Comparing Predicted Risks on an Absolute Scale
A predicted risk can be easier to interpret when it is compared with another predicted risk on the same probability scale. For example, a model might predict the probability of an event of interest by three years for two specified covariate profiles. Subtracting these predictions gives an absolute contrast: the difference in predicted cumulative incidence. Its size is expressed as a difference in probability, or equivalently in percentage points.
The contrast answers a focused prediction question: at the chosen horizon, how far apart are the model’s predicted cumulative incidences for these two profiles? It does not by itself describe an average difference in a population, and it does not show what would happen if a person’s covariates were changed. Those distinctions matter especially when the profiles differ in several characteristics.
The subtraction order is part of the definition: profile A minus profile B. A positive value means the predicted cumulative incidence is higher for profile A; a negative value means it is lower. A value of zero means the two fitted predictions are equal at that horizon, not that every feature of their risk processes is the same.
What the Contrast Represents
A cumulative incidence is the probability of experiencing event type \(k\) by time \(t\) before a competing event occurs. It is therefore not interchangeable with a cause-specific hazard, which concerns the instantaneous event rate among people still event-free. A risk difference compares two cumulative-incidence predictions, incorporating how both the event of interest and competing events affect the probability of remaining event-free.
The profiles must be specified clearly. They may differ in one covariate or in several. If they differ in several, the subtraction gives a contrast between the two complete profiles; it does not isolate the contribution of any one covariate. Nor is it automatically an average across people. A standardized risk difference instead compares risks averaged over a chosen target population. The profile contrast and standardized contrast answer different questions.
A model-based contrast is also not automatically causal. A causal interpretation would require a causal question and additional assumptions appropriate to that question. In ordinary prediction, the profiles are descriptions used to obtain conditional model predictions. Their difference is a model-based association on the absolute-risk scale, not an effect of intervening to move an individual from one profile to another.
Worked Example: Calculating a Difference From Discrete-Time Predictions
Consider a hypothetical fitted prediction model with two event types: event \(k\), the event of interest, and a competing event. For two successive intervals, suppose the model predicts the following conditional probabilities of each event among people who are event-free at the start of that interval. Within an interval, the event types are mutually exclusive.
| Profile | Interval | Event \(k\) | Competing event |
|---|---|---|---|
| A | 1 | 0.12 | 0.08 |
| A | 2 | 0.20 | 0.10 |
| B | 1 | 0.08 | 0.12 |
| B | 2 | 0.10 | 0.15 |
For a profile, the probability of remaining event-free through an interval is one minus the two event probabilities in that interval. Thus profile A has probability \(1-0.12-0.08=0.80\) of remaining event-free after interval 1. Its predicted cumulative incidence for event \(k\) through interval 2 is $$ \widehat{F}_k(2\mid z_A)=0.12+(0.80)(0.20)=0.28. $$ The first term is the probability of event \(k\) in interval 1. The second is the probability of remaining event-free through interval 1 and then experiencing event \(k\) in interval 2.
For profile B, the probability of remaining event-free after interval 1 is \(1-0.08-0.12=0.80\), giving $$ \widehat{F}_k(2\mid z_B)=0.08+(0.80)(0.10)=0.16. $$ With profile A first in the subtraction, the fitted risk difference is $$ \widehat{\Delta}_k(2;z_A,z_B)=0.28-0.16=0.12. $$ Thus, in this hypothetical model, profile A has a predicted cumulative incidence 0.12 higher than profile B through interval 2, or 12 percentage points higher. This is a comparison of the two predicted risks; it is not a causal effect of changing a person’s profile.
The competing-event predictions are part of the calculation. For example, they determine the probability of being event-free and therefore still at risk of event \(k\) in the next interval. Ignoring them would not produce these cumulative-incidence predictions.
Why Competing Events Matter to the Difference
The discrete-time recursion makes explicit why a difference in cumulative incidence is not simply a difference in event-\(k\) probabilities at each time. Let \(S_{j-1}(z)\) denote the probability of remaining free of all event types through the start of interval \(j\), and let \(p_{k,j}(z)\) be the conditional probability of event \(k\) in interval \(j\). Then $$ F_{k,j}(z)=F_{k,j-1}(z)+S_{j-1}(z)p_{k,j}(z). $$ The increment in cumulative incidence depends on both the conditional probability of event \(k\) and the probability of still being event-free.
For two profiles, subtract their recursions. Adding and subtracting \(S_{j-1}(z_A)p_{k,j}(z_B)\) gives $$ \begin{aligned} F_{k,j}(z_A)-F_{k,j}(z_B) ={}&F_{k,j-1}(z_A)-F_{k,j-1}(z_B)\\ &+S_{j-1}(z_A)\{p_{k,j}(z_A)-p_{k,j}(z_B)\}\\ &+\{S_{j-1}(z_A)-S_{j-1}(z_B)\}p_{k,j}(z_B). \end{aligned} $$