Deriving Collider Bias From Follow-Up Selection

Why Follow-Up Can Create an Association

A study may begin with two factors that are independent in the source population, yet find that they are associated among participants who remain under observation. This can happen because inclusion in the observed follow-up sample depends on both factors. The association in that selected sample need not reflect a relationship between the factors in the original population.

For example, suppose both access to care and a participant’s health status affect whether a scheduled visit is observed. Among those with an observed visit, participants with less access may be disproportionately those whose health status made attendance more likely. Conditioning on having an observed visit can therefore connect the two causes, even if they were independent before selection. This is a consequence of selection, not evidence that one factor caused the other.

The causal diagram for the basic mechanism is small:

Here \(A\) and \(B\) are factors in the source population, and \(R=1\) indicates that a person is retained in follow-up (or otherwise selected for the analysis). The arrowheads meet at \(R\), making \(R\) a collider on this path. The path between \(A\) and \(B\) is blocked before conditioning on \(R\), but conditioning on \(R=1\) can open it.

Deriving the Induced Association

Let \(A\) and \(B\) be binary, and suppose they are independent in the source population. Write \(r_{ab}=P(R=1\mid A=a,B=b)\), for \(a,b\in\{0,1\}\). Assume these four probabilities are positive. The selection probability may depend on both factors, and no particular functional form is required.

Because \(A\) and \(B\) are independent before selection, their source-population joint probabilities factor into their marginal probabilities. In the selected group, Bayes’ rule gives a joint probability proportional to the source probability multiplied by the chance of selection:

For binary variables, the odds ratio summarizes their association. When the selected cell probabilities are substituted into the odds ratio, the source-population marginal probabilities and the common normalizing factor cancel:

Thus, even though the source-population odds ratio is 1, the selected-sample odds ratio is determined by the pattern of selection probabilities. If \(r_{11}r_{00}<r_{10}r_{01}\), the selected association is negative; if \(r_{11}r_{00}>r_{10}r_{01}\), it is positive. Equality means the selected odds ratio remains 1.

This result also supplies a short justification for a common intuition about selection. If the probability of selection follows an additive pattern, \(r_{ab}=a_0+b_0a+c_0b\), then

When both factors increase selection probability (\(b_0>0\) and \(c_0>0\)), this difference is negative, so the selected-sample odds ratio is below 1. Informally, among selected people, having one factor can partly make up for not having the other. This direction is not universal: a sufficiently strong interaction in the selection probabilities can change it.

How the Selection Mechanism Controls the Direction

The sign of the induced association depends on all four selection probabilities, not just on whether each factor predicts follow-up. In particular, the way the joint presence of both factors affects selection matters. The next examples use hypothetical probabilities to show why it is important to calculate rather than assume a direction.

Interpreting the Result in Follow-Up Data

In a longitudinal study, \(R=1\) might indicate that a biomarker measurement is available at a particular visit. The factors \(A\) and \(B\) could represent distinct determinants of being observed, such as access to care and health status. If the analysis uses only people with a measurement, it conditions on the common effect \(R\). A relationship between the determinants in that subset may then arise even if they were independent in the source population.

The same logic applies when selection is defined by being event-free and observable at a visit. The event-free observation indicator can be affected by multiple causes. Restricting the analysis to that indicator’s selected value can connect those causes. The key question is not whether a variable is called “dropout,” “attendance,” or “eligibility,” but whether analysis conditions on a common effect of factors that may otherwise be unrelated.

The calculation above has clear assumptions. The source-population independence of \(A\) and \(B\) is needed for the simple odds-ratio formula; if they are already associated, their source association also contributes to the selected association. The \(r_{ab}\) values must correspond to the selection event being analyzed, and the relevant cells must have positive probability to define the odds ratio. The diagram represents causal assumptions; observed data alone do not establish that its arrows or source independence are correct.

Diagnostics, Alternatives, and Limitations

Collider bias is a warning about a design or analysis restriction, not a pattern that can always be diagnosed from a single selected data set. Useful checks include comparing measured characteristics between people retained and not retained, examining whether candidate causes of follow-up predict observation, and checking whether the analysis conditions on an event or visit indicator influenced by multiple variables. These checks can flag plausible selection mechanisms, but they cannot prove independence in the source population or rule out unmeasured causes.

The main alternatives address different questions:

• Analyze the full eligible cohort where possible. Avoiding an unnecessary restriction on follow-up can prevent this particular collider path from being opened, though missing outcomes may still require handling.
• Model selection or use inverse-probability weighting. When important predictors of selection are measured, a suitable model for observation probabilities can support analyses that reweight observed participants toward the target population. This requires adequate measurement, a correctly specified selection model, and positivity: every relevant covariate pattern must have a nonzero chance of being observed. It does not automatically remove bias from unmeasured selection factors.
• Use a joint model when the scientific problem calls for it. A joint modeling approach can represent dependence between longitudinal measurements and event or observation processes under explicit assumptions. It does not make selection assumptions unnecessary, and its conclusions still depend on the adequacy of the specified model.

These approaches are not interchangeable cures. A complete-case analysis conditions directly on availability; weighting relies on measured predictors and a selection model; joint modeling represents dependence through its chosen structure. Each targets a different way of addressing incomplete or selected follow-up, and each can fail if its assumptions are implausible.

A further limitation is that the simple derivation concerns two binary factors and a binary selection indicator. With continuous factors, multiple visits, or selection depending on time-varying history, the induced association may be more complex. The basic lesson still applies: selection can alter relationships, and the direction and size cannot be inferred from the word “collider” alone.

Common Pitfall

A common mistake is to say, “\(A\) and \(B\) are independent in the cohort, so they will still be independent among participants with observed follow-up.” That reasoning ignores that selection probabilities can vary jointly with \(A\) and \(B\). The correction is to distinguish the source population from the selected analysis sample and, when the factors are binary, inspect \(r_{11}r_{00}\) relative to \(r_{10}r_{01}\). A second mistake is to assume that the resulting association is necessarily negative. The interaction example shows that it can be positive. Neither direction, by itself, establishes a causal effect between the factors.