From a Treatment Study to a Causal Risk Contrast
A treatment can affect both the event of primary interest and the event that prevents it from occurring. For example, a therapy might reduce disease progression while changing the risk of death before progression. A useful causal summary therefore needs to say which event is being contrasted, at what time horizon, and what happens to the competing event under each treatment strategy.
This workshop focuses on a treatment-policy question: what would the absolute risk of each event be by a fixed time \(t\) if everyone in a target population followed treatment strategy 1, compared with the risk if everyone followed strategy 0? The competing event is retained as part of the outcome process under each strategy. This is not the risk that would have occurred if the competing event had been removed.
The potential outcome \(J^a\) distinguishes the target event from its competing event. A person who experiences the competing event before the target event does not later contribute a target event in this first-event formulation. The cumulative incidence \(F_k^a(t)\) is consequently a probability in a process where both event types can occur; it is not a net risk calculated as if competing events were absent.
Identification: What Must Be Assumed?
The causal risks are defined in terms of potential outcomes, which are not jointly observed for each person. In an observational study, identification requires assumptions linking those potential outcomes to the observed treatment and event data. Let \(X\) denote measured baseline covariates sufficient to control confounding.
Under consistency, exchangeability, and positivity, the causal cumulative-incidence risk is identified by standardization:
For continuous covariates, the sum is an integral over the target population’s covariate distribution. The conditional probability is a cumulative incidence: it accounts for the competing event as an event type, not as ordinary censoring. If the study has loss to follow-up, its censoring assumptions and estimation method must also support estimation of that conditional cumulative incidence.
The derivation is the law of total probability applied to the potential outcome \(I(T^a\leq t,J^a=k)\). First condition its expectation on \(X\). By exchangeability, within each covariate profile its mean under strategy \(a\) equals the observed outcome probability among people with \(A=a\). Consistency identifies that observed probability with the corresponding potential-outcome probability, and positivity ensures the treated or untreated data needed for that profile exist. Averaging over the target distribution of \(X\) gives the expression above.
Worked Example: A Randomized Study With Complete Follow-Up
Worked Example: A Randomized Study With Complete Follow-Up
In a hypothetical randomized study, 200 participants are assigned to each of two treatment strategies. By the prespecified horizon, the treatment group has 36 target events, 26 competing events, and 138 participants with neither event. The control group has 50 target events, 14 competing events, and 136 participants with neither event. Assume complete follow-up through the horizon and a well-defined treatment-policy question.
Random assignment supports exchangeability, and both strategies have positive probability. Consistency and no interference are also assumed. Since follow-up is complete, censoring does not complicate this calculation. The observed proportions estimate the corresponding treatment-policy risks.
For the target event, the two risks are \(36/200=0.180\) and \(50/200=0.250\). Thus the risk difference is:
The estimated target-event risk is 7 fewer per 100 participants under treatment by this horizon. The competing-event risks are \(26/200=0.130\) and \(14/200=0.070\), so their risk difference is \(0.130-0.070=0.060\), or 6 more competing events per 100 participants. The combined risk of either event is \(62/200=0.310\) versus \(64/200=0.320\), a difference of \(-0.010\), or 1 fewer first event of either type per 100. These are separate contrasts: the competing-event risk does not enter the arithmetic for the target-event risk difference.
In a nonrandomized study, the same arithmetic applied to crude treatment-group proportions generally does not identify causal risks. The covariate distributions of the groups may differ, which is why standardization is needed.
Worked Example: Standardizing Absolute Risks
Worked Example: Standardizing to a Common Population
Consider a hypothetical observational study with a binary baseline covariate \(X\), classified as lower or higher baseline risk. The target population is 60% lower-risk and 40% higher-risk. Suppose conditional cumulative-incidence estimates at the chosen horizon are as follows.
| Baseline risk group | Target risk, strategy 0 | Target risk, strategy 1 | Competing risk, strategy 0 | Competing risk, strategy 1 |
|---|---|---|---|---|
| Lower | 0.100 | 0.080 | 0.050 | 0.080 |
| Higher | 0.400 | 0.320 | 0.200 | 0.240 |
Assume the conditional estimates are valid under consistency, conditional exchangeability given \(X\), positivity in both risk groups, and the required censoring conditions. Standardize each strategy to the same 60%/40% target population. For the target event:
The standardized target-event risk difference is \(0.176-0.220=-0.044\): an estimated 4.4 fewer target events per 100 people by the horizon. The risk ratio is \(0.176/0.220=0.800\), corresponding to a 20% lower risk relative to strategy 0. The absolute contrast is usually easier to translate into expected event counts for a population.
For the competing event, the standardized risks are \((0.60)(0.050)+(0.40)(0.200)=0.110\) under strategy 0 and \((0.60)(0.080)+(0.40)(0.240)=0.144\) under strategy 1. The competing-event risk difference is \(0.144-0.110=0.034\), or 3.4 more per 100. The event-free risks are \(1-0.220-0.110=0.670\) and \(1-0.176-0.144=0.680\), respectively. These calculations show why reporting the target-event contrast alone may not describe how the overall event experience changes.
For illustration, suppose 50% of the observed strategy-0 group but only 30% of the strategy-1 group were higher-risk. Their crude target-event risks would be \((0.50)(0.100)+(0.50)(0.400)=0.250\) and \((0.70)(0.080)+(0.30)(0.320)=0.152\). Their crude difference, \(-0.098\), is not the standardized causal contrast of \(-0.044\). The groups were averaged over different covariate distributions; standardization instead asks what each strategy’s risk would be in the same target population.
Worked Example: Contrasts Depend on the Event Definition
Worked Example: Target Event Versus Either Event
Suppose a hypothetical randomized study estimates target-event risks of 0.120 under strategy 0 and 0.090 under strategy 1 by a fixed horizon. The competing-event risks are 0.040 and 0.070, respectively. Assume randomization, consistency, complete outcome ascertainment, and no interference.
The target-event risk difference is \(0.090-0.120=-0.030\), or 30 fewer target events per 1,000 people under strategy 1. Its risk ratio is \(0.090/0.120=0.750\), a 25% lower target-event risk relative to strategy 0. For the competing event, the risk difference is \(0.070-0.040=0.030\), or 30 more per 1,000.
If the outcome is instead defined as either first event, the risks are \(0.120+0.040=0.160\) and \(0.090+0.070=0.160\). The composite risk difference is \(0.160-0.160=0\). The target-event contrast and the composite contrast answer different questions; neither is a correction to the other. The choice of outcome should follow the clinical question, and the component risks should be visible when a composite could conceal opposing changes.
Estimation Checks, Alternatives, and Limitations
In practice, conditional cumulative incidences can be estimated within covariate strata or using an outcome model, then averaged over the target population’s covariates. With many covariates, analysts may use a flexible model and predict each person’s risk under each strategy before averaging predictions. The model and the standardization target should be reported. Uncertainty estimates should account for estimation of the conditional risks and, when relevant, the target covariate distribution; a confidence interval quantifies sampling uncertainty under the analysis assumptions, not uncertainty that those assumptions are true.
Several checks can reveal when an estimate is unreliable. Inspect treatment overlap across important covariate profiles; very few observations under one strategy indicate a positivity problem. Examine whether censoring is concentrated in particular groups or occurs before the target horizon, and check whether the censoring adjustment is supported by observed follow-up. Assess model fit and calibration for the conditional event risks, including whether predictions are plausible and whether predicted event probabilities are compatible with the competing-event probabilities. Large changes under reasonable model specifications can signal model dependence. These diagnostics cannot prove exchangeability, which depends on having measured the relevant confounders.
| Approach | What it summarizes | How it differs from this risk contrast |
|---|---|---|
| Cause-specific hazard contrast | Relative instantaneous event rate among those still event-free | It is not itself an absolute probability of the event by time \(t\). |
| Subdistribution-hazard contrast | A model-based contrast associated with the cumulative-incidence process | It is a hazard-scale parameter, not a risk difference or risk ratio. |
| Hypothetical competing-event elimination | Risk under a specified intervention that removes or changes the competing event | It targets a different intervention from retaining competing events under each treatment strategy. |
A common mistake is to treat people who experience the competing event as if they were ordinary censored observations when calculating the target-event risk, then interpret the resulting quantity as the observed-world probability of the target event. The correction is to estimate the target cumulative incidence while representing the competing event as a distinct event type. Another mistake is to condition on remaining free of the competing event and call the resulting comparison a population causal effect; that selects post-treatment survivors and generally changes the question. Define the event, treatment strategies, horizon, and target population before calculating a contrast.
Check Your Understanding
For each question, give a brief answer before reading the solution that follows it.
- In a treatment-policy estimand, how is a competing event handled?
- A standardized target-event risk is 0.14 under strategy 1 and 0.20 under strategy 0. What are the risk difference and risk ratio, and how would the difference be described per 100 people?
- Why might crude treatment-group risks differ from standardized risks?
- What does a cause-specific hazard contrast summarize that a cumulative-incidence risk difference does not?
- Name two checks relevant to the validity or stability of a standardized risk estimate.
Solutions. (1) It remains a distinct event in the outcome process under each strategy; it is not removed by the treatment-policy estimand. (2) The difference is \(0.14-0.20=-0.06\), or 6 fewer target events per 100 people by the stated horizon. The ratio is \(0.14/0.20=0.70\), a 30% lower risk relative to strategy 0. (3) The treatment groups may have different covariate distributions, so their crude risks average conditional risks over different populations. (4) A cause-specific hazard contrast summarizes relative instantaneous rates among those still event-free; it does not directly give an absolute probability by a fixed horizon. (5) Examples include checking treatment overlap by covariate profile, censoring and follow-up support, conditional-risk model calibration, and sensitivity to model specification.