the trial population, the posterior median HR for death or progression was 0.78 (95% credible interval 0.60 to 1.01). At 2 years, posterior mean progression-free survival was 83% with treatment and 79% with control; the posterior mean treatment-minus-control difference was 4 percentage points (95% credible interval −1 to 9 percentage points).”

The HR summary suggests a lower event hazard in the treatment group at its median, but its interval includes 1. The difference interval includes 0, so these summaries do not place all of the central 95% posterior interval on the benefit side. Neither interval is a frequentist confidence interval. The survival difference is specific to 2 years, the stated event, and the trial population; it should not be presented as a constant benefit at every follow-up time.

Posterior summaries from the same model need not look algebraically identical. For instance, the difference between two posterior mean survival probabilities equals the posterior mean of their draw-wise difference when the same draws and weights are used. But the median of a difference generally does not equal the difference of medians, and a credible interval for the difference must be calculated from the difference draws. Report exactly which summary was calculated rather than implying that one was obtained by subtracting unrelated marginal summaries.

Worked Example: Translating a Survival Difference Into Expected Counts

Suppose another hypothetical trial reports 3-year survival of 0.70 under control and 0.77 under treatment in its target population. The posterior mean survival difference is 0.07, with a 95% credible interval from 0.01 to 0.13. Explain the absolute scale while retaining the time and population context.

The point difference is \(0.77-0.70=0.07\), or 7 percentage points. Multiplying by 1,000 gives \(0.07\times1{,}000=70\). Thus, per 1,000 comparable patients, the posterior mean difference corresponds to 70 more patients alive at 3 years in the treatment group than in the control group, if the reported survival probabilities apply to those patients and the treatment contrast is interpreted causally under the trial design and analysis assumptions.

The interval transforms in the same way: \(0.01\times1{,}000=10\) and \(0.13\times1{,}000=130\). The corresponding interval is 10 to 130 more patients alive per 1,000 at 3 years. This is a restatement of uncertainty in the survival difference, not a prediction that exactly 70 additional individuals will survive in a particular group of 1,000. A full report would also name the event definition and target population; “alive” is appropriate here only because the example specifies overall survival.

Choosing the Time Horizon and Checking the Report

The horizon should be clinically interpretable and supported by the trial’s follow-up. A very late time may be dominated by model extrapolation or by few participants still under observation. A very early time may miss a delayed treatment effect. If several horizons are reported, identify them all and avoid presenting only the most favorable one without context. The estimand, time origin, handling of censoring, and analysis population should be described consistently with the trial analysis.

Before finalizing a report, check that the relative and absolute summaries refer to the same event, treatment contrast, population, and time origin. Check that the HR has been oriented correctly: if the numerator is treatment and the denominator is control, HR below 1 points toward a lower treatment hazard. Check the sign convention for the survival difference, stated here as treatment minus control. Also check that the time is explicit and that percentages and percentage points are not confused.

The HR and survival summaries answer related but different questions. The HR is a relative comparison of hazards under its model assumptions; the survival probabilities give outcomes at specified times. A survival difference is an absolute contrast at one horizon. If proportional hazards is not credible, a single HR may be an inadequate summary of a time-varying relative effect; survival probabilities at chosen times can still be useful, but should be reported from an analysis that supports them rather than inferred from a single HR.

Common Pitfalls and Alternatives

Calling the HR a survival ratio. An HR of 0.75 does not mean survival is 25% higher. Report survival probabilities or their difference to describe outcomes at a specified time.
Reporting survival without a horizon. “Treatment survival was 80%” is incomplete. State, for example, “80% at 2 years,” along with event and population.
Omitting the control outcome. A 5-percentage-point difference is easier to interpret alongside the treatment and control probabilities from which it is calculated.
Mixing summaries or intervals. Calculate posterior differences draw by draw. Do not subtract unrelated endpoints of marginal credible intervals to create an interval for the difference.
Assuming proportional hazards without checking. If the relative effect changes over time, a single HR can conceal that pattern. Report time-specific survival estimates and explain the limits of a single relative summary.
Overstating a population-specific estimate. Survival probabilities apply to the stated population and analysis assumptions. They are not automatically transportable to patients with different baseline risk or care.

Other summaries, such as a time-specific survival ratio or restricted mean survival time, answer different questions and may be useful when aligned with the estimand. They do not remove the need to define the event, horizon or interval of follow-up, and target population. The purpose of pairing an HR with survival probabilities is not to declare one scale universally best, but to make the relative and absolute aspects of the time-to-event result visible together.

Key takeaway: Report the posterior HR with its treatment-versus-control orientation and uncertainty, then report survival in both groups and the treatment-minus-control difference at a named time. Specify the event and target population, and do not interpret a relative hazard as an absolute survival effect.

Check Your Understanding

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

A report gives a treatment HR of 0.80. What does this say, and what does it not say, about survival at 2 years?
At 18 months, posterior mean survival is 0.72 under control and 0.78 under treatment. Calculate the treatment-minus-control difference in percentage points.
Why should a posterior credible interval for a survival difference be calculated from draw-wise differences rather than by subtracting marginal interval endpoints?
Name four details that should accompany a reported time-specific survival probability.

Solutions. (1) Under the model and its assumptions, the treatment hazard is estimated to be 20% lower relative to control at the HR point estimate. The HR does not mean survival is 20% higher at 2 years; the time-specific survival probabilities are needed to describe that outcome. (2) \(0.78-0.72=0.06\), or 6 percentage points. (3) The treatment and control survival quantities are dependent, and the interval for their difference depends on their joint posterior draws. Marginal interval endpoints do not preserve that dependence and do not generally give a credible interval for the difference. (4) State the event, time horizon and time unit, target population, and treatment group or comparison. The time origin and relevant analysis definition should also be clear when they affect interpretation.