Presenting Adjusted Absolute-Risk Predictions

What an Adjusted Absolute-Risk Prediction Describes

A relative effect estimate does not, by itself, tell a reader how often an event is predicted to occur. An adjusted absolute-risk prediction supplies that scale: for example, the predicted probability of a specified event by three years, under a stated covariate profile or for a stated target population. To be interpretable, the prediction must also make clear what happens to the risk calculation when a competing event occurs.

The central reporting decision is whether a prediction describes a particular covariate profile or an average over a population's covariate distribution. These are different estimands, even when they come from the same fitted model. Naming the estimand, prediction horizon, event, competing-event context, and uncertainty makes the result understandable without requiring readers to infer these details from a model table.

Definition: Let \(F_k(t\mid x)\) denote the event-specific cumulative incidence: the probability of experiencing event type \(k\) by time \(t\), before any competing event, conditional on covariate profile \(x\). A profile-specific prediction reports \(F_k(t\mid x)\) for a stated profile. A standardized prediction averages \(F_k(t\mid x)\) over a specified target population's covariate distribution.

For a first-event analysis, event types are mutually exclusive as the first observed event. A competing event precludes the event of interest from being the first event. Thus, the event-specific cumulative incidence is not the probability of the event in a hypothetical world where competing events are removed. It is the probability of the event occurring by the horizon in the presence of the competing events included in the analysis.

From the Model to the Reported Risk

For continuous event times, let \(\lambda_j(u\mid x)\) be the cause-specific hazard at time \(u\) for event type \(j\), conditional on profile \(x\). If the modeled causes are \(1,\ldots,J\), the event-free survival probability is

S(u\mid x)=\exp\left\{-\int_0^u\sum_{j=1}^{J}\lambda_j(v\mid x)\,dv\right\}, \qquad F_k(t\mid x)=\int_0^t S(u\mid x)\lambda_k(u\mid x)\,du.

The factor \(S(u\mid x)\) matters: only individuals who have not yet experienced any modeled event can experience event \(k\) at time \(u\). Each competing cause contributes to the total hazard in \(S\), and therefore affects the event-specific probability. A fitted model may generate these probabilities by combining cause-specific hazards or by directly modeling a subdistribution. Whichever model is used, the reported quantity should be the predicted cumulative incidence for the named event, not a hazard ratio presented as though it were an absolute risk.

A profile-specific prediction uses a single specified \(x\), such as age 65 years, treatment A, and a stated disease category. A standardized prediction for a target population with profiles \(x_1,\ldots,x_n\) and weights \(w_i\) is

\widehat{F}_{k,\mathrm{std}}(t)=\sum_{i=1}^{n} w_i\widehat{F}_k(t\mid x_i), \qquad \sum_{i=1}^{n}w_i=1.

With equal weights, this is the average of the individual predicted risks. Other weights may represent a defined target distribution, such as survey weights or prespecified population proportions. State what the weights represent. A standardized prediction is not automatically an estimate for the study sample: that interpretation depends on the chosen target population and weights.

Assumptions and scope: These predictions depend on the fitted model adequately representing the relevant event process and covariate effects over the stated horizon. They also rely on appropriate handling of censoring; in particular, censoring must be independent of future event time conditional on the variables used by the analysis, or otherwise handled by a valid method. The target population must be defined, and its covariate profiles must be supported by the data and model. A standardized prediction describes an adjusted association; a causal interpretation requires additional design and identification assumptions.

Worked Predictions and Standardization

Worked Example: Reporting Risk for a Covariate Profile

Suppose a hypothetical fitted competing-risk model predicts three-year risks for a patient profile defined as age 65 years, treatment A, and a specified baseline disease category. The predicted cumulative incidence of the primary event is 0.18, and the predicted cumulative incidence of death before the primary event is 0.09. The predicted probability of remaining free of either first event through three years is 0.73.

The probabilities are coherent because \(0.18+0.09+0.73=1.00\). The primary-event prediction is 18%, not 18% among people who would survive without a competing event. It is the predicted probability of the primary event by three years, with death before that event treated as a competing event.

Suppose the model-based 95% confidence interval for the primary-event risk is 0.13 to 0.24. A suitable report is: “For the stated covariate profile, the model-predicted three-year cumulative incidence of the primary event was 18% (95% confidence interval, 13% to 24%), with death before the primary event treated as a competing event.” This interval describes uncertainty in the estimated risk under the analysis and model; it is not a statement that there is a 95% probability that this fixed risk lies in this particular interval.

Worked Example: Standardizing to a Target Population

In a hypothetical target population, 40% of individuals have profile 1 and 60% have profile 2. At a five-year horizon, a fitted model predicts primary-event cumulative incidences of 0.12 and 0.25 for these profiles, respectively. The target population's profile proportions are the prespecified standardization weights.

The standardized prediction is

\widehat{F}_{\mathrm{std}}(5) =0.40(0.12)+0.60(0.25) =0.048+0.150 =0.198.

Thus, the predicted five-year cumulative incidence is 19.8% for this target population, with competing events handled as in the fitted model. It is not the unweighted average of the two profile risks, which would be 18.5%; the target has more individuals with profile 2. Nor does 19.8% describe either profile alone. A report should name the target population or its weighting definition, identify the event and competing causes, and state the five-year horizon.

The arithmetic averages predicted risks, not covariate values. In general, predicting at the mean covariate profile need not equal averaging predictions across the target population, especially when the prediction function is nonlinear.

Uncertainty and Comparisons

A point prediction is incomplete when the goal is to communicate how precisely it has been estimated. Uncertainty for a profile-specific risk should account for estimation of the model parameters used to produce that risk. Uncertainty for a standardized risk should also reflect the construction of the standardization, with the details depending on whether the target covariate distribution is treated as fixed or estimated from sampled individuals.

A confidence interval can be obtained using a method appropriate to the fitted model, such as a delta method on a suitable scale or a bootstrap that repeats the relevant estimation steps. Because a probability lies between zero and one, a symmetric interval on the risk scale can extend outside those bounds. A transformed-scale interval or another method respecting the parameter space may be preferable. The method should be stated when needed to make the analysis reproducible.

For a comparison of two standardized risks, uncertainty must account for their dependence if they were estimated from the same individuals or model. Do not infer a confidence interval for a risk difference by subtracting unrelated interval endpoints. Estimate the contrast in each bootstrap resample, or use a variance calculation that includes the covariance between the two risk estimates.

Worked Example: Reporting a Standardized Risk Difference

Consider a hypothetical analysis standardizing two treatment strategies to the same target population at four years. The estimated cumulative incidence of the primary event is 0.22 under strategy A and 0.14 under strategy B. A bootstrap that refits the models and recalculates both standardized risks in each resample gives a 95% confidence interval of 0.02 to 0.14 for the risk difference, defined as A minus B.

The estimated risk difference is

\widehat{F}_{A,\mathrm{std}}(4)-\widehat{F}_{B,\mathrm{std}}(4) =0.22-0.14=0.08.

This is an estimated difference of 8 percentage points at four years. The interval corresponds to 2 to 14 percentage points. A context-complete report is: “After standardization to the stated target population, the four-year cumulative incidence of the primary event was 22% under strategy A and 14% under strategy B, a risk difference of 8 percentage points (95% confidence interval, 2 to 14 percentage points). Competing events were included as specified in the model.” This result alone does not establish that changing strategies would cause the difference; that conclusion would require the relevant causal assumptions.

How to Present the Prediction Clearly

A concise report should answer four questions: what probability, for whom, by when, and in the presence of which competing events? It should then give uncertainty and, when relevant, identify the standardization target and method. Use the event name rather than a generic label such as “failure,” and give the time unit (for example, years) and the risk unit (probability, percentage, or percentage points for a contrast).

For tables and figures, make the estimand legible in the labels or accompanying text. A row labeled only “Adjusted risk: 18%” leaves the reader to guess the event, horizon, profile, target population, and competing-event handling. If multiple horizons are shown, label each one. If predictions are presented for several profiles, state the covariate values defining each profile. If predictions are standardized, state the target distribution rather than merely calling the risks “adjusted.”

Name the event and horizon.
Specify the event type and the time by which its cumulative incidence is predicted, including the time unit.

Name who the prediction describes.
Give the covariate profile for a conditional prediction, or define the target population and weights for a standardized prediction.

State the competing-event context.
Name the competing events or refer precisely to the event definitions already given in the report; make clear they are included in the cumulative-incidence prediction.

Give uncertainty and interpretation.
Report the interval and confidence level, identify a contrast's direction and units, and avoid causal language unless justified.

Common Mistakes and Diagnostics

Calling a profile-specific risk “standardized.” Reporting one model prediction at a covariate profile is not an average over a target population. Name the profile or calculate and describe the target-weighted average.
Omitting the horizon. A risk without a time point is not a complete absolute-risk prediction. Include, for example, “by three years.”
Reporting a net risk as a competing-risk probability. Removing competing events or treating them as ordinary censoring answers a different question. Report the cumulative incidence with competing causes represented in the analysis.
Leaving the target population implicit. A standardized risk depends on whose covariate distribution is used. State the population or the weighting rule.
Presenting the prediction as causal by default. Adjustment alone does not guarantee removal of confounding or justify a treatment effect interpretation. Use association language unless the causal assumptions are supported.

Before reporting, check that predicted event probabilities are within \([0,1]\), that event-specific risks and event-free probability are coherent for a given profile and horizon when they are jointly calculated, and that each displayed estimate corresponds to its stated target and time point. Small discrepancies from rounding may occur. If the predictions are implausible or change sharply under reasonable model choices, revisit model fit, covariate support, censoring assumptions, and the prediction calculation rather than relying on polished wording alone.

Key takeaway: An adjusted absolute-risk prediction is interpretable only when the report identifies the event, horizon, covariate profile or standardization target, competing-event context, and uncertainty. “Adjusted risk” alone does not supply those details.

Check Your Understanding

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

A model predicts a 12% risk of event A by two years for a profile, with event B as a competing event. What key context should accompany the 12%?
A target population has profile proportions 25% and 75%. The predicted risks are 0.10 and 0.18. Calculate the standardized risk.
Explain why averaging predictions over target profiles can differ from predicting at the average covariate values.
A report gives a risk difference of 6 percentage points and its confidence interval. What additional information is needed to interpret the contrast?
Why is it incorrect to describe a competing-risk prediction as the probability of event A “if nobody experienced event B”?

Solutions. (1) State that the 12% is the profile-specific cumulative incidence of event A by two years, with event B treated as a competing event, and give its uncertainty. (2) \(0.25(0.10)+0.75(0.18)=0.025+0.135=0.160\), or 16.0%. (3) The prediction function can be nonlinear, so the average of predictions need not equal the prediction at average covariates; standardization averages risks over the target distribution. (4) Give the event, horizon, competing-event context, target or profiles being compared, direction of subtraction, and uncertainty method or interval. (5) That statement removes the competing event and describes a different hypothetical quantity; a competing-risk cumulative incidence is the probability of event A by the horizon in the presence of event B as a competing event.