Why Event Structure Changes a Dose Summary
A dose-level response percentage can look straightforward until a patient dies, progresses, develops severe toxicity, or stops treatment before response is assessed. These events affect what “response by day 28” means. If death before response is counted as a competing event, the summary answers how often response occurs before death. If response is instead assessed regardless of treatment discontinuation, a patient who stops treatment may still contribute a later response. Neither definition is universally correct; the event definition must match the clinical question.
The same issue arises for toxicity. A toxicity can be a terminal first event for a particular analysis, but it may also be followed by recovery, treatment discontinuation, or response. Treating every toxicity as if it permanently prevents later benefit may therefore misrepresent the clinical course. Explicit event definitions are especially important in joint dose-finding, where efficacy and harm can happen at different times and in different orders.
Choose the Event Definition Before Summarizing
Suppose the outcome of interest is tumor response by day 28. Death before response is generally a competing event for the probability of response before death: a patient who has died cannot subsequently respond by day 28. Progression may also be competing if the endpoint is specifically response before progression. By contrast, treatment discontinuation is not automatically competing. If response follow-up continues after discontinuation, a later response can still count under a treatment-policy definition. If assessment stops at discontinuation and the endpoint requires response while receiving treatment, discontinuation may end the opportunity to meet that endpoint.
Toxicity requires the same careful distinction. For a first-event endpoint defined as “response before grade 3 or higher toxicity,” such toxicity is competing by definition. For an endpoint that records whether response and toxicity have each occurred by day 28, a patient may experience both; forcing those outcomes into mutually exclusive first-event categories would discard information. A multistate representation can retain the order, for example, moving from on-treatment and event-free to toxicity, then to discontinuation or response.
Before calculating a dose-level summary, specify the time origin, horizon, event definitions, and rules for simultaneous or tied events. State whether post-discontinuation outcomes count and how an event occurring at the same recorded time as another is classified. If the protocol records only dates, a prespecified tie rule or a joint-event category may be needed; choosing an order after seeing the outcomes can bias a comparison.
Worked Example: A First-Event Summary at One Dose
Hypothetical data. At one dose, 20 patients have complete follow-up through day 28. The endpoint is explicitly defined as “tumor response before progression, death, or grade 3 or higher toxicity.” The first-event counts are 8 responses, 4 progressions, 2 deaths, 3 severe toxicities, and 3 patients with none of these events by day 28.
Under this endpoint, response before any listed competing event is \(8/20=0.400\), or 40.0%. Progression, death, and severe toxicity together occur first for \(4+2+3=9\) patients, giving \(9/20=0.450\), or 45.0%. The remaining \(3/20=0.150\), or 15.0%, have no listed event by the horizon.
These proportions sum to one because the endpoint assigns each patient to exactly one first-event category or to no event by day 28. The 40.0% is not the probability of ever responding under continued follow-up, nor the probability of responding if competing events could somehow be prevented. It is the observed proportion with response first under the stated definition. If toxicity were not defined as preventing a later response, this first-event summary would answer a different question from an analysis that followed patients after toxicity.
Cumulative Incidence With Competing Events
When event times are available, the cumulative incidence function for event type \(j\), written \(F_j(t)\), is the probability that an individual experiences event \(j\) by time \(t\) and before any competing event. Its value depends on the incidence of all event types: patients who experience a competing event leave the event-free risk set and cannot later have the first event of interest.
For each event type \(j\), the cause-specific hazard at time \(u\) describes the instantaneous event rate among people still free of every event immediately before \(u\). With independent censoring and correctly observed event times and types, the cumulative incidence can be expressed in continuous time as the event-free survival probability just before each time, multiplied by the cause-specific hazard increment and accumulated over time:
Here \(S(u-)\) is the probability of remaining free of all event types just before \(u\), and \(\Lambda_j\) is the cumulative cause-specific hazard for event type \(j\). This expression shows why a cause-specific hazard alone is not an absolute event probability: the number still at risk also depends on the other causes.
For discrete event times, the corresponding calculation adds the probability of surviving event-free to each time, multiplied by the fraction of the risk set that experiences the event of interest at that time. This is the basis of the competing-risk cumulative incidence calculation. It assumes that event types are classified consistently and that loss to follow-up does not selectively remove patients with different future event risks, conditional on the information used in the analysis. Administrative censoring at a fixed horizon is commonly compatible with this calculation; informative loss to follow-up requires additional methods or sensitivity analysis.
Worked Example: Calculating Response Cumulative Incidence
Hypothetical data. At a dose, 20 patients are followed for three intervals. Response is the event of interest; progression or death is the competing event. In interval 1, 3 respond and 2 have progression or death, leaving 15 event-free. In interval 2, 4 respond and 3 have progression or death, leaving 8 event-free. In interval 3, 2 respond and 2 have progression or death; the final 4 remain event-free through the horizon. There is no loss to follow-up.
The response cumulative incidence is the sum of the response fraction at each interval multiplied by the probability of having remained event-free up to that interval. At the first interval, the contribution is \(3/20=0.150\). At the second, \(15/20\) remain event-free and \(4/15\) of that risk set respond, so the contribution is \((15/20)(4/15)=0.200\). At the third, the contribution is \((8/20)(2/8)=0.100\).
Thus the estimated probability of response before progression or death by the end of interval 3 is 45.0%. The corresponding competing-event contributions are \(2/20=0.100\), \((15/20)(3/15)=0.150\), and \((8/20)(2/8)=0.100\), totaling 35.0%. The remaining 20.0% are event-free at the horizon. The three results sum to 100.0%, as expected when there is complete follow-up and the categories are exhaustive.
A common alternative is to censor patients when progression or death occurs and calculate \(1-\)Kaplan–Meier for response. That calculation gives \(1-(17/20)(11/15)(6/8)=0.5325\), or 53.3%. It does not estimate the actual probability of response before a competing event in this cohort. By treating competing events as if the patients remained capable of later response, it estimates a different, “net” quantity under assumptions that effectively remove those competing events. For the clinical question posed here, the cumulative incidence of 45.0% is the relevant summary.
First Events and Multistate Summaries
A first-event analysis is useful when the clinical endpoint is explicitly about which event happens first. It is compact and produces mutually exclusive categories, but it cannot describe what happens after the first event. A multistate model retains transitions and can represent, for example, event-free on treatment, response, toxicity, discontinuation, progression, and death. Death is typically absorbing; other states need not be. A patient may move from toxicity to discontinuation, or from toxicity to response if the endpoint and follow-up permit it.
A multistate summary can report the probability of occupying each state at a particular time, or the probability of following a specified transition path. These are different from a first-event probability. For example, the probability of being in the response state at day 28 may exclude patients who responded earlier and then progressed, unless the state definition retains that history. State definitions must therefore say whether response is a current status, an event ever experienced, or a transition that remains recorded after it occurs.
Worked Example: Discontinuation Changes the Efficacy Summary
Hypothetical data. Two randomized dose groups each contain 20 patients, with follow-up through day 28 continuing after treatment discontinuation. At the lower dose, 7 patients respond before discontinuation; 3 discontinue first, of whom 2 later respond and 1 does not; 2 experience progression or death before response; and 8 have none of these events. At the higher dose, the corresponding counts are 6 responses before discontinuation, 6 discontinuations first (4 later responses and 2 without response), 2 progressions or deaths before response, and 6 with none of these events.
If the endpoint is response before discontinuation, the dose-level proportions are \(7/20=0.350\) at the lower dose and \(6/20=0.300\) at the higher dose. This endpoint treats discontinuation as competing, so later responses do not count. If the endpoint is response by day 28 regardless of discontinuation, the later responses count: the totals are \(7+2=9\) and \(6+4=10\), giving \(9/20=0.450\) and \(10/20=0.500\), respectively.
The ranking of the two dose groups changes under these definitions: the lower dose has the larger response-before-discontinuation proportion, while the higher dose has the larger response-regardless-of-discontinuation proportion. Neither calculation is inherently erroneous. They address different estimands and should not be presented as interchangeable measures of efficacy. Because these are hypothetical summaries, they do not establish that dose caused the observed difference; interpretation also depends on the design and analysis assumptions.
Diagnostics, Alternatives, and Limitations
A dose-level summary is only as defensible as its event structure. Check that event times, event types, and follow-up status are recorded consistently across doses. Tabulate the number of patients in each first-event category and the number still event-free at the horizon. In time-to-event summaries, inspect the risk set over time and report how much censoring occurs. If follow-up is shorter in one dose group, a raw proportion by a common horizon may not be comparable without accounting for censoring.
The main alternatives answer distinct questions. A fixed-horizon first-event proportion is transparent when all patients are observed through that horizon and the endpoint is explicitly first-event based. A cumulative incidence function uses event times and accounts for competing events when follow-up is censored. A cause-specific hazard describes the event rate among those still free of all events, rather than the absolute probability of the event. A multistate analysis is appropriate when sequences and transitions matter. A composite endpoint can count several outcomes together, but it can hide clinically important differences between response, toxicity, and death unless its components are also shown.
No single event representation resolves ambiguous clinical priorities. A first-event rule may discard a response that follows manageable toxicity; a treatment-policy summary may count responses after stopping for reasons related to prognosis; and a multistate model can become difficult to interpret if states or transitions are too numerous. Small dose cohorts also make event-specific estimates imprecise. Report counts alongside percentages, avoid overinterpreting small numerical differences, and consider sensitivity to plausible alternative event definitions.
Specify the response or harm outcome, time horizon, and whether outcomes after treatment discontinuation count.
Decide which events truly prevent the outcome, which can be followed by later outcomes, and how ties are handled.
Use a first-event proportion, cumulative incidence, cause-specific hazard, or multistate summary according to the estimand.
Verify event ordering and follow-up, then present event counts and the precise interpretation for each dose.
Check Your Understanding
Answer each question using the stated event definitions. Solutions follow the questions.
- For response before death, a patient dies without responding by day 28. Should death be treated as a competing event or ordinary censoring? Explain.
- In a cohort of 40 patients, 10 respond before any competing event by day 28. What is the fixed-horizon first-event proportion?
- Why can a patient who discontinues treatment still contribute a response under a treatment-policy definition?
- In a multistate analysis, why might toxicity not be an absorbing state?
Solutions. (1) Death is a competing event: it occurs before response and makes later response impossible. Censoring it as if the patient could remain at risk for response would not describe the observed probability of response before death. (2) The proportion is \(10/40=0.250\), or 25.0%; it is the fraction with response first by day 28 under the specified definition. (3) A treatment-policy definition follows the outcome regardless of treatment stopping, so an observed response after discontinuation still counts if it occurs within the defined follow-up horizon. (4) Toxicity need not prevent later states: a patient may recover, discontinue treatment, or respond afterward. It is absorbing only if the model’s state definition or endpoint makes further transitions impossible.