Why Event-Free Survival Can Change Who Is Compared
At a fixed time point, analysts sometimes restrict an analysis to people who have had neither the outcome of interest nor a competing event. They may then compare the risk of a later outcome between treatment groups. That comparison can be useful descriptively, but it need not compare like with like: treatment may affect who remains event-free, while characteristics that predict later outcomes may also affect event-free survival.
The resulting problem is selection bias from conditioning on event-free survival. It can occur even when treatment was randomized and therefore was independent of baseline prognostic factors before selection. Randomization does not guarantee that the selected groups remain comparable after treatment-dependent eligibility is imposed.
In potential-outcome notation, let \(S^a\) be event-free status at \(t\) under treatment \(a\). Under random assignment and consistency, the selected risk in arm \(a\) corresponds to \(P(Y^a=1\mid S^a=1)\), where \(Y^a\) is the later outcome under \(a\) for those eligible to have that outcome. The conditioning sets, \(S^1=1\) and \(S^0=1\), can contain different people. Thus their contrast is not automatically a treatment effect for one common population. In particular, it is not the total effect in the original randomized population.
The Causal Diagram and the Open Path
Let \(U\) represent a baseline prognostic characteristic, such as frailty. Suppose treatment affects the chance of being event-free at the landmark, and frailty affects both event-free status and the later outcome. A simplified causal diagram is:
Here \(S\) is a collider: it is a common effect of \(A\) and \(U\). Before conditioning, randomization can make \(A\) independent of \(U\). Restricting the analysis to \(S=1\), however, can associate treatment with frailty within the selected sample. Since frailty predicts \(Y\), that induced association can distort the treatment-group comparison. A direct arrow from \(A\) to \(Y\) may also be present; the diagram highlights the additional path created by selection.
Worked Example: A Selected Risk Difference
Worked Example: A Selected Risk Difference
Consider a hypothetical randomized study with 1,000 people assigned to each treatment. Half have low frailty (\(U=0\)) and half high frailty (\(U=1\)) in each arm. Treatment does not change the later-outcome risk within either frailty group, but it changes the probability of being event-free at the landmark. Assume the following probabilities:
| Treatment | Event-free probability, low frailty | Event-free probability, high frailty | Later outcome risk among event-free, low frailty | Later outcome risk among event-free, high frailty |
|---|---|---|---|---|
| Control (\(A=0\)) | 0.80 | 0.40 | 0.10 | 0.80 |
| Treatment (\(A=1\)) | 0.40 | 0.10 | 0.10 | 0.80 |
In the control arm, the expected event-free counts are \(500(0.80)=400\) low-frailty people and \(500(0.40)=200\) high-frailty people. The expected later-outcome cases among them are \(400(0.10)=40\) and \(200(0.80)=160\). The selected risk is therefore
In the treatment arm, the expected event-free counts are \(500(0.40)=200\) low-frailty people and \(500(0.10)=50\) high-frailty people. The expected outcome cases are \(200(0.10)=20\) and \(50(0.80)=40\), giving
The selected risk difference, treatment minus control, is \(0.2400-0.3333=-0.0933\), or about 9.33 percentage points lower in the treatment group. Yet within each frailty group the stipulated later-outcome risks are identical across treatments. The difference arises because high-frailty people make up \(200/600=0.3333\) of event-free controls but only \(50/250=0.2000\) of event-free treated people. The selected contrast is real as a comparison of these two selected groups; it does not isolate a treatment effect for a shared population.
Why Selection Changes the Prognostic Mix
The mechanism can be shown directly with conditional probabilities. Suppose treatment is randomized, so that \(A\) and \(U\) are independent before conditioning. Among event-free people in treatment arm \(a\), Bayes' rule gives:
The first factor is the baseline odds of high versus low frailty. The second is the relative event-free probability for high- versus low-frailty people under treatment \(a\). If this relative probability differs between treatment arms, the selected frailty odds differ between arms. In the worked example, that relative probability is \(0.40/0.80=0.50\) for control and \(0.10/0.40=0.25\) for treatment. With baseline frailty odds equal to 1, the selected high-to-low odds are 0.50 for control and 0.25 for treatment. These correspond to high-frailty proportions \(0.50/(1+0.50)=0.3333\) and \(0.25/(1+0.25)=0.2000\), as calculated from the expected counts.
Worked Example: Selection Can Reverse the Direction
Worked Example: Selection Can Reverse the Direction
In a second hypothetical example, half the people have low frailty and half high frailty. The later-outcome risks among eligible people are 0.10 for low frailty and 0.70 for high frailty under either treatment. Event-free probabilities are 0.80 and 0.20 for low- and high-frailty controls, respectively; under treatment, they are 0.40 and 0.80.
Among controls, the event-free proportions of the original group are \(0.5(0.80)=0.40\) low frailty and \(0.5(0.20)=0.10\) high frailty. Thus the selected risk is
Among treated people, the corresponding proportions are \(0.5(0.40)=0.20\) low frailty and \(0.5(0.80)=0.40\) high frailty. Their selected risk is
The selected risk difference is \(0.5000-0.2200=0.2800\), or 28.0 percentage points higher under treatment, although the outcome risks within frailty groups are unchanged. Here treatment selects a group with a greater proportion of high-frailty people. This example shows that selection bias has no fixed direction: depending on how treatment changes event-free status across prognostic groups, the selected contrast can appear harmful or beneficial.
When the Selection Mechanism May Be Less Concerning
Worked Example: Same Selection Probabilities Across Arms
Suppose again that half the population has low frailty and half high frailty, and the later-outcome risks are 0.10 and 0.70, respectively, regardless of treatment. This time, event-free probabilities are 0.80 for low frailty and 0.40 for high frailty in both arms.
In either arm, the event-free proportions are \(0.5(0.80)=0.40\) low frailty and \(0.5(0.40)=0.20\) high frailty. The selected risk in either arm is
The selected risk difference is \(0.3000-0.3000=0\). The groups are selected, and their frailty composition differs from that of the original population, but the composition is the same in both treatment arms. In this example, selection does not create a treatment-group contrast. This is a special case, not proof that conditioning is harmless in general: other prognostic factors, treatment effects on outcomes, or treatment-by-factor interactions may still matter.
Diagnostics, Alternatives, and Limitations
Selection bias cannot usually be diagnosed from a single model coefficient or removed simply by adding more covariates to a regression restricted to event-free people. Useful checks and questions include:
- Does treatment affect the probability of remaining event-free to the landmark? Compare event-free proportions by treatment, with uncertainty and the landmark clearly stated.
- Are measured prognostic characteristics distributed differently across treatment groups within the selected sample? Such differences can reveal selection, although balance on measured factors cannot establish balance on unmeasured ones.
- Could important causes of both event-free status and the later outcome be unmeasured? If so, the collider path may remain even after adjustment for recorded variables.
- Does the analysis describe a selected group, or is it being interpreted as a causal effect for everyone randomized? Those are different claims and should be reported distinctly.
The main alternative depends on the scientific question. For an overall treatment-policy question, compare outcomes from the original treatment groups without restricting eligibility to post-treatment event-free status; the treatment-specific cumulative incidence and total-effect approaches address outcomes in the presence of competing events. If the question is explicitly about people who would be event-free under either treatment, that is a principal-stratum question associated with survivor average causal effects; membership in that shared stratum is not observed for each person. A hypothetical effect under competing-event elimination asks a different question and requires additional assumptions. None of these estimands is obtained automatically by conditioning on observed event-free survival.
A landmark analysis restricted to those event-free at \(t\) can still be useful for prognosis or for describing subsequent outcomes among the people actually eligible under each treatment. Its limitation is interpretive: treatment groups may represent different selected populations. Moreover, when an earlier event makes a later outcome undefined, the selected risks do not by themselves reveal what the outcome would have been for those excluded. Reweighting or standardizing can address measured differences only under a well-specified target and adequate assumptions; it cannot recover information about unmeasured selection factors by arithmetic alone.
Common Mistake
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In the diagram \(A\to S\leftarrow U\to Y\), why can conditioning on \(S=1\) associate randomized treatment \(A\) with \(U\)?
- Suppose the baseline odds of high versus low frailty are 2. Under control, the high-to-low event-free probability ratio is 0.5; under treatment it is 0.25. What are the high-to-low frailty odds among selected people in each arm?
- In the first worked example, what was the selected risk difference, and why is it not automatically a total treatment effect?
- Does finding similar measured baseline covariates among event-free people prove there is no selection bias? Explain.
Solutions. (1) \(S\) is a collider of \(A\) and \(U\). Conditioning on a common effect can open the path \(A\to S\leftarrow U\to Y\), creating an association between \(A\) and \(U\) in the selected sample. (2) The conditional-odds formula multiplies the baseline odds by the arm-specific event-free probability ratio: control \(2(0.5)=1.00\), treatment \(2(0.25)=0.5000\). Thus selected high-frailty proportions are \(1/(1+1)=0.5000\) and \(0.5/(1+0.5)=0.3333\), respectively. (3) It was \(-0.0933\), or 9.33 percentage points lower in the treated selected group. The groups can contain different people because treatment changes event-free status, so the contrast is not automatically an effect in a common population. (4) No. Balance on measured variables does not rule out imbalance in unmeasured prognostic factors or establish that selection is harmless.