Follow-Up Is Part of the Causal Structure
Longitudinal studies often intend to measure an outcome at a later visit but cannot measure it for everyone. Participants may withdraw, become unreachable, or die before that visit. These events are not merely gaps in a data table: they determine whose later measurements are available. A causal diagram should therefore represent follow-up explicitly, so that the consequences of restricting analysis to observed participants can be assessed.
The central issue is selection. If two variables affect whether a participant is observed, their common effect is a collider. Conditioning on being observed can open a path between those variables, even when they were independent before selection. The opened path may connect treatment to an outcome through a prognostic factor, producing an association that is not a causal effect.
The indicators \(R_t\), \(S_t\), and \(Q_t\) have different meanings. Loss to follow-up is not death, and an outcome that is unrecorded because of dropout is not automatically the same as an outcome that is undefined because the participant died. A DAG may include separate nodes for survival and observation, with a combined observation indicator used only as a shorthand for who can contribute a measurement.
Drawing Selection Nodes in a Follow-Up DAG
Place each follow-up indicator at the time when it determines whether later measurements are available. Causes of dropout should point into the relevant \(R_t\) node; causes of survival should point into \(S_t\). For example, treatment \(A\) may affect both the outcome \(Y\) and retention \(R\), while a health history \(L\) may affect \(Y\) and retention as well. A useful fragment is \(A\rightarrow R\leftarrow L\rightarrow Y\), together with \(A\rightarrow Y\). If the analysis is restricted to \(R=1\), it conditions on the collider \(R\) and opens the path \(A\rightarrow R\leftarrow L\rightarrow Y\).
The arrows into a selection node should reflect the study’s causal assumptions, not just associations noticed in the observed data. A variable that predicts dropout because it predicts illness may be a cause of both the outcome and dropout. An unmeasured cause can also point to both nodes. The earlier material on latent causes and unmeasured confounding is relevant here: a selection node can have unmeasured parents, and the resulting open path may not be repairable by conditioning on the variables recorded in the study.
The d-separation rule for a collider applies to selection just as it does to other nodes. Without conditioning on \(R\), the path through \(R\) is blocked at the collider. Conditioning on \(R\), or on a descendant of \(R\), can open it. In practice, a complete-case analysis conditions on having the required observed data; it can therefore condition on one or more such nodes even if the analyst never writes “condition on \(R=1\)” in a model.
Worked Example: Sequential Loss to Follow-Up
Suppose \(R_1=1\) means observed at the first follow-up, and \(R_2=1\) means observed at the second. Participants cannot return after being lost, so \(R_2=1\) requires \(R_1=1\). Among those observed at visit 1, assume the probability of returning for visit 2 depends on baseline treatment \(A\) and a measured health history \(L\):
| \(A\) | \(L\) | \(P(R_2=1\mid R_1=1,A,L)\) |
|---|---|---|
| 0 | 0 | 0.90 |
| 0 | 1 | 0.50 |
| 1 | 0 | 0.60 |
| 1 | 1 | 0.20 |
Assume for this calculation that \(A\) and \(L\) are independent binary variables, each with probability \(0.5\), and that \(P(R_1=1)=0.8\) in every \((A,L)\) group. Then \(P(R_2=1\mid A,L)=P(R_1=1)P(R_2=1\mid R_1=1,A,L)\). The four probabilities of complete follow-up are therefore \(0.8(0.90)=0.72\), \(0.8(0.50)=0.40\), \(0.8(0.60)=0.48\), and \(0.8(0.20)=0.16\), respectively.
Among participants with \(A=0\) who are observed at visit 2, the probability that \(L=1\) is \[ \frac{P(L=1)P(R_2=1\mid A=0,L=1)} {P(L=0)P(R_2=1\mid A=0,L=0)+P(L=1)P(R_2=1\mid A=0,L=1)} =\frac{0.5(0.40)}{0.5(0.72)+0.5(0.40)} =\frac{0.40}{1.12}\approx0.357. \] For \(A=1\), the corresponding probability is \(0.16/(0.48+0.16)=0.25\). In the full population, \(L\) had probability \(0.5\) in both treatment groups. The observed groups no longer have the same distribution of \(L\): conditioning on complete follow-up selected them differently.
A Result on Selection-Induced Association
The mechanism can be stated exactly for two binary variables. This result is useful because it shows that selection bias is not merely a possibility suggested by a graph; it follows from the conditional distribution when selection probabilities vary jointly across groups.
Proof. Write \(p_a=P(A=a)\) and \(q_u=P(U=u)\). Independence gives \(P(A=a,U=u)=p_aq_u\). By Bayes’ rule, the selected cell probability is proportional to \(p_aq_ur_{au}\); the common normalizing constant cancels when forming an odds ratio. Thus \[ \operatorname{OR}_{AU\mid R=1} =\frac{(p_1q_1r_{11})(p_0q_0r_{00})} {(p_1q_0r_{10})(p_0q_1r_{01})} =\frac{r_{11}r_{00}}{r_{10}r_{01}}. \] When this ratio differs from 1, the selected joint distribution does not factor into its selected marginal distributions, so \(A\) and \(U\) are associated given \(R=1\). \(\square\)
Worked Example: A Collider Opens a Noncausal Path
Let \(A\) and \(U\) be independent Bernoulli variables with probability \(0.5\) each. Let \(R=1\) indicate that the outcome is observed, with selection probabilities \(r_{00}=0.9\), \(r_{01}=0.3\), \(r_{10}=0.3\), and \(r_{11}=0.9\). The selection odds ratio is \[ \frac{r_{11}r_{00}}{r_{10}r_{01}} =\frac{0.9(0.9)}{0.3(0.3)} =9. \] Thus the variables become associated among those with observed outcomes.
The selected joint probabilities before normalization are \(0.25(0.9)=0.225\) for \((A,U)=(0,0)\), \(0.25(0.3)=0.075\) for \((0,1)\), \(0.075\) for \((1,0)\), and \(0.225\) for \((1,1)\). Their sum is \(0.6\). Dividing by \(0.6\), the selected joint probabilities are \(0.375, 0.125, 0.125, 0.375\).
Suppose the outcome is \(Y=U\), and the graph contains \(A\rightarrow R\leftarrow U\rightarrow Y\), but no causal arrow from \(A\) to \(Y\). Among selected participants, \(P(Y=1\mid A=1,R=1)=0.375/(0.125+0.375)=0.75\), while \(P(Y=1\mid A=0,R=1)=0.125/(0.375+0.125)=0.25\). The difference is \(0.50\), although \(A\) and \(Y=U\) were independent before selection. The selection restriction opened the path through the collider \(R\).
Survival and Loss to Follow-Up Are Different Selection Processes
When a later outcome is available only for participants who are alive and still in contact, the observed group is described by \(Q_t=1\), not merely by \(R_t=1\) or \(S_t=1\). A DAG should keep the mechanisms separate when they have different causes. For example, treatment and frailty may both affect survival, while frailty and access to care may affect continued contact. Collapsing these into one node can hide which paths are opened and which assumptions concern death versus dropout.
Worked Example: Conditioning on Survival
Consider a randomized binary treatment \(A\) and a binary frailty variable \(U\), independent with probability \(0.5\) each. Suppose the probability of surviving to a later visit is \(0.9\) for \((A,U)=(0,0)\), \(0.5\) for \((0,1)\), \(0.5\) for \((1,0)\), and \(0.1\) for \((1,1)\). Among survivors with \(A=0\), the frailty probability is \[ P(U=1\mid A=0,S=1) =\frac{0.5(0.5)}{0.5(0.9)+0.5(0.5)} =\frac{0.25}{0.70}\approx0.357. \] Among survivors with \(A=1\), it is \[ P(U=1\mid A=1,S=1) =\frac{0.5(0.1)}{0.5(0.5)+0.5(0.1)} =\frac{0.05}{0.30}\approx0.167. \] Thus survival conditioning changes the frailty composition of the treatment groups. In the graph \(A\rightarrow S\leftarrow U\rightarrow Y\), conditioning on \(S=1\) opens a noncausal path from treatment to a later outcome \(Y\) affected by frailty.
This example concerns an outcome measured among survivors. If the outcome is not defined after death, the survivor-only comparison does not by itself estimate a treatment effect for the original randomized population. The causal target must specify how death is handled; a DAG identifies the selection structure but does not choose that target.
Diagnostics, Alternatives, and Limits
A practical audit starts by asking who is absent at each visit and why. Draw a node for each distinct observation or survival event, then list plausible causes of that event. Look especially for common causes of selection and the outcome, and for treatment or treatment history that affects selection. If the analysis keeps only complete records, mark the corresponding selection node as conditioned on and trace every path it opens. This is a diagnostic of the causal structure, not a statistical test that can establish the arrows from observed data alone.
A complete-case analysis is straightforward but changes the analyzed population to those observed. Treating dropout as if it were an ordinary covariate can also be misleading: adjustment for a selection variable can open a collider path rather than close a confounding path. Inverse-probability-of-censoring weighting is a standard alternative for appropriately defined censoring settings, but it requires assumptions about measured predictors of censoring and adequate probabilities of remaining observed. A DAG helps state these assumptions; it does not guarantee they hold.
- Check timing. A cause of dropout must precede the dropout decision it is drawn to explain. Avoid arrows that imply information from a later visit caused an earlier loss.
- Keep censoring and death distinct. A missing measurement and an undefined post-death outcome require different interpretations.
- Do not infer safety from a balanced observed sample. Similar observed covariate summaries do not rule out selection through unmeasured causes.
- Check positivity conceptually. If some histories make continued follow-up impossible, the data cannot directly represent those histories at later visits; no diagram removes that limitation.
The main comparison is therefore not between “biased” and “unbiased” labels attached to particular diagrams. It is between different ways of defining and analyzing the population: all eligible participants, those who survive, those who remain observed, or a target population reconstructed under explicit assumptions. These populations can answer different questions. Selection nodes make the distinction visible and help prevent a descriptive comparison among observed cases from being mistaken for a causal effect in the full study population.
Check Your Understanding
Answer each question using the follow-up DAG principles above. Short solutions follow the questions.
- If \(A\rightarrow R\leftarrow U\rightarrow Y\), what happens to the path between \(A\) and \(Y\) when analysis is restricted to \(R=1\)?
- In the selection theorem, what condition on the four selection probabilities preserves independence of \(A\) and \(U\) among selected individuals?
- Why should a DAG distinguish a participant who died before a visit from one who was alive but lost to follow-up?
- If \(Q_t=S_tR_t\), what does \(Q_t=1\) mean?
- Name one reason a complete-case result may not represent the eligible study population.
Solutions. (1) Conditioning on collider \(R\) opens the path \(A\rightarrow R\leftarrow U\rightarrow Y\). (2) Independence is preserved when \(r_{11}r_{00}=r_{10}r_{01}\), so the selected odds ratio is 1. (3) Death may make the outcome undefined, whereas loss to follow-up leaves it unobserved; they imply different selection mechanisms and causal targets. (4) The participant is both alive and observed at visit \(t\). (5) Selection can change the distribution of treatment and prognostic factors, or open a noncausal path through the observation indicator.