Subgroup-Specific Absolute Risk

Why Average Risk Within a Subgroup?

A predicted risk for one person answers an individual question: given this person’s covariates, what is the model-predicted probability of experiencing a particular cause of failure by a specified time? A subgroup summary answers a different question: what is the average of those predicted probabilities across the people in a defined subgroup? The distinction matters when a report describes “the five-year risk in subgroup A.” That phrase could refer to the risk for a particular profile chosen to represent A, or to an average across all members of A. These quantities need not be the same.

Here the focus is the subgroup average formed by predicting each subgroup member’s risk and then averaging those predictions. The resulting summary is tied to the covariate distribution of the subgroup being averaged over. It is not a prediction for an invented “typical” person, and it is not an average of hazard values.

Definition: Let $F_k(t\mid x)$ be the cumulative incidence of cause $k$ by time $t$, conditional on covariate profile $x$, in a competing-risks model. For a subgroup $g$ containing $n_g$ individuals with covariate profiles $x_i$, the subgroup-average predicted risk at time $t$ is $$ \widehat R_{gk}(t)=\frac{1}{n_g}\sum_{i\in I_g}\widehat F_k(t\mid x_i), $$ where $I_g$ is the set of subgroup members and $\widehat F_k(t\mid x_i)$ is the model-predicted cumulative incidence for cause $k$ for member $i$. This is the arithmetic mean of the individual predicted probabilities.

The definition requires a particular cause, time horizon, subgroup membership rule, and set of individual covariate profiles. For example, “cause 1 by five years among patients in subgroup A” defines a different summary from “any failure by five years among subgroup A.” The predicted cumulative incidence for cause $k$ accounts for competing events through the model: it is not the probability obtained by pretending that competing events cannot occur.

Conditions for Interpreting the Average

The arithmetic is simple, but the interpretation depends on what the inputs represent. Before averaging, check the following conditions.

Consistent prediction target. Every term must refer to the same cause $k$, time horizon $t$, time origin, and endpoint definition. A mix of one-year and five-year predictions does not estimate one clearly defined subgroup risk.
Competing-risk predictions. Each $\widehat F_k(t\mid x_i)$ must be a cumulative incidence from a competing-risks prediction procedure that accounts for the other causes. Cause-specific hazards can be used to construct such predictions, but averaging hazards is not the same operation as averaging cumulative incidences.
Meaningful covariate inputs. The profiles $x_i$ must correspond to the intended members of the subgroup, with covariates measured and coded as required by the fitted model. Missing or incorrectly coded predictors can make the individual predictions inappropriate.
Suitable model and population. The fitted model must be appropriate for the prediction task, and the subgroup’s covariate profiles must be within a range where its predictions are credible. If the observed subgroup is meant to represent a larger population, the data or any weighting scheme must also support that interpretation.
Clear subgroup definition. Membership must be determined consistently. If the subgroup is defined by a covariate, specify its categories or threshold; if it is defined by another rule, apply that rule to every record being averaged.

For an empirical subgroup with one record per person, the formula gives equal weight to every member. If a different target distribution is intended, weights may be needed, but then the weighted average and its target must be stated explicitly. The unweighted formula should not be described as a population-wide risk unless the subgroup records justify that interpretation.

Why the Average Has This Form

The averaging formula follows directly from the meaning of cumulative incidence. Let $Y_{ik}(t)$ equal 1 if individual $i$ experiences cause $k$ by time $t$, and 0 otherwise. Conditional on that individual’s covariates, the probability that $Y_{ik}(t)=1$ is $F_k(t\mid x_i)$. The average of these indicators is the subgroup’s observed proportion experiencing cause $k$ by $t$. Its conditional expectation is the average of the individual probabilities:

E\left\{\frac{1}{n_g}\sum_{i\in I_g}Y_{ik}(t)\,\middle|\,x_i,\ i\in I_g\right\} =\frac{1}{n_g}\sum_{i\in I_g}E\{Y_{ik}(t)\mid x_i\} =\frac{1}{n_g}\sum_{i\in I_g}F_k(t\mid x_i).

The first equality uses linearity of expectation; it does not require the indicators to be independent. Replacing each unknown conditional probability with its model prediction gives the plug-in estimate $\widehat R_{gk}(t)$. This reasoning explains why the individual predictions are averaged: the subgroup-average probability is the average of the members’ cause-specific event probabilities. It does not establish that a fitted model is correct or that the records represent a broader population.

Worked Example: Averaging Five-Year Risks in a Subgroup

Suppose a hypothetical clinical dataset has four members of subgroup A. A fitted competing-risks model gives each person’s predicted probability of cause 1 by five years as shown below. Assume the four profiles have complete model covariates and the predictions all use the same cause definition and five-year horizon.

Member	Predicted cause 1 risk by 5 years	Predicted cause 2 risk by 5 years
1	0.08	0.04
2	0.12	0.06
3	0.20	0.08
4	0.32	0.10

The subgroup-average predicted risk for cause 1 is

\widehat R_{A,1}(5) =\frac{0.08+0.12+0.20+0.32}{4} =\frac{0.72}{4} =0.18.

Thus the estimated subgroup-average five-year risk of cause 1 is 0.18, or 18%. The corresponding average for cause 2 is $(0.04+0.06+0.08+0.10)/4=0.07$, or 7%. For each person, the two cause-specific cumulative incidences sum to 0.12, 0.18, 0.28, and 0.42, respectively, all no greater than 1. The subgroup averages sum to $0.18+0.07=0.25$; this is the average predicted probability of failure from either of these two causes by five years, provided these are the only causes represented. It is not the risk for every member: individual predicted risks for cause 1 range from 8% to 32%.

Subgroup Average Versus Representative Profile

A prediction for one covariate profile, $\widehat F_k(t\mid x^\ast)$, answers a profile-specific question. The subgroup average, $\widehat R_{gk}(t)$, averages predictions over the profiles of subgroup members. A profile can be chosen for clinical or descriptive reasons, or it might be constructed from subgroup summaries. Either way, one profile supplies one prediction; it does not automatically represent the average of the subgroup’s individual predictions.

To keep the quantities distinct, report the profile and the subgroup average separately when both are useful. The profile prediction should be described with its covariate values. The subgroup average should identify the subgroup, the cause, the horizon, and the members or target distribution over which averaging was done. A value for one carefully selected person is not converted into a subgroup-average estimate simply by labelling that person “representative.”

Worked Example: A Profile Prediction Is Not the Subgroup Average

Consider a hypothetical subgroup of three people. At a specified three-year horizon, a competing-risks model predicts cause 1 risks of 0.10, 0.20, and 0.40 for their respective covariate profiles. A separate profile of interest, chosen to summarize the subgroup’s covariates, has a model-predicted cause 1 risk of 0.22. Assume all four predictions use the same fitted model, endpoint, and horizon.

The subgroup-average prediction is calculated from the three members:

\widehat R_{g,1}(3) =\frac{0.10+0.20+0.40}{3} =\frac{0.70}{3} \approx 0.2333.

The representative-profile prediction is 0.22, or 22%; the subgroup-average prediction is approximately 0.2333, or 23.33%. Both are legitimate model outputs, but they answer different questions. The first is conditional on the stated profile; the second averages the three members’ individual predictions. In this example they are close, but closeness does not make their definitions interchangeable. To report an average over these three subgroup members, use 23.33%, not the profile’s 22%.

Subgroup Summaries and the Pooled Average

A useful arithmetic check is that an unweighted pooled average across several disjoint subgroups can be recovered from subgroup averages by weighting each subgroup mean by its size. This identity is about how the records are partitioned; it does not say that a pooled average is the best summary for every question.

\frac{1}{n}\sum_g\sum_{i\in I_g}\widehat F_k(t\mid x_i) =\sum_g\frac{n_g}{n}\widehat R_{gk}(t), \qquad n=\sum_g n_g.

Worked Example: Checking Two Subgroup Averages Against the Pooled Average

Suppose a hypothetical cohort has three members in subgroup A, with predicted cause 1 risks by two years of 0.10, 0.20, and 0.30. Subgroup B has two members with predicted risks of 0.25 and 0.35. The model, cause, and horizon are the same for all five people.

For subgroup A, the average is $(0.10+0.20+0.30)/3=0.60/3=0.20$. For subgroup B, it is $(0.25+0.35)/2=0.60/2=0.30$. The pooled average must weight these subgroup averages by their sizes:

\widehat R_{\mathrm{pooled},1}(2) =\frac{3}{5}(0.20)+\frac{2}{5}(0.30) =0.12+0.12 =0.24.

Directly averaging all five risks gives $(0.10+0.20+0.30+0.25+0.35)/5=1.20/5=0.24$, confirming the result. Taking the unweighted mean of the two subgroup averages would instead give $(0.20+0.30)/2=0.25$, which is not the pooled mean for these five records because the subgroups have different sizes. The subgroup-specific estimates remain 0.20 and 0.30; the pooled calculation does not replace them.

Practical Checks and Common Pitfalls

Before interpreting a subgroup estimate, inspect both the records and the predictions. Check that each intended subgroup member appears once in an unweighted average, that the subgroup count $n_g$ is correct, and that no records from another subgroup were included. Verify the prediction columns use the same event coding and horizon. Individual cumulative incidences should be probabilities between 0 and 1; where multiple causes are modeled, their cumulative incidences should collectively respect the probability bound. Unexpected values or inconsistent cause labels can indicate a prediction or data-processing problem.

The estimated average also depends on which subgroup members are included. A subgroup with a narrow covariate range may yield a different average from a broader subgroup defined by a looser rule. A small or unusual set of profiles can make the average sensitive to individual predictions. These are reasons to state the subgroup definition and the number of members, not reasons to replace the average with a profile prediction without changing the scientific question.

Pitfall: treating a representative profile as the average. The wrong reasoning is, “This profile summarizes the subgroup, so its predicted risk must be the subgroup’s average risk.” The correction is to calculate the prediction for every subgroup member and average those predictions when the target is the subgroup-average risk. A profile prediction is reported as conditional on that profile.
Pitfall: averaging risks from mismatched targets. Combining predictions for different horizons or different causes does not produce a clearly defined subgroup-specific cause risk. Align the event, time, and subgroup before averaging.
Pitfall: averaging hazard predictions instead. A mean hazard is not the defined average cumulative incidence. First obtain each person’s predicted cumulative incidence for the chosen cause and horizon, then average those probabilities.
Pitfall: claiming more than the model and records support. The arithmetic mean of model predictions is a model-based estimate for the profiles being averaged. It is not automatically a validated population risk or an observed event proportion.

Compared with a profile-specific prediction, the subgroup average summarizes the actual set of covariate profiles in the subgroup. Compared with a pooled average, it preserves the subgroup distinction rather than combining members across groups. Its advantage is that the summary matches a subgroup-average question; its limitation is that it depends on the chosen subgroup and its represented covariate distribution, as well as on the adequacy of the prediction model.

Key takeaway: For subgroup $g$, cause $k$, and time $t$, calculate each member’s predicted cumulative incidence and take their mean. State the subgroup, cause, horizon, and averaging set. Do not substitute a prediction for one representative profile or an average hazard for this subgroup-average risk.

Check Your Understanding

Work through each question, then check the solutions below.

Three subgroup members have predicted cause 1 risks of 0.05, 0.15, and 0.25 by four years. What is the subgroup-average predicted risk?
A model predicts a cause 1 risk of 0.18 for a stated profile. Is that value, by itself, the subgroup-average risk for the group from which the profile was chosen? Explain.
Subgroup A has 8 members and an average predicted risk of 0.10; subgroup B has 2 members and an average predicted risk of 0.30. What is the pooled average across the 10 members?
Why is the average of predicted cause-specific hazards not the subgroup-average cumulative incidence defined here?

Solutions. (1) $(0.05+0.15+0.25)/3=0.45/3=0.15$, so the subgroup-average predicted risk is 15% by four years. (2) No. The value 0.18 is conditional on one profile. The subgroup average requires the predictions for all subgroup members to be averaged; the profile value could coincide with that average, but it is not guaranteed to. (3) Weight by subgroup size: $(8/10)(0.10)+(2/10)(0.30)=0.08+0.06=0.14$. The pooled average is 14%. (4) The defined quantity averages each member’s predicted cumulative incidence for the cause by the horizon. A hazard is a different time-specific quantity; its average is not the arithmetic mean of those cumulative-incidence predictions.