Why Compare More Than One Risk Measure?
A treatment can change the chance of one event type while leaving the chance of experiencing any event nearly unchanged. It can also produce the same relative reduction in risk in two settings where the absolute reductions are quite different. These are not contradictions: the measures summarize different features of the event experience.
This tutorial compares cause-specific and any-event risks at a fixed follow-up horizon. The central calculation is straightforward, but a meaningful interpretation depends on defining the outcome, treatment strategies, population, and horizon consistently. A risk for a particular cause includes the competing-event process that occurs under that treatment strategy; it is not the risk that would have occurred if competing events had been removed.
The outcomes by time \(t\) are mutually exclusive: a person has cause 1 first, cause 2 first, or no event by \(t\). Thus, in a complete table, the three proportions in each treatment group sum to one. A treatment contrast compares risks in the same target population, over the same horizon, under the two strategies. Write the risk difference (RD) and risk ratio (RR) for cause \(k\) as:
The any-event contrasts are defined in the same way, replacing \(F_k^a(t)\) with \(F_{\mathrm{any}}^a(t)\). An RD is an absolute difference in probability, often reported in percentage points or events per 1,000 people. An RR is a relative comparison of probabilities. If the reference risk is zero, the RR is undefined, even though the RD may still be defined.
Conditions for a Causal Interpretation
A two-arm table provides causal contrasts when its treatment-specific risks identify the risks under the two strategies in the target population. For a randomized study, this requires valid randomization, consistent implementation of the treatment strategies, and outcome ascertainment appropriate to the stated horizon. For an observational study, the table’s crude risks generally need not be causal: identification also requires the relevant exchangeability and positivity conditions, typically addressed by standardization or another appropriate method. The earlier material on exchangeability, positivity, consistency, and censoring assumptions gives the formal causal framework.
The worked tables below are explicitly hypothetical randomized trials. Assume treatment assignment is randomized, each strategy has a well-defined version, every participant is followed through the stated horizon, and the first event and its cause are correctly recorded. These conditions make the displayed group proportions estimates of the corresponding strategy-specific risks. In a real study with loss to follow-up before the horizon, simple proportions would not generally suffice; censoring must be handled under suitable assumptions.
Worked Example: A Cause-Specific Reduction With No Change in Any-Event Risk
In a hypothetical randomized trial, 1,000 participants are assigned to each strategy and followed completely for one year. The table shows their first event by the end of follow-up.
| Strategy | Cause 1 | Cause 2 | No event by 1 year | Total |
|---|---|---|---|---|
| Reference (\(a=0\)) | 80 | 120 | 800 | 1,000 |
| Comparison (\(a=1\)) | 50 | 150 | 800 | 1,000 |
The randomized assignment, complete one-year follow-up, and complete classification of the first event satisfy the assumptions stated above for using these proportions as risk estimates. For cause 1, the estimated risks are \(50/1{,}000=0.05\) under the comparison strategy and \(80/1{,}000=0.08\) under the reference strategy. Therefore:
At one year, the comparison strategy has an estimated 3-percentage-point lower risk of a first cause 1 event, or 30 fewer events per 1,000 participants, than the reference strategy. The cause 1 risk is 0.625 times the reference risk, a 37.5% relative reduction. These are different ways of expressing the same cause 1 risks.
For any event, the reference risk is \((80+120)/1{,}000=0.20\), and the comparison risk is \((50+150)/1{,}000=0.20\). Thus \(\widehat{RD}_{\mathrm{any}}=0.20-0.20=0\) and \(\widehat{RR}_{\mathrm{any}}=0.20/0.20=1\). The cause 1 reduction is exactly offset by an increase in cause 2 events in this table. The any-event contrast says there is no estimated change in the probability of experiencing either event by one year; it does not say the cause-specific risks are unchanged.
How the Contrasts Relate
When the two causes are mutually exclusive and exhaustive, the any-event risk is the sum of the two cause-specific risks. Subtracting the reference risk from the comparison risk therefore gives an exact decomposition of the any-event RD:
This identity explains how opposing cause-specific differences can cancel in the any-event RD. Risk ratios do not add in the same way. If the reference any-event risk is positive, then the any-event RR is a baseline-risk-weighted average of the cause-specific RRs:
The weights are the proportions of reference-group events due to each cause, and they sum to one. Consequently, the any-event RR is generally not the unweighted average of the cause-specific RRs. If a cause has zero reference risk, its cause-specific RR is undefined, so the weighted-average expression should not be used in that form; the any-event RR can still be calculated if its reference risk is positive.
Worked Example: Same Relative Reduction, Different Absolute Risk Difference
Consider another hypothetical randomized trial with complete follow-up to one year and 1,000 participants per strategy. There are 20 cause 1 events and 30 cause 2 events under the reference strategy, compared with 10 cause 1 events and 30 cause 2 events under the comparison strategy. The remaining participants have no event by one year. Assume correct randomization, well-defined strategies, and complete event classification and follow-up, as in the previous example.
For cause 1, the risks are \(10/1{,}000=0.01\) and \(20/1{,}000=0.02\). Hence:
The comparison strategy has a 1-percentage-point lower cause 1 risk, equivalent to 10 fewer cause 1 events per 1,000 participants, and half the reference risk. For any event, the risks are \((10+30)/1{,}000=0.04\) and \((20+30)/1{,}000=0.05\). The any-event RD is \(0.04-0.05=-0.01\), and the any-event RR is \(0.04/0.05=0.80\). Thus, cause 1 risk is halved, while any-event risk is reduced by 20% relative to the reference risk. The distinction arises because cause 2 risk is unchanged, so it contributes to any-event risk but not to cause 1 risk.
Worked Example: Opposing Cause-Specific Changes With an Unchanged Composite Risk
In a third hypothetical randomized trial, each strategy is assigned to 1,000 participants, all followed through one year. The reference group has 100 cause 1 events and 100 cause 2 events. The comparison group has 150 cause 1 events and 50 cause 2 events. Assume randomization, consistency, complete follow-up, and correctly classified first events.
For cause 1, the risks are \(0.15\) under the comparison strategy and \(0.10\) under the reference strategy, giving \(\widehat{RD}_1=0.15-0.10=0.05\), or 5 percentage points, and \(\widehat{RR}_1=0.15/0.10=1.50\). For cause 2, the corresponding risks are \(0.05\) and \(0.10\), so \(\widehat{RD}_2=0.05-0.10=-0.05\) and \(\widehat{RR}_2=0.05/0.10=0.50\).
The any-event risk is \((150+50)/1{,}000=0.20\) under the comparison strategy and \((100+100)/1{,}000=0.20\) under the reference strategy. Therefore, \(\widehat{RD}_{\mathrm{any}}=0\) and \(\widehat{RR}_{\mathrm{any}}=1\). The any-event measure summarizes whether either event occurred; it does not distinguish how the events are distributed across causes. Here, the 5-percentage-point increase for cause 1 and the 5-percentage-point decrease for cause 2 cancel in the total risk.
The any-event RR also illustrates the weighted-average identity. Each cause accounts for half of the reference events, so \(RR_{\mathrm{any}}=0.5(1.50)+0.5(0.50)=1.00\), not the unweighted average in general merely by coincidence; in this example the weights happen to be equal.
Choosing and Checking the Measure
A cause-specific risk contrast addresses the probability of a first event of that cause by the chosen horizon under each treatment strategy. An any-event contrast addresses the probability of a first event of either listed cause. The any-event outcome is a composite: it can be useful when the question concerns avoiding any event, but it discards information about which cause occurred. Neither measure is universally preferable; the choice follows the clinical question and the estimand specified in advance.
| Measure | Question answered | Useful scale | What it does not say |
|---|---|---|---|
| Cause-specific RD | How much does the strategy change the probability of cause \(k\) by \(t\)? | Percentage points or events per 1,000 | It does not summarize all event types. |
| Cause-specific RR | How large is the cause \(k\) risk relative to the reference risk? | Ratio, or percent relative change | It does not give the absolute difference by itself. |
| Any-event RD or RR | How does the probability of either event by \(t\) compare? | Absolute difference or relative ratio | It does not show which cause accounts for the comparison. |
Before interpreting a calculation, check the table and the estimand:
- Verify that the event categories are mutually exclusive first events and that all relevant causes are included in “any event.”
- Use the same follow-up horizon and target population for both strategies. A one-year risk contrast is not interchangeable with a contrast at another time.
- Use the number assigned to each strategy as the denominator for a risk when follow-up and outcome classification are complete. Do not substitute person-time denominators, which produce rates rather than risks.
- Confirm that missing outcomes and censoring have been handled appropriately. Simply treating participants lost before \(t\) as event-free can bias the estimated risks.
- For observational data, do not interpret an unadjusted table as causal without justification of identification assumptions and appropriate adjustment.
A common mistake is to calculate a lower cause 1 risk and conclude that “the treatment prevents events overall.” That conclusion does not follow: cause 2 risk may rise, as in the first example, and any-event risk may be unchanged. The correction is to state the endpoint and measure precisely—for example, “the one-year risk of a first cause 1 event was 3 percentage points lower,” followed by the separately calculated any-event contrast if that question matters. Conversely, a null any-event contrast does not establish that the cause-specific risks are equal.
These risk contrasts are fixed-horizon summaries. They do not by themselves describe when events occur, quantify uncertainty, or establish that treatment caused an observed difference unless the causal identification assumptions hold. A risk ratio can also appear large when the underlying risk is small, while a modest ratio can correspond to an important absolute difference when baseline risk is high. Reporting both RD and RR, with the horizon and cause named, often makes the result easier to interpret.
Check Your Understanding
Use the definitions and identities above to answer each question. The solutions are included after the questions.
- In a hypothetical trial, cause 1 occurs in 30 of 600 reference participants and 18 of 600 comparison participants by a fixed horizon. Calculate the cause 1 RD and RR, and interpret each.
- In that trial, suppose cause 2 occurs in 30 participants in each group. Calculate the any-event risks and contrasts.
- Why is the any-event RR generally not the simple average of the cause-specific RRs?
- A table reports a cause-specific RR of 0.70. Can the absolute risk reduction be determined from this ratio alone? Explain.
Solutions. 1. The cause 1 risks are \(30/600=0.05\) and \(18/600=0.03\). Thus \(RD_1=0.03-0.05=-0.02\), a 2-percentage-point lower risk under the comparison strategy, and \(RR_1=0.03/0.05=0.60\), or 40% lower relative risk. 2. The any-event risks are \((30+30)/600=0.10\) for the reference strategy and \((18+30)/600=0.08\) for the comparison strategy. Therefore \(RD_{\mathrm{any}}=-0.02\) and \(RR_{\mathrm{any}}=0.08/0.10=0.80\). 3. The any-event RR is a weighted average of cause-specific RRs, with weights equal to each cause’s share of reference-group events; those weights need not be equal. 4. No. An RR of 0.70 gives a relative comparison, but the absolute reduction also depends on the reference risk: if that risk is \(r\), the comparison risk is \(0.70r\) and the RD is \(-0.30r\). The horizon, cause, and reference risk are needed to state the absolute difference in context.