Worked Example With Progression-Free Survival Selection

Why Use Progression-Free Survival to Select Arms?

In a time-to-event MAMS design, an interim decision may use progression-free survival (PFS) to decide which experimental arms continue to the next stage. PFS is often observed sooner than overall survival (OS), so it can provide a timely basis for selection. The decision is consequential: dropping an arm limits its later evidence, while retaining it does not establish that it improves OS.

PFS and OS are different endpoints. PFS is the time from a defined origin, such as randomization, to the first documented disease progression or death, according to the protocol’s definitions. OS is the time to death from any cause. PFS can therefore be favorable even when OS is not, and an interim PFS result is not a substitute for the later OS analysis.

Assumptions and a Calculation Rule

For the examples, assume randomized comparisons of each experimental arm with a concurrent control, a common PFS time origin, protocol-defined progression assessments, and a prespecified interim analysis. The log-rank calculations assume independent participants, appropriately handled right censoring, and censoring that is non-informative for the event process within the comparison. In practice, the analysis plan must also address the assessment schedule, missing evaluations, and any stratification used in randomization.

Use \(O-E\) for the observed minus expected number of PFS events in the experimental arm under the log-rank null hypothesis of equal hazards. Negative values indicate fewer observed events than expected in that arm, which is favorable when the event is progression or death. Let \(V\) be the variance of \(O-E\) under the null. The log-rank score statistic and a common large-sample approximation to the log hazard ratio are:

Here \(HR\) is the hazard in the experimental arm divided by the hazard in the control arm; an \(HR\) below 1 favors the experimental arm. The approximation for the hazard ratio is a score-based summary, not necessarily the same estimate as a fitted Cox model. A two-sided log-rank p-value is approximated by \(2\{1-\Phi(|Z_{\text{PFS}}|)\}\), where \(\Phi\) is the standard normal cumulative distribution function.

Suppose a protocol retains an arm only if at least 40 PFS events have accrued in its comparison and \(Z_{\text{PFS}}\leq-0.50\). This is an illustrative, hypothetical rule, not a universal threshold. The event-count condition guards against making a decision from very sparse information; the statistic condition requires at least some evidence of a favorable direction and magnitude. The exact thresholds and analysis must be chosen before observing the interim outcomes.

From Retention to Later OS Evidence

If A continues, the final OS analysis compares time to death in A and control using the analysis specified in the protocol. The later OS estimate should be reported with its uncertainty and the planned inferential procedure. A PFS-based decision does not turn an OS p-value into the probability that the treatment works, and a favorable interim PFS result does not make a later OS estimate automatically favorable.

For a simple illustration, the log-rank score approximation gives an OS log hazard ratio \( \widehat{\log(HR)}\approx (O-E)/V\), with approximate standard error \(SE\approx1/\sqrt{V}\). A large-sample 95% confidence interval on the log scale is the estimate plus or minus \(1.96SE\); exponentiating its endpoints gives a confidence interval for the hazard ratio. This approximation illustrates the arithmetic. The trial’s actual analysis may use a prespecified Cox model or another appropriate method.

Why PFS Selection Can Disagree With OS

PFS is an intermediate outcome, not a guarantee of the final outcome. A treatment might delay progression without extending life; later treatments, treatment toxicity, crossover, or the relationship between progression and death can all affect OS. Conversely, a modest or uncertain PFS effect does not logically rule out an OS effect. The relationship depends on the disease, treatment, subsequent care, endpoint definitions, and follow-up.

Derivation and Interpretation of the Score Approximation

The approximation \( \widehat{\log(HR)}\approx(O-E)/V\) can be understood through a local expansion of the log-rank score. Write \(\beta=\log(HR)\), so the null corresponds to \(\beta=0\). At the null, the score is approximately \(U=O-E\), and its information is approximately \(V\). A first-order expansion of the score around zero is \(U(\beta)\approx U(0)-V\beta\). Setting the score to zero gives the approximate estimate:

Because \(V>0\), the sign of the approximate log hazard ratio matches the sign of \(O-E\). Thus fewer-than-expected experimental-arm events yield a negative log hazard ratio and an estimated hazard ratio below 1. This derivation relies on the large-sample score approximation; sparse events, heavy censoring, or a strongly non-proportional hazard pattern can make a single hazard-ratio summary inadequate.

Diagnostics, Alternatives, and Limitations

Before acting on an interim result, examine whether the endpoint data and the analysis assumptions are credible. Useful checks include the event accrual and censoring pattern by arm, the timing and completeness of progression assessments, and whether censoring or missing assessments differ materially between groups. For a Cox-model analysis, inspect whether proportional hazards is a plausible summary; crossing or strongly changing hazards can make one hazard ratio obscure important time patterns. Confirm that progression definitions and assessment schedules are applied consistently, since differences in assessment can affect when progression is recorded.

The main alternative is to base selection on another prespecified intermediate outcome, such as early response, or to wait for more mature OS information. Each has a different trade-off: response may be available earlier but may be less informative about survival; waiting for OS may delay decisions; PFS includes both progression and death and can be more clinically informative than response, but it remains distinct from OS. The appropriate endpoint depends on the clinical question and the assumed relationship between intermediate and final outcomes. Earlier material on selection when intermediate and final effects disagree explains why that relationship should not be treated as certain.

Selection also affects how later evidence should be summarized. A retained arm was chosen partly because its interim result looked favorable, so its interim effect can overstate the underlying effect on average. Use the prespecified MAMS analysis and error-control plan for confirmatory claims; do not present a selected interim estimate as if the arm had been chosen independently of its data. Selection rules are also sensitive to threshold choice and event timing, so their operating characteristics should be assessed under plausible PFS and OS relationships.

• Do not equate retention with efficacy. Retention says that the arm met a rule, not that a final hypothesis was rejected.
• Do not equate dropping with no benefit. A noisy interim estimate may miss a beneficial arm, and dropping may reduce the amount of later evidence.
• Do not treat PFS as OS. They have different event definitions and can show different treatment effects.
• Do not ignore data quality. Unequal assessment timing, informative censoring, or sparse events can make an apparently precise selection decision misleading.