19 Continuous or Time-to-Event? Choosing the Endpoint Model
Prerequisites: Chapters 2, 8, and 18.
19.1 Learning objectives
By the end of this chapter you should be able to:
- Explain why dichotomizing a continuous trajectory into a time-to-event endpoint discards information, and quantify the loss.
- Describe the historical reasons two therapeutic areas converged on different default analyses.
- Identify the conditions under which a time-to-event endpoint is nonetheless the right choice.
- Recognize interval censoring induced by a visit schedule and its consequences.
19.2 Orientation
Two therapeutic areas with equally serious diseases and equally sophisticated statisticians have arrived at opposite default analyses. Oncology analyzes time to event with Cox regression. Alzheimer disease analyzes change in a continuous score with MMRM. The difference is not arbitrary and it is not purely statistical; it is the product of the endpoints available, the regulatory history, and a genuine efficiency argument.
The question this chapter answers is what happens when both options are available for the same trial. In Alzheimer disease, a patient’s CDR score can be analyzed as a continuous trajectory, or dichotomized into ‘conversion to CDR at or above 1.0’ and analyzed as a time to event. Both are legitimate. They are not equally efficient.
19.3 Provenance
This chapter follows the research compendium 13-mmrm-vs-survival-ad (project mmrmsurvival), which combines a historical review of how the two fields diverged with an ADEMP simulation comparing MMRM against Cox regression applied to CDR data from the same simulated trials, in an MCI-to-mild-dementia population over 24 months.
19.4 The statistician’s contribution
(Judgment 1.) Whether the clinical question is about level or about crossing a threshold. These are different questions. ‘Does the treatment slow decline’ is a question about a trajectory. ‘Does the treatment delay progression to dementia’ is a question about a threshold crossing. Both are legitimate and the second is often more meaningful to patients even though it is less efficient.
(Judgment 2.) Whether the threshold is clinically principled or statistically convenient. A threshold chosen because it is a recognized clinical transition is defensible. A threshold chosen because it yields the most events is data-driven and inadmissible.
(Judgment 3.) What the visit schedule does to a time-to-event endpoint. If conversion can only be detected at scheduled visits, the data are interval-censored, and a standard Cox model analyzing time-to-visit-of-detection is fitting the wrong likelihood.
19.5 How the two fields diverged
The compendium’s historical review identifies three reinforcing forces behind Alzheimer trials’ convergence on MMRM with change-score estimands.
The regulatory requirement for continuous co-primaries. The 1990 FDA guidance for Alzheimer drug development established a requirement for both a cognitive and a functional co-primary endpoint, and the instruments available for both are continuous scales. The endpoint structure was fixed by regulation before the analysis debate began.
The fall of LOCF. Last observation carried forward was the standard handling of dropout in this literature until empirical work demonstrated its bias, and the National Research Council panel codified the case against it. What replaced it was the likelihood-based longitudinal analysis, which is to say MMRM.
The efficiency argument. Modeling repeated continuous data is more efficient than dichotomizing into a time-to-event endpoint, and this was demonstrated specifically for Alzheimer endpoints.
Oncology arrived at a different equilibrium for reasons equally structural. Its primary endpoint, death, is inherently binary and irreversible; the proportional hazards framework has deep regulatory precedent; and no single continuous severity measure plays the role that a cognitive scale plays in Alzheimer disease. There is no continuous trajectory to model.
The lesson generalizes. The default analysis in a therapeutic area is usually the product of what its endpoints permit and what its regulators established decades ago, not of a comparison anyone recently ran.
19.6 The efficiency comparison
The compendium simulates trials in an MCI population over 24 months, generating CDR trajectories, and analyzes the same simulated data two ways: MMRM on the continuous CDR sum of boxes, and Cox regression on time to first conversion to CDR at or above 1.0.
The expected and confirmed result is that MMRM yields substantially greater power, and the mechanism is information use. The continuous analysis uses the full trajectory of scores at every visit for every patient. The survival analysis reduces that trajectory to a single binary indicator and a time, and for the many patients who never cross the threshold within 24 months it reduces to a censoring indicator carrying almost no information about the treatment effect.
The advantage is most pronounced where conversion rates are modest and censoring is heavy, which is exactly the MCI-to-mild-dementia setting: most participants in a 24-month trial will not reach the threshold, so the Cox model is fitting to a minority of the sample.
A rough way to see the magnitude: dichotomizing a continuous variable at a threshold typically costs 30 to 50% of the information, and dichotomizing a trajectory into a first-crossing time costs more, because the timing of the crossing is measured coarsely and the pre-crossing trajectory is discarded entirely.
19.7 Caveats to the comparison
The compendium is explicit about three.
The comparison is asymmetric by construction. MMRM uses all visits; Cox uses the first conversion event. That is the point of the comparison, and it also means the result is not a statement that one method is better than the other in general, only that using more information yields more power.
The visit schedule limits the survival endpoint. True conversion may occur between visits, so what is observed is interval-censored: the event happened somewhere between the last clean visit and the first visit showing conversion. A standard Cox model treating the detection visit as the event time is fitting the wrong likelihood. The consequences are usually modest with frequent visits and can be material with annual ones; interval-censoring methods are available and are rarely used.
Relative efficiency depends on the threshold. A threshold that most patients cross gives the survival analysis more events and narrows the gap; a threshold few cross widens it. Since the threshold also determines what the endpoint means clinically, it should not be chosen to optimize power, and the sensitivity of the conclusion to the threshold should be reported.
19.8 Practical guidance
Use the continuous longitudinal analysis when a validated continuous measure exists, when the disease progresses gradually, when the trial is powered on modest effects, and when the audience can interpret the scale.
Use the time-to-event analysis when the event is discrete and irreversible, when the timing is what matters clinically, when the event rate is high enough that censoring is not dominant, or when regulatory precedent requires it.
Use both, hierarchically, when the trial must be both efficient and communicable, which is most of the time.
Consider the intermediate options. A responder analysis at a fixed time point is a dichotomization without the timing, and is usually the worst of both worlds. A joint model of the longitudinal trajectory and the time to event uses all the information and is the statistically correct answer to the question ‘what does the whole record say’, at the cost of complexity and of a model whose assumptions are harder to check. Restricted mean time in a state, or the difference in area under the severity curve, is a summary that keeps the continuous information while producing a time-scaled quantity that communicates well.
19.9 Worked example: endpoint strategy for an MCI trial
A 24-month trial in MCI due to Alzheimer disease, 600 patients, visits at 6, 12, 18, and 24 months.
Primary endpoint. Change in CDR-SB from baseline to month 24, analyzed by categorical-time MMRM as in Chapter 18, since the mechanism suggests delayed onset.
Key secondary. Time to conversion to CDR global score at or above 1.0, analyzed by Cox regression, tested only if the primary is significant, under the hierarchical procedure.
Power. Simulated for both. At the design effect size, the MMRM primary has 90% power; the Cox secondary has 62%. The team accepts that the secondary may fail to reach significance even when the primary succeeds, and the SAP states in advance that this pattern is expected and does not undermine the primary conclusion. Saying this in advance is what prevents an awkward conversation at the results meeting.
Interval censoring. With visits every six months, the conversion time is known only to within a six-month window. A sensitivity analysis using an interval-censoring model is pre-specified. If the two agree, the simpler analysis is reported; if they differ, the interval-censored analysis is the correct one.
Threshold sensitivity. The conversion definition is fixed at CDR at or above 1.0 on clinical grounds. A sensitivity analysis at a CDR-SB-based threshold is pre-specified to demonstrate that the conclusion is not an artifact of the cut point.
Communication plan. The abstract reports both the CDR-SB difference and, if the secondary is significant, the delay in conversion. If the secondary is not significant, the abstract reports the primary and states the secondary’s result without a claim.
19.10 Collaborating with an LLM on endpoint choice
Prompt 1: ‘Should our endpoint be continuous or time-to-event?’
What to watch for. Models give a reasonable summary of the trade-off and rarely mention the interval-censoring issue induced by the visit schedule, which is the technical detail most likely to be missed in a real protocol.
Verification. Ask how often the outcome is assessed and whether the event can be detected between assessments.
Prompt 2: ‘Compare the power of MMRM and Cox on the same simulated data.’
What to watch for. Simulations in which the two analyses are applied to different generated data, which makes the comparison meaningless. Both must be applied to the same replicates.
Verification. Confirm the code generates one dataset per replicate and derives the survival endpoint from the same trajectories the MMRM analyzes.
Prompt 3: ‘Justify dichotomizing our continuous outcome at this threshold.’
What to watch for. Models will supply a justification, because that is what was asked. The useful prompt is the opposite one: ask for the argument against.
Verification. Ask what fraction of patients cross the threshold and what the analysis discards.
19.11 Principle in use
Derive both endpoints from the same trajectories when comparing them. Anything else compares simulations, not analyses.
Never choose the threshold to maximize events. It is a clinical definition, and choosing it from the data invalidates the analysis.
State in advance that a less powerful secondary may fail. It prevents a true positive primary from being undermined by a predictable secondary null.
19.12 Exercises
Simulate 24-month CDR trajectories for two arms. Analyze with MMRM and with Cox on threshold crossing, and compute the power of each at the same effect size.
Vary the conversion threshold and plot the power of the survival analysis against the proportion of patients who convert. Where is it maximized, and why is that not a reason to choose it?
Implement an interval-censored analysis of the same simulated data and compare the estimated hazard ratio and its standard error to the naive Cox result.
Compute the difference in area under the CDR-SB trajectory between arms as an alternative summary. Compare its power to both other analyses.
Fit a joint longitudinal and time-to-event model to one simulated dataset and describe what it estimates that neither separate analysis does.
19.13 Further reading
- The compendium
13-mmrm-vs-survival-ad. - Mallinckrodt et al. (2008), on the MMRM convergence.
- National Research Council (2010), on the case against LOCF.
- Fleming & Harrington (1991), for the survival machinery.
- Ard & Edland (2011) and Donohue et al. (2014), on design and analysis considerations specific to Alzheimer trials.
- The
survival,icenReg, andJMR packages.