Risk Differences and Ratios for Competing Events

Why the Time Horizon Belongs in the Contrast

A treatment comparison of competing-event risks is incomplete unless it names both the event type and the time at which risk is assessed. A treatment may be associated with a lower probability of one event type by one year but a higher probability by three years. Neither comparison is automatically the single “overall” effect: each describes a different time-specific outcome.

Let \(A\) denote treatment assignment, with \(A=1\) for treatment and \(A=0\) for control. Let \(T^a\) be the time to the first event under assignment \(a\), and let \(J^a\) denote its type. For a specified event type \(k\), define \(Y_k^a(t)=1\) if, under assignment \(a\), event \(k\) occurs by time \(t\) and before any competing event; otherwise define \(Y_k^a(t)=0\). The treatment-specific cumulative incidence is \(F_k^a(t)=P\{Y_k^a(t)=1\}\).

Definition: At a fixed horizon \(t\), the risk difference comparing treatment with control for event type \(k\) is \(RD_k(t)=F_k^1(t)-F_k^0(t)\). The risk ratio is \(RR_k(t)=F_k^1(t)/F_k^0(t)\), provided \(F_k^0(t)>0\). The risk difference is an absolute contrast, measured in probability units; the risk ratio is a unitless relative contrast.

The order of subtraction and division matters: these definitions compare treatment with control. A negative risk difference indicates lower risk under treatment, while a risk ratio below 1 indicates lower risk relative to control. A risk difference of \(-0.04\) means four fewer events of type \(k\) per 100 people by time \(t\), on average, under treatment than control. A risk ratio of \(0.80\) means the treatment risk is 80% of the control risk, or 20% lower relative to control. It does not mean a 20-percentage-point reduction.

Assumptions for a Causal Interpretation

The contrasts are causal when the treatment-specific risks represent the relevant potential outcomes for the population being studied. In a randomized study, random assignment supports comparability of treatment groups. In an observational study, identification from the observed data requires additional assumptions; the treatment groups cannot simply be treated as exchangeable because they were observed.

Consistency: each person’s outcome under the treatment they actually receive agrees with the corresponding potential outcome, and the treatment versions are sufficiently well defined.
Exchangeability: conditional on measured baseline covariates \(X\), treatment assignment is independent of the potential event outcomes. Randomization supports this condition without conditioning on confounders, subject to the trial design.
Positivity: at every covariate pattern included in the target population, each treatment being compared has a positive probability of assignment.
No interference: one person’s assignment does not change another person’s potential event outcome, unless interference is explicitly built into the estimand.
Appropriate follow-up: censoring must be handled using assumptions appropriate to the analysis. For example, independent censoring conditional on treatment and relevant covariates can justify standard censoring adjustments. A competing event is an observed outcome, not ordinary loss to follow-up.

If these conditions hold, treatment-specific cumulative incidences can identify the corresponding counterfactual risks in the target population. Without them, the calculated contrasts may still describe the observed groups, but a causal interpretation is not warranted. The introductory material on treatment-specific cumulative incidence provides the causal-outcome context; here the focus is on contrasting those risks.

Constructing and Interpreting the Contrasts

Worked Example: Absolute and Relative Effects at Two Years

Suppose a hypothetical randomized study estimates that, by two years, the cumulative incidence of event type \(k\) is 0.14 in the treatment group and 0.20 in the control group. Assume randomization was implemented as intended, outcomes were classified correctly, and censoring was handled under a defensible analysis assumption.

The estimated risk difference is

\widehat{RD}_k(2)=0.14-0.20=-0.06.

Thus, the treatment group has an estimated six-percentage-point lower risk of event type \(k\) by two years than the control group. Equivalently, this is six fewer events per 100 people by that horizon, as a difference between group risks.

The estimated risk ratio is

\widehat{RR}_k(2)=\frac{0.14}{0.20}=0.70.

The estimated two-year risk under treatment is 70% of the control risk, or 30% lower relative to control. These statements refer to the same two-year risks but express different scales. The absolute difference is \(-0.06\); the relative reduction is 30%.

A cumulative incidence is nondecreasing in time for each treatment group, but a contrast between two nondecreasing curves need not be constant or even keep the same sign. In continuous time, the event-type-specific cumulative incidence can be written as the integral of the event-\(k\) cause-specific hazard weighted by the probability of still being free of every event immediately before that time. Consequently, risks for competing event types contribute to the accumulation of \(F_k^a(t)\). This is one reason not to treat a risk contrast as a time-independent treatment property.

Worked Example: The Contrast Changes With Follow-Up

Consider hypothetical treatment-specific risks for the same event type: at one year, \(F_k^1(1)=0.10\) and \(F_k^0(1)=0.08\); at three years, \(F_k^1(3)=0.18\) and \(F_k^0(3)=0.20\). Each group’s risk increases over time, as a cumulative incidence should.

At one year,

RD_k(1)=0.10-0.08=0.02,\qquad RR_k(1)=\frac{0.10}{0.08}=1.25.

By one year, the treatment risk is two percentage points higher, and 1.25 times the control risk. At three years,

RD_k(3)=0.18-0.20=-0.02,\qquad RR_k(3)=\frac{0.18}{0.20}=0.90.

By three years, the treatment risk is two percentage points lower, and 10% lower relative to control. The contrast changes direction across these horizons. This does not violate the fact that both cumulative incidence curves increase: it is their difference and ratio, not either curve alone, that changes direction.

The conclusion is necessarily time-specific. It would be inaccurate to report simply that treatment “increases” or “decreases” the risk without indicating the horizon. The hypothetical values do not establish why the contrast changes; interpretation requires considering the full event process and the study design.

Uncertainty for Risk Differences and Ratios

Point estimates alone do not show how much sampling uncertainty is present. Let \(\widehat F_1\) and \(\widehat F_0\) estimate the treatment and control cumulative incidences for event type \(k\) at the same horizon. If their estimated variances are \(v_1\) and \(v_0\), and their covariance is \(c\), the delta method gives the following approximate variances:

\widehat{\operatorname{Var}}\{\widehat{RD}_k(t)\} =v_1+v_0-2c, \qquad \widehat{\operatorname{Var}}\{\log \widehat{RR}_k(t)\} \approx \frac{v_1}{\widehat F_1^{\,2}} +\frac{v_0}{\widehat F_0^{\,2}} -\frac{2c}{\widehat F_1\widehat F_0}.

The second expression follows by applying the delta method to \(g(F_1,F_0)=\log(F_1/F_0)\), whose gradient is \((1/F_1,-1/F_0)\). If the estimates come from independent treatment groups, their covariance may be zero; it should not be assumed zero when the design or estimation procedure induces dependence. Variance estimates should also reflect the method used to estimate cumulative incidence under censoring.

Worked Example: A Confidence Interval for a Risk Ratio

In a hypothetical study, the estimated three-year risks are 0.12 under treatment and 0.20 under control. The estimated risk ratio is \(0.12/0.20=0.60\). Suppose the estimated standard error of the log risk ratio is 0.2380. This standard error is assumed to come from an appropriate variance calculation for the study design and risk estimator.

An approximate 95% confidence interval is first calculated on the log scale. Since \(\log(0.60)=-0.5108\),

-0.5108\ \pm\ 1.96(0.2380) =-0.5108\ \pm\ 0.46648 =(-0.97728,\,-0.04432).

Exponentiating both endpoints gives the interval on the risk-ratio scale:

\left(\exp(-0.97728),\exp(-0.04432)\right) \approx (0.376,\ 0.957).

The estimated three-year treatment risk is 60% of the control risk. The approximate 95% confidence interval ranges from 0.376 to 0.957, corresponding to a treatment risk between 37.6% and 95.7% of the control risk under the model and assumptions used. Because the interval excludes 1, it is inconsistent with a null risk ratio of 1 at the corresponding two-sided 5% level under this approximation. A frequentist 95% confidence interval does not mean there is a 95% probability that this fixed risk ratio lies in this particular interval.

Choosing a Scale and Checking the Analysis

Contrast	What it communicates	Key limitation
Risk difference	Absolute excess or reduction in risk by the stated horizon	Its numerical size depends on the population’s baseline risk
Risk ratio	Risk under treatment relative to risk under control by the stated horizon	It can be unstable or undefined when the control risk is near or equal to zero

Neither scale is universally preferable. Risk differences can be useful for expressing the absolute size of a change in event probability, while risk ratios express a relative comparison. The two are linked by \(RD_k(t)=F_k^0(t)\{RR_k(t)-1\}\): for a fixed relative contrast, the absolute difference depends on the control risk. Reporting both can give complementary information, provided each is clearly tied to the same event type, population, and horizon.

Before interpreting results, check that treatment and control risks use the same event definition, time origin, population, and follow-up horizon. Check that competing events have not been incorrectly treated as ordinary censoring, and that uncertainty calculations match the design and estimator. A very small control risk makes a ratio especially sensitive to estimation error; if the control risk is zero, the ratio as defined is not finite and another summary or a different analysis plan is needed.

Common Mistakes and Limitations

Dropping the time horizon. A risk difference or ratio is a function of \(t\). State the horizon with the estimate and do not imply it holds at every time.
Confusing relative and absolute changes. An \(RR\) of 0.70 means 30% lower relative risk, not a 30-percentage-point reduction. Calculate the risk difference separately for the absolute change.
Interpreting a descriptive contrast as causal. The formulas alone do not remove confounding. A causal claim requires suitable identification assumptions, including exchangeability when treatment is not randomized.
Treating a competing event as censoring. A competing event is part of the outcome process for cumulative incidence. Reclassifying it as loss to follow-up changes the target quantity and can invalidate the risk estimate.
Using a ratio when its denominator is zero or unstable. The risk ratio requires a positive control risk and can be highly variable when that risk is small.

These risk contrasts describe event probabilities by a chosen horizon; they do not by themselves explain the mechanisms behind a difference or imply that the same contrast applies to another population or follow-up period. Cause-specific hazard comparisons answer a different question and should not be substituted for these risk contrasts.

Key takeaway: Define the event type and time horizon first. Then report treatment and control cumulative incidences alongside the risk difference and, when defined, the risk ratio. Their meanings are complementary, and their causal interpretation depends on the design and identification assumptions.

Check Your Understanding

Answer each question, then compare with the solutions below.

At five years, the treatment and control risks for an event are 0.09 and 0.15. Calculate the risk difference and risk ratio, and interpret both.
Why must a risk difference for competing-event risk include a time horizon?
If \(F_k^0(t)=0\), can the risk ratio \(F_k^1(t)/F_k^0(t)\) be interpreted as a finite number? What remains definable?
Give two assumptions needed to interpret observed treatment-specific risks causally in an observational study.
A reported risk ratio is 0.80. Is this an absolute reduction of 20 percentage points? Explain.

Solutions. (1) \(RD=0.09-0.15=-0.06\), a six-percentage-point lower treatment risk; \(RR=0.09/0.15=0.60\), so the treatment risk is 60% of control, or 40% lower relative to control, by five years. (2) The cumulative incidences and their contrast can change with follow-up; the same two groups can have different contrasts at different times. (3) No. Division by zero does not define a finite risk ratio, but the risk difference \(F_k^1(t)-F_k^0(t)\) remains defined. (4) Examples include conditional exchangeability given baseline covariates, positivity, consistency, and appropriate assumptions for censoring. (5) No. It is a 20% relative reduction; the absolute reduction must be calculated from the two risks and may be smaller or larger than 20 percentage points.