Absolute Risk As a Competing-Risk Prediction Target

What Absolute Risk Means in a Competing-Risk Setting

A clinical prediction such as “the risk of relapse within two years” is incomplete unless it says who is at risk, when the clock starts, and what happens if another event occurs first. A patient who dies before relapse cannot subsequently experience relapse as the first event. The relevant probability therefore concerns not just whether relapse occurs, but whether it occurs by the specified time before a competing event.

The target is a probability for a defined population and a defined event process. It is not, by itself, a particular statistical model or calculation procedure. The earlier material on cumulative incidence functions introduced the probability of a particular event type occurring by time $t$; here the emphasis is on specifying that probability as the clinical prediction target.

Definition: Let $T$ be the time from a specified time origin to the first event of any type, and let $J$ identify the type of that first event. For a target population with probability law $P^*$, the horizon-specific absolute risk for event type $k$ by time $\tau$ is $$ F_k^*(\tau)=P^*(T\leq \tau,\ J=k). $$ It is the probability that a member of the target population experiences event $k$ as the first event, by time $\tau$. A competing event is a different event type that occurs first and prevents the target event from subsequently occurring as the first event.

The notation makes the parts of the question explicit. $k$ names the event type of interest; $\tau$ is the prediction horizon, in the time units measured from the stated origin; and $P^*$ refers to the population for whom the prediction is intended. For example, “relapse within two years after diagnosis, before death, among adults newly diagnosed with the disease” identifies an event type, a horizon, a time origin, a competing event, and a population.

The event $T\leq\tau$ alone is not enough: it includes first events of every type. Likewise, the event $J=k$ alone is not enough: it does not restrict the event to the chosen horizon. Both conditions are required. If the target event happens after a competing first event, it is not counted in $F_k^*(\tau)$.

Work Through the Target Definition

Worked Example: Relapse Risk Within Two Years

Suppose the target population is adults newly diagnosed with a particular cancer, and time zero is the date of diagnosis. Let $J=1$ denote relapse as the first event and let death before relapse be a competing event. In a hypothetical, fully observed group of 1,000 people, by two years 82 have relapsed first, 61 have died first, and 857 have had neither first event.

The target is $P^*(T\leq2\text{ years},J=1)$. Under the stated complete follow-up and representative-sample setup, its empirical proportion is $$ \frac{\text{number with relapse first by 2 years}}{\text{number in the target group}} =\frac{82}{1000}=0.082. $$

Thus the estimated two-year absolute risk is 0.082, or 8.2%. It describes the proportion expected to experience relapse first by two years in the defined target population. The 61 people who died first are not counted as relapses; neither are the 857 who had no first event by the horizon.

Worked Example: A Post-Surgery Complication

Consider a hypothetical target population of 500 patients undergoing a specified operation. Time zero is the end of surgery, event type $J=1$ is a serious surgical complication, and death before that complication is a competing event. By 30 days, 21 patients have had the complication first, 9 have died first, and 470 have had neither event first.

The 30-day target is $P^*(T\leq30\text{ days},J=1)$. With the hypothetical complete ascertainment described above, the calculation is $$ \frac{21}{500}=0.042. $$

The corresponding estimated absolute risk is 4.2% by 30 days after surgery. The denominator is all 500 members of the defined target group, not only the 491 who did not die first. Restricting the denominator to those who avoided death would describe a different quantity.

Worked Example: The Population Is Part of the Target

Suppose a hypothetical study reports 20 first events of type $k$ by one year among 1,000 people recruited from a community screening program. A separate hypothetical referral clinic reports 80 such first events among 1,000 people. In both settings, the event definition and the time origin are otherwise the same.

For the screening-program population, the empirical risk is $$ \frac{20}{1000}=0.020, $$ or 2.0% by one year. For the referral-clinic population, it is $$ \frac{80}{1000}=0.080, $$ or 8.0% by one year. $$

These figures do not contradict one another: they refer to different target populations. The clinic result should not automatically be presented as the risk for people in the screening program. A prediction is meaningful only when its intended population is stated and its data and analysis support that intended use.

Why This Probability Has This Form

The definition can be justified directly from the event sequence. Under the competing-risk setup, each person has a first-event time $T$, and the type of that first event is $J$. For event type $k$ to be the first event by $\tau$, two conditions must hold: the first event has occurred by $\tau$, and its type is $k$. Their intersection is exactly $\{T\leq\tau,J=k\}$. Taking its probability in the target population gives the absolute-risk target.

\begin{aligned} \{\text{event }k\text{ occurs first by }\tau\} &=\{T\leq\tau\}\cap\{J=k\},\\ P^*(\text{event }k\text{ occurs first by }\tau) &=P^*(T\leq\tau,\ J=k) =F_k^*(\tau). \end{aligned}

This justification also clarifies why both the event type and the horizon belong in the definition. Dropping $J=k$ asks about any first event by $\tau$, while dropping $T\leq\tau$ asks about event type $k$ without the stated time restriction. Neither is the specified horizon-specific absolute risk.

Target Probability, Data, and Assumptions

The probability $F_k^*(\tau)$ is a population quantity. An observed proportion or a model-based prediction is an estimate of that quantity, not the target itself. If every member of a representative sample has event status known through $\tau$, the proportion with $T\leq\tau$ and $J=k$ estimates the target. In studies with right censoring before $\tau$, some participants’ first-event status by the horizon is unknown; they cannot simply be counted as event-free. Competing-risk methods, such as those covered in the earlier material on cumulative incidence, are needed to estimate the probability appropriately.

The target’s definition does not require a statistical model. But a valid estimate or prediction requires that the data and analysis correspond to the target. In particular, event types must be defined and classified consistently; the time origin and horizon must be applied consistently; and the study population and follow-up information must support the intended target population and time range. In an analysis with censoring, the assumptions used by the selected estimation procedure must also be credible. These are conditions for learning about the target from data, rather than additional terms in the probability’s definition.

Before interpreting a reported risk, check the following:

Population: Who is included in the target population, and do the data represent them?
Time origin and units: When does follow-up begin, and is the horizon measured in days, months, or years from that point?
Event definition: What counts as event $k$, and how are other first events classified?
Follow-up: Is event status known through the horizon, or is estimation with censoring required?
Interpretation: Does the reported number refer to the stated population and to event $k$ occurring first by the stated horizon?

Choosing the Right Quantity to Report

Absolute risk is not interchangeable with every number associated with an event. A cause-specific hazard describes an event rate among people still at risk at a particular time; it is not itself the probability of the event by a horizon. A proportion experiencing any first event has a different event definition from the risk of a particular type. Similarly, changing the target event, the time origin, the horizon, or the target population changes the question even if the underlying dataset stays the same.

A useful report states the target in words before showing a number: “the probability of [event type] occurring first by [horizon] after [time origin] in [target population].” This wording helps clinicians and readers distinguish an absolute probability from a rate, a relative comparison, or a probability for a different population. It also makes it possible to judge whether the estimate answers the clinical question that motivated the prediction.

Common Mistakes and Limitations

Leaving the horizon implicit. “Relapse risk” could mean risk by one year, five years, or another time. State $\tau$ and its units.
Leaving out the time origin. “Within two years” is ambiguous unless the starting point, such as diagnosis or surgery, is specified.
Counting a competing first event as the target event. If death occurs before relapse, that person did not have relapse as the first event. Keep the event type explicit.
Treating censoring as proof of no event. Someone lost to follow-up before $\tau$ may have an unknown event status by $\tau$; appropriate methods are needed rather than automatic classification as event-free.
Applying a number to an unsupported population. A risk estimated in a referral clinic need not describe a community population with a different case mix or care setting.

A precisely stated target does not guarantee that a prediction is accurate. The data may be sparse near the horizon, event classifications may be unreliable, or the study population may differ from the intended population. Those limitations affect how well an analysis estimates or predicts $F_k^*(\tau)$; they do not change what the target means. This tutorial defines the quantity, while selecting a horizon and constructing particular prediction models require additional decisions.

Check Your Understanding

For each question, identify or calculate the requested competing-risk absolute-risk target. Solutions follow the questions.

For adults newly diagnosed with a disease, time zero is diagnosis, $J=1$ denotes relapse first, and death before relapse is competing. Write the target for relapse by two years.
In a hypothetical, fully observed target group of 400 people, 28 experience event type $k$ first by six months. Calculate the six-month empirical absolute risk.
Why should a person censored at four months not automatically be counted as having no event by a one-year horizon?
A report says “the one-year risk is 6%.” Name two details needed to determine exactly which absolute-risk target this refers to.

Solutions. (1) The target is $P^*(T\leq2\text{ years},J=1)$, with $T$ measured from diagnosis. (2) The empirical risk is $28/400=0.070$, or 7.0% by six months. (3) Censoring at four months means the person’s first-event status after that point is unknown; they could have experienced an event before one year. (4) The report should specify at least the event type and the target population; it should also state the time origin and clarify how competing first events are classified.