Total Effects on Cause-Specific Risk

What Does a Total Effect on Cause-Specific Risk Mean?

When an intervention can affect more than one event type, its effect on one chosen event cannot be described by looking at that event in isolation. A person who experiences a competing event can no longer experience the event of interest first. The cause-specific risk therefore depends on the treatment’s effects on the event of interest and on the competing events that occur naturally under that treatment.

For example, consider an intervention intended to prevent a disease event when death from another cause can occur first. The total effect on disease risk compares the probability of disease by a specified time under intervention with the probability under control. In each case, other causes of death are allowed to happen as they would under the assigned treatment. This is a causal contrast in risks, not a comparison of instantaneous event rates.

Definition: Let \(A\) denote treatment, with levels \(a=1\) and \(a=0\). For each person, let \(T^a\) be the event time that would occur under treatment \(a\), and let \(J^a\) be its event type. For a chosen cause \(k\) and time \(\tau\), the treatment-specific cumulative incidence is \(F_k^a(\tau)=P(T^a\leq \tau,\ J^a=k)\). A total causal effect on cause-specific risk is a prespecified contrast between \(F_k^1(\tau)\) and \(F_k^0(\tau)\). Both risks allow competing events to occur naturally under their respective treatments.

The word total refers to the whole effect of assigning treatment on this cause-specific risk. It includes any treatment effects that operate through changes in the occurrence of competing events. It does not mean the probability of having any event, and it does not mean the risk of the chosen event in a hypothetical world where competing events have been removed.

A contrast must specify the cause, the time horizon, the treatment strategies being compared, and the scale on which the risks are contrasted. For instance, a contrast can express how many more or fewer people per 100 would experience cause \(k\) by \(\tau\) under treatment than under control. The estimand is about the population risks under interventions; it is not automatically identified by comparing the groups observed in a particular data set.

Why Competing Events Are Part of the Total Effect

Under treatment \(a\), write \(S^a(u-)\) for the probability of remaining free of every event just before time \(u\), and let \(\Lambda_k^a\) be the cumulative cause-specific hazard for cause \(k\). The cumulative incidence can be expressed as \(F_k^a(\tau)=\int_0^\tau S^a(u-)\,d\Lambda_k^a(u)\). When the hazards have rates \(\lambda_j^a(u)\), this becomes an integral of the event-free probability multiplied by the cause-\(k\) rate.

This expression follows by considering small time intervals. A cause-\(k\) event at time \(u\) requires both that the person has remained event-free up to \(u\), and that cause \(k\) occurs at \(u\). The probability of the first condition is \(S^a(u-)\); the conditional increment in cause-\(k\) event probability is represented by \(d\Lambda_k^a(u)\). Summing those contributions over follow-up gives the cumulative incidence. Since \(S^a\) depends on all event types, changing a competing-event process can change \(F_k^a\), even if the cause-\(k\) hazard itself does not change.

The causal estimand exists as a feature of the potential outcomes, whether or not a particular statistical model is used. To identify it from observed data, the treatment comparison and follow-up must support the relevant counterfactual risks. A common identification set of assumptions is:

Consistency: a person who receives treatment \(a\) has the event time and cause represented by \(T^a,J^a\). Treatment strategies must be sufficiently well defined that “treatment \(a\)” does not hide materially different versions.
No interference: one person’s treatment does not alter another person’s potential event time or cause.
Exchangeability: conditional on measured baseline covariates \(L\), treatment assignment is independent of the potential event outcomes. Randomization supports this condition by design; in an observational study it requires that relevant common causes of treatment and outcome have been measured and adequately handled.
Positivity: each treatment being compared has positive probability at every covariate pattern in the target population.
Appropriate handling of censoring: losses to follow-up must not be treated as event-free outcomes. For standard censoring adjustment, censoring must be independent of the event process conditional on the variables used in the adjustment, with adequate follow-up support through \(\tau\).

Under consistency, exchangeability, and positivity, the treatment-specific risk can be identified by standardizing conditional risks over the target population’s baseline covariate distribution:

F_k^a(\tau) = \int P(T\leq \tau,\ J=k\mid A=a,L=l)\,dP(l).

Here the conditional risk must account appropriately for censoring. In a randomized study, the treatment groups are exchangeable by design, so a treatment-arm cumulative incidence can estimate the corresponding intervention risk, subject to consistency, follow-up, and censoring conditions. In an observational study, the crude treatment-arm risks need not estimate the standardized causal risks when the covariate distributions differ.

Worked Example: Interpreting the Potential-Outcome Risks

Suppose a hypothetical target population contains 1,000 people. At a prespecified horizon \(\tau\), under intervention, 120 would experience the chosen event, 180 would experience a competing event first, and 700 would remain event-free. Under control, the corresponding numbers would be 90, 260, and 650. These are population counts under each intervention, not claims that each person’s outcomes under both treatments are observed.

The intervention risk for the chosen cause is \(120/1000=0.12\), or 12%. The control risk is \(90/1000=0.09\), or 9%. On the risk-difference scale, the contrast is \(\,0.12-0.09=0.03\,\), or 3 percentage points by \(\tau\). In this hypothetical population, the intervention increases the chosen cause’s cumulative incidence by 3 percentage points.

The competing-event risks are 18% under intervention and 26% under control. Those differences are not a reason to remove competing events from the chosen-cause comparison: they are part of the natural outcomes under each treatment and affect who remains able to experience the chosen event. The calculation does not tell us which individuals would experience the chosen event under both treatments.

Estimating the Contrast in a Randomized Example

With observed data, a treatment-specific cumulative incidence can be estimated from the event counts and risk sets while retaining competing events as competing outcomes. In the simple example below, follow-up has two event times, there is no censoring before \(\tau\), and all 100 participants in each arm are followed through \(\tau\). The treatment is randomized, the event types are consistently classified, and there is no interference. These conditions support a causal interpretation of the arm-specific risks in this hypothetical study.

Worked Example: Contrasting Two Cumulative Incidences

In the intervention arm, at time 1 there are 8 events of cause \(k\) and 10 competing events among 100 people. At time 2, among the 82 people still event-free, there are 20 events of cause \(k\) and 8 competing events. In the control arm, at time 1 there are 10 events of cause \(k\) and 20 competing events among 100 people. At time 2, among the 70 still event-free, there are 14 events of cause \(k\) and 7 competing events. Take \(\tau=2\).

For these discrete event times, each contribution to the cause-\(k\) cumulative incidence is the probability of having remained event-free to that time multiplied by the fraction of the current risk set experiencing cause \(k\). Thus, in the intervention arm,

\widehat F_k^1(2) = \frac{8}{100} + \left(\frac{82}{100}\right)\left(\frac{20}{82}\right) = 0.08+0.20 = 0.28.

The corresponding control calculation is

\widehat F_k^0(2) = \frac{10}{100} + \left(\frac{70}{100}\right)\left(\frac{14}{70}\right) = 0.10+0.14 = 0.24.

The estimated risk contrast, intervention minus control, is \(0.28-0.24=0.04\), or 4 percentage points by time 2. Under the stated randomized-study assumptions, this estimates a total causal effect on cause-\(k\) risk. It is not an effect on an instantaneous rate. The competing-event counts are included in the risk-set changes: for example, the 20 competing events in the control arm leave fewer people event-free at time 2.

As a check on the arithmetic, the intervention arm has \(8+10+20+8=46\) observed events and 54 people event-free at time 2. Its estimated cumulative incidences are 28% for cause \(k\) and \(10/100+(82/100)(8/82)=18\%\) for the competing cause; together with 54% event-free, these sum to 100%. The analogous control proportions are 24%, 27%, and 49%, also summing to 100%.

Covariate Adjustment and Standardization

In an observational study, treatment groups may differ in baseline prognosis. Adjustment should target the risks under each treatment in the same population, rather than comparing conditional risks from different covariate mixes. Standardization does this by estimating a conditional cumulative incidence for each treatment and averaging it over one explicitly chosen target distribution of \(L\).

Worked Example: Standardizing Over a Shared Population

Suppose a hypothetical population is half in a lower-risk covariate stratum and half in a higher-risk stratum. The cause-specific cumulative incidences by \(\tau\), within the strata, are:

Baseline stratum	Risk under intervention	Risk under control
Lower-risk	0.10	0.05
Higher-risk	0.40	0.30

Using the target population’s 50%/50% stratum weights, the standardized intervention risk is \(0.50(0.10)+0.50(0.40)=0.25\). The standardized control risk is \(0.50(0.05)+0.50(0.30)=0.175\). Their risk-difference contrast is \(0.25-0.175=0.075\), or 7.5 percentage points by \(\tau\).

Now suppose the observed intervention group is 80% higher-risk and the observed control group is 20% higher-risk. Their crude risks would be \(0.80(0.40)+0.20(0.10)=0.34\) and \(0.20(0.30)+0.80(0.05)=0.10\), respectively. The crude contrast of 24 percentage points differs from the standardized contrast because the groups have different covariate mixes. The standardized result has a causal interpretation only if the measured covariates suffice for exchangeability, positivity holds, and consistency and censoring assumptions are also reasonable.

Interpretation, Diagnostics, and Alternatives

A useful interpretation names the event, treatment strategies, population, time horizon, and contrast scale. For example: “By 2 years, assignment to the intervention changes the probability of experiencing cause \(k\) before any competing event by an estimated 4 percentage points, compared with control.” This describes an average population contrast. It does not say that treatment caused the event for a particular person, nor does it imply a change in the event’s instantaneous rate.

Several checks help reveal when the analysis may be going wrong:

Confirm that the event of interest and every competing event have been coded as distinct outcomes, and that the time origin and horizon are the same across treatment groups.
Check whether follow-up and censoring differ substantially by treatment or covariates. Sparse follow-up near \(\tau\) makes risks at that horizon weakly supported by the data.
For observational analyses, inspect covariate overlap and whether each treatment occurs across the covariate patterns used for standardization. Severe lack of overlap undermines positivity and can make standardized estimates unstable.
Check that estimated cause-specific cumulative incidences are nonnegative and that the sum across causes does not exceed one. A violation can indicate an estimation or coding error.
Consider whether the adjustment model captures important covariate relationships. A fitted model cannot repair unmeasured confounding or a treatment group with no comparable people in part of the target population.

A cause-specific hazard model addresses the instantaneous rate among people who remain event-free; its coefficient is not itself a total effect on cumulative incidence. A model for the subdistribution hazard uses a different quantity and likewise requires care before its coefficient is interpreted as a risk contrast. Combining all event types into an any-event outcome answers a different question, while eliminating competing events asks about a different hypothetical world. These approaches can be useful for their own targets, but they should not be substituted for the treatment-specific cause-\(k\) risks defined here.

The main limitation is that a causal interpretation relies on assumptions that data alone generally cannot verify, especially exchangeability in observational studies and the adequacy of censoring adjustment. The risk contrast is also specific to its chosen cause, follow-up horizon, treatment definition, and target population; it should not be generalized beyond those specifications without further justification.

Key takeaway: A total effect on cause-specific risk contrasts the cumulative incidence of one chosen event under two treatment interventions, allowing competing events to occur naturally under each. Because competing events affect the chance of remaining event-free, they are part of this total-effect estimand rather than events to be deleted.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

What does “total” mean in a total causal effect on cause-specific risk?
In a hypothetical randomized study, the estimated risks for cause \(k\) by 3 years are 0.16 under treatment and 0.21 under control. State the risk-difference contrast in context.
Why can treatment affect cause-\(k\) cumulative incidence even if it does not change the cause-\(k\) hazard?
An observational analysis standardizes risks to the intervention group’s covariate distribution. What population does this target, and what assumption is needed for a causal interpretation?

Solutions. (1) The contrast includes all pathways by which treatment changes the chosen cause’s risk, including changes in competing events; those events are allowed to occur naturally under each treatment. (2) Treatment minus control is \(0.16-0.21=-0.05\): treatment is associated with an estimated 5-percentage-point lower cause-\(k\) risk by 3 years; randomization and the other identification conditions support interpreting this as a causal contrast. (3) A change in competing events changes the probability of remaining event-free and therefore changes how many people can later experience cause \(k\). (4) It targets the intervention group’s covariate distribution. A causal interpretation requires conditional exchangeability given the measured covariates, as well as consistency, positivity, and appropriate handling of censoring.