Two Routes From Dropout to an Outcome Distribution
When outcomes are missing after dropout, a model must say what distribution those outcomes would have had. One possibility is that dropout depends directly on the outcome itself: people with different, possibly unobserved, outcome values have different dropout probabilities. Another is that dropout depends on an observed covariate that also predicts the outcome. The second mechanism can produce different outcome distributions among dropouts and completers even when, within covariate strata, dropout carries no additional information about the outcome.
The distinction is consequential. Under shared-covariate dropout, the model extrapolates the observed outcome distribution within covariate strata to dropouts, then averages across the covariate distribution among dropouts. Under direct outcome-dependent dropout, it also changes the outcome distribution within those strata. These are different extrapolations, not just different labels for the same assumption.
The equations below describe one dropout transition. In a longitudinal analysis, \(X\) can be augmented by observed history available up to that transition; the key comparison remains whether dropout depends on an outcome that is unobserved at the transition. The shared-covariate statement requires the covariates being conditioned on to be observed for dropouts. If a supposedly shared covariate is itself missing, this simple conditional argument does not apply without an additional model.
What Each Model Extrapolates
Write \(f(y\mid x)\) for the outcome distribution conditional on \(X=x\), and \(\pi(y,x)=P(D=1\mid Y=y,X=x)\) for the dropout probability. For a discrete outcome, \(f\) is a probability mass function; for a continuous outcome, it is a density. By conditioning on dropout, the outcome distribution among dropouts within a covariate stratum is
For a discrete outcome, replace the integral in the denominator with a sum over possible outcome values. This formula follows from Bayes' rule: the numerator is the joint contribution of an outcome value and dropout, and the denominator sums that contribution over all outcome values. It shows exactly where the models differ.
Even under the shared-covariate model, the marginal distribution among dropouts need not match the marginal distribution among observed cases. For discrete \(X\),
The weights \(P(X=x\mid D=1)\) can differ from the covariate distribution among observed cases. Thus the dropout group can have a different marginal outcome distribution because its covariate composition differs, without any within-stratum outcome dependence in dropout.
Worked Example: Shared-Covariate Dropout With a Continuous Outcome
Suppose a hypothetical population has two equally common covariate groups, \(X=0\) and \(X=1\). The outcome is normally distributed within each group: \(Y\mid X=0\sim N(50,100)\) and \(Y\mid X=1\sim N(70,100)\), where the second argument is the variance. Dropout probabilities are 0.20 for \(X=0\) and 0.60 for \(X=1\), with no further dependence on \(Y\). Find the mean and variance of \(Y\) among dropouts.
The population proportions in the two groups are each 0.50. Their dropout contributions are \(0.50(0.20)=0.10\) and \(0.50(0.60)=0.30\), so \(P(D=1)=0.40\). Consequently,
The shared-covariate assumption says that within each group the outcome distribution is unchanged by dropout. The mean among dropouts is therefore
For a mixture, the variance is the weighted within-group variance plus the variance of the group means around the mixture mean:
The dropout-group standard deviation is \(\sqrt{175}=13.23\) outcome units. The variance exceeds the within-group variance of 100 because the dropout group combines groups with different means. No outcome-dependent dropout within \(X\) was needed to produce this broader marginal distribution.
Worked Example: Direct Outcome Dependence Within a Covariate Stratum
In a separate hypothetical setting, \(Y\) is binary, with \(P(Y=1\mid X=x)=0.40\) in a particular covariate stratum. Suppose \(P(D=1\mid Y=1,X=x)=0.60\) and \(P(D=1\mid Y=0,X=x)=0.20\). Calculate the outcome prevalence among dropouts and among observed cases in this stratum.
The joint probability of \(Y=1\) and dropout is \(0.40(0.60)=0.24\); the joint probability of \(Y=0\) and dropout is \(0.60(0.20)=0.12\). Thus the dropout probability is \(0.24+0.12=0.36\), and
For observed cases, the corresponding joint probabilities are \(P(Y=1,D=0\mid X=x)=0.40(0.40)=0.16\) and \(P(Y=0,D=0\mid X=x)=0.60(0.80)=0.48\). Hence
The outcome prevalence among dropouts is 66.67%, compared with 25.00% among observed cases in this stratum. This difference arises within the stratum because dropout probability varies directly with \(Y\). It is not explained by a change in the covariate mix.
Why the Distinction Matters for Inference
A shared-covariate model uses the observed outcomes to estimate \(f(y\mid x)\), then transports that conditional distribution to dropouts with the same \(x\). Its marginal prediction for dropouts also needs the covariate distribution among dropouts. That distribution can generally be estimated when \(X\) is observed for everyone. In contrast, an outcome-dependent model modifies the conditional outcome distribution among dropouts using the outcome-specific dropout probabilities. When \(Y\) is missing, the dependence of dropout on that missing value cannot generally be read directly from the observed outcomes.
This is the central extrapolation distinction. Observed outcomes can inform the relationship between \(Y\) and \(X\), and observed covariates can show which \(X\) values are common among dropouts. But the observed data alone do not, in general, determine whether outcomes differ within \(X\) because dropout directly selected on \(Y\). An outcome-dependent model therefore needs a defensible specification or sensitivity assumptions for that dependence; it should not be treated as though its key selection relationship were automatically established by the observed cases.
Worked Example: Covariate Composition Can Change a Binary Outcome Rate
Consider a hypothetical population with equal numbers in low-risk and high-risk covariate groups. The outcome is a binary indicator, with \(P(Y=1\mid X=\text{low})=0.80\) and \(P(Y=1\mid X=\text{high})=0.40\). Dropout probabilities are 0.20 in the low-risk group and 0.60 in the high-risk group, and dropout depends on \(X\) only. Find the outcome rate among dropouts and observed cases.
Among dropouts, the group weights are proportional to \(0.50(0.20)=0.10\) and \(0.50(0.60)=0.30\). Normalizing gives weights 0.25 and 0.75. The dropout-group outcome rate is
Among observed cases, the group contributions are \(0.50(0.80)=0.40\) for the low-risk group and \(0.50(0.40)=0.20\) for the high-risk group. The observed-group weights are \(0.40/0.60=2/3\) and \(0.20/0.60=1/3\). Therefore,
The predicted outcome rate is 50.00% among dropouts and 66.67% among observed cases. The difference is entirely attributable to the larger share of high-risk covariate values among dropouts. Within each risk group, the model predicts the same outcome rate for dropouts and observed cases.
Assumptions, Checks, and Comparisons
The shared-covariate extrapolation is most credible when the relevant covariates are measured before dropout, observed for dropouts, and sufficiently predictive of both dropout and the outcome. It also requires a correctly specified conditional outcome distribution \(f(y\mid x)\). In a longitudinal setting, conditioning only on baseline \(X\) may be inadequate if observed outcome history predicts both later outcomes and dropout; the conditioning set should represent the observed information used by the model.
A practical check is whether covariate values among dropouts are represented among cases with observed outcomes. If a covariate stratum occurs among dropouts but has few or no observed outcomes, the conditional distribution for that stratum is weakly supported or unavailable. Extrapolating \(f(y\mid x)\) there requires additional assumptions. Also examine whether the chosen covariates plausibly capture important common predictors of dropout and outcome. Such checks can reveal a fragile shared-covariate extrapolation, but they cannot prove that dropout does not depend on the missing outcome.
For an outcome-dependent model, inspect whether the implied missing-outcome distribution is substantively plausible and whether conclusions change under reasonable choices for the outcome–dropout association. Plausibility is a check on assumptions, not proof that the association is correct. Both models can reproduce aspects of the observed data while making different predictions about missing outcomes. Agreement in observed-data fit does not establish agreement in extrapolation.
| Feature | Shared-covariate dropout | Outcome-dependent dropout |
|---|---|---|
| Dropout probability | Depends on observed \(X\), not additionally on \(Y\) given \(X\) | May depend on \(Y\) after conditioning on \(X\) |
| Outcome distribution within \(X\) among dropouts | Same as \(f(y\mid x)\) | Reweighted according to outcome-specific dropout probabilities |
| Marginal difference from observed cases | Can result from different covariate composition | Can result from both covariate composition and within-stratum selection |
| Main extrapolation concern | Missing covariate overlap or inadequate conditional outcome model | Unidentified outcome–dropout dependence and its specification |
Neither model is universally preferable. If relevant shared predictors are observed and conditional exchangeability is credible, the shared-covariate model gives a relatively direct extrapolation. If dropout plausibly depends on an unobserved current outcome even after accounting for observed information, the shared-covariate assumption may be inadequate, and an outcome-dependent model may be an appropriate sensitivity specification. These alternatives should be compared as assumptions about the missing outcomes, not ranked solely by how well they describe observed cases.
Common Mistake
A frequent error is to conclude that shared-covariate dropout implies the same marginal outcome distribution among dropouts and observed cases. That conclusion ignores the covariate mix. The correct statement is conditional: under the model, \(f(y\mid D=1,x)=f(y\mid x)\). Marginally, the two groups average over different distributions of \(X\), so their outcome distributions can differ. The reverse error is also possible: observing different marginal outcome rates does not, by itself, demonstrate direct outcome-dependent dropout. A covariate-composition shift may explain the difference.
Check Your Understanding
Answer each question, then compare with the solutions below.
- Under a shared-covariate model, what equality holds between the outcome distributions among dropouts and all individuals within a fixed \(X=x\)?
- Suppose \(P(Y=1\mid X=0)=0.30\), \(P(Y=1\mid X=1)=0.70\), and \(P(X=1\mid D=1)=0.80\). Under shared-covariate dropout, calculate \(P(Y=1\mid D=1)\).
- In one stratum, \(P(Y=1)=0.50\), \(P(D=1\mid Y=1)=0.40\), and \(P(D=1\mid Y=0)=0.10\). Calculate the outcome prevalence among dropouts.
- Does a difference in marginal outcome rates between dropouts and observed cases establish direct outcome-dependent dropout? Explain.
Solutions. 1. The equality is \(f(y\mid D=1,X=x)=f(y\mid X=x)\); it holds within the stratum, not necessarily marginally. 2. The remaining dropout-group weight is \(P(X=0\mid D=1)=0.20\), so the rate is \(0.20(0.30)+0.80(0.70)=0.06+0.56=0.62\). 3. The joint probabilities for \(Y=1,D=1\) and \(Y=0,D=1\) are \(0.50(0.40)=0.20\) and \(0.50(0.10)=0.05\). Thus \(P(Y=1\mid D=1)=0.20/(0.20+0.05)=0.80\). 4. No. A shift in the distribution of observed covariates can change the marginal outcome distribution even when dropout is independent of the outcome conditional on those covariates.