Deriving the Competing-Risk G-Formula

Why the Competing-Risk G-Formula Needs Every Cause

A cause-specific event probability at a particular time is conditional on remaining event-free up to that time. It is not, by itself, the probability of experiencing that cause by the end of follow-up. To obtain a cumulative incidence, we must combine the probability of reaching each time without any event with the probability of the event of interest at that time. Because any competing cause also removes a person from the event-free group, its conditional probability enters that calculation too.

The discrete-time g-formula makes this combination explicit. For each treatment level and baseline covariate profile, it chains together the conditional probabilities for all causes and times. It then averages the resulting treatment-specific risk over the covariate distribution of the target population. This is the competing-risk counterpart of standardization for a total causal effect.

Define \(S^a(t\mid x)=P(T>t\mid\) the intervention setting \(A=a,X=x)\), with \(S^a(0\mid x)=1\). The treatment-specific cumulative incidence for cause \(k\) through time \(t\), conditional on \(X=x\), is \(F_k^a(t\mid x)=P(T\le t,J=k\mid\) the intervention setting \(A=a,X=x)\). In this discrete-time setup, the corresponding g-formula is

For a discrete baseline covariate distribution, the population risk under treatment \(a\) is

For continuous covariates, replace the sum over \(x\) by an integral with respect to the target population distribution of \(X\). A causal risk difference, for example, is \(F_k^1(t)-F_k^0(t)\). It compares the probability of cause \(k\) by time \(t\) under two treatment interventions, allowing competing events to occur as they naturally would under each intervention.

Why the Formula Holds

The derivation uses a simple recursion. For an individual with covariates \(x\) under treatment \(a\), being event-free after time \(t\) requires being event-free before time \(t\), then having no event of any cause at \(t\). Therefore, \(S^a(t\mid x)=S^a(t-1\mid x)\{1-\sum_j q_j(t\mid a,x)\}\). Starting from \(S^a(0\mid x)=1\) and applying this recursion successively gives the product expression for survival.

The probability of cause \(k\) at time \(s\), conditional on \(x\), is the probability of being event-free just before \(s\) multiplied by the conditional probability of cause \(k\) at \(s\). Thus the probability of cause \(k\) at that time is \(S^a(s-1\mid x)q_k(s\mid a,x)\). The events \(\{T=s,J=k\}\) for \(s=1,\ldots,t\) are mutually exclusive, so their probabilities add. This gives \(F_k^a(t\mid x)\). Averaging these conditional risks over the baseline distribution of \(X\) gives the population g-formula.

These conditions connect observed conditional probabilities to the intervention risks; the product-and-sum calculation alone does not remove confounding or repair unsupported extrapolation. In practice, the event probabilities may be estimated with a discrete-time model, separately within covariate strata, or by another suitable approach. Whatever the estimation method, the probabilities must describe all event causes among those still event-free. The causal interpretation comes from the identification assumptions, not from the algebraic form of the formula.

Worked Calculations

Reading the Result and Checking the Calculation

The g-formula is useful because a treatment can change both the event of interest and the competing event, thereby changing who remains at risk at later times. The formula accounts for this changing event-free fraction at every time point. It produces the probability of the cause of interest in the presence of competing events, under each treatment intervention and in the stated target population.

Several checks help identify problems in an implementation:

• For each treatment, covariate profile, and time, every \(q_j(t\mid a,x)\) must be between 0 and 1, and their sum across causes must not exceed 1.
• The calculated event-free probability must be nonincreasing over time, and each cause-specific cumulative incidence must be nondecreasing.
• At each horizon, the sum of the cause-specific cumulative incidences and the event-free probability should equal 1, apart from rounding.
• Examine whether treatment and censoring information support the covariate profiles being standardized to. Sparse data, extreme model predictions, and poor calibration of event probabilities can undermine the calculation.

The principal alternative summaries answer different questions or use different information. A cause-specific hazard describes the conditional event probability among people still event-free; it is not a cumulative risk. A net-risk calculation that removes the competing event as if it were censoring does not give the observed-world cumulative incidence. An unadjusted cumulative-incidence estimate summarizes the observed treatment groups, while standardization uses conditional risks and a common target covariate distribution to address measured baseline confounding. Inverse probability weighting is another way to adjust for treatment assignment, but its estimation and diagnostics are separate topics.

The formula does not eliminate the need to choose an appropriate target population, measure adequate confounders, or assess whether the fitted conditional probabilities are credible. It also does not identify risks for covariate profiles with no treatment support. When censoring depends on evolving observed history, a model using only baseline covariates may not be enough to identify the desired risks. These are limitations of the identification and estimation strategy, not issues that can be fixed by rearranging the g-formula.