Binary Outcomes Under Current-Outcome Selection

Why Current-Outcome Selection Needs a Joint Model

For a repeated binary outcome, dropout at a visit may depend on the binary outcome that would have been recorded at that visit. For example, a participant who would have experienced a symptom may be more likely to miss the assessment. If dropout occurs before measurement, that current outcome is not observed, so its relationship with dropout cannot be handled by simply comparing observed outcomes among those who remain. A selection model represents both the outcome process and the dropout process, then sums over the unobserved current outcome.

The time ordering used here is explicit: at each scheduled visit, a participant is at risk of dropout; a current outcome is conceptually defined; dropout or retention is generated conditional on that outcome and the available history; and the outcome is recorded only if the participant is retained. The current outcome for a participant who drops out is therefore a latent variable in the likelihood, not a value to be filled in with an assumed observation.

Definition: Let $Y_{it}\in\{0,1\}$ be the binary outcome for participant $i$ at visit $t$. Let $D_{it}=1$ mean that the participant drops out before the visit-$t$ outcome is recorded, and $D_{it}=0$ mean that the participant remains and the outcome is observed. Dropout is monotone: after the first $D_{it}=1$, no later outcomes are observed. The outcome $Y_{it}$ is defined in the joint model even when $D_{it}=1$, but is then unobserved.

Write $X_i$ for measured covariates and $H_{it}=(X_i,Y_{i1},\ldots,Y_{i,t-1})$ for the covariates and outcome history available immediately before visit $t$. A convenient outcome model is a logistic regression conditional on this history:

P(Y_{it}=1\mid H_{it})=\operatorname{expit}(\eta_{it}),\qquad \eta_{it}=\beta_{0t}+X_i^\top\beta_X+\beta_L Y_{i,t-1},

where $\operatorname{expit}(u)=1/(1+\exp(-u))$. At the first visit the lagged-outcome term is omitted. Visit-specific intercepts allow the outcome prevalence to change over time; the model can be extended with other prespecified history terms.

A logistic selection equation for dropout is

P(D_{it}=1\mid Y_{it},H_{it}) =\operatorname{expit}(\gamma_{0t}+X_i^\top\gamma_X+\gamma_Y Y_{it}+\gamma_H^\top H_{it}^{*}),

where $H_{it}^{*}$ denotes any chosen history terms in the dropout equation. The coefficient $\gamma_Y$ compares the conditional odds of dropout for $Y_{it}=1$ versus $Y_{it}=0$, holding the included covariates and history fixed. It is a conditional association parameter, not a causal effect. The logistic link is one modeling choice; whatever link and terms are chosen, the probabilities must be defined for both possible current outcomes.

Observed-Data Likelihood When Dropout Hides the Current Outcome

For a participant retained at visit $t$, both $Y_{it}$ and $D_{it}=0$ are observed. Their contribution at that visit, conditional on the history, is the outcome probability multiplied by the retention probability. For a participant who drops out at visit $t$, only $D_{it}=1$ is observed. The joint contribution must sum over the two possible values of the unobserved outcome.

\begin{aligned} L_{it}^{\text{retained}} &=f_Y(y_{it}\mid H_{it})\{1-q_{it}(y_{it},H_{it})\},\\ L_{it}^{\text{dropout}} &=\sum_{y=0}^{1} f_Y(y\mid H_{it})q_{it}(y,H_{it}), \end{aligned}

Here $f_Y(y\mid H_{it})=P(Y_{it}=y\mid H_{it})$ and $q_{it}(y,H_{it})=P(D_{it}=1\mid Y_{it}=y,H_{it})$. The summation is essential: the dropout indicator is observed, but the current outcome is not.

Likelihood result: If a participant has observed outcomes $y_{i1},\ldots,y_{i,k-1}$ and first drops out before visit $k$, the observed-data likelihood contribution is $$ \left\{\prod_{t=1}^{k-1} f_Y(y_{it}\mid H_{it})[1-q_{it}(y_{it},H_{it})]\right\} \sum_{y=0}^{1} f_Y(y\mid H_{ik})q_{ik}(y,H_{ik}). $$ Each earlier observed visit contributes both its outcome probability and its probability of retention. A participant observed at every visit through $T$ contributes the product of these outcome-and-retention terms for $t=1,\ldots,T$.

Worked Likelihood and Probability Calculations

Worked Example: Marginal Dropout at One Risk-Set Visit

Suppose that, among participants entering a visit, the model gives $P(Y=1)=0.40$. The dropout probabilities are $P(D=1\mid Y=1)=0.30$ and $P(D=1\mid Y=0)=0.10$. Calculate the dropout probability and the outcome prevalence among those retained. These are probabilities conditional on entering this visit's risk set.

By summing over the two possible current outcomes, the visit-specific dropout probability is

$$ P(D=1)=0.40(0.30)+0.60(0.10)=0.12+0.06=0.18. $$

The probability of being retained with $Y=1$ is $0.40(1-0.30)=0.28$. The probability of retention is $1-0.18=0.82$, so

P(Y=1\mid D=0)=\frac{0.40(0.70)}{0.82} =\frac{0.28}{0.82}=0.3415.

Thus 18% of this visit's risk set is expected to drop out, and the outcome prevalence among retained participants is about 34.2%, lower than the 40% prevalence before selection. These calculations are conditional on entering the risk set; they do not include the probability of surviving any earlier visit.

Worked Example: Interpreting a Logistic Selection Equation

Suppose the current-outcome coefficient in a dropout model is $\gamma_Y=1.20$, and the intercept for a specified covariate and history profile is $\gamma_0=-2.00$. There are no other terms varying between the two outcome values. Calculate the two dropout probabilities and their marginal value when $P(Y=1)=0.40$.

For $Y=0$, the linear predictor is $-2.00$, giving

q_0=\operatorname{expit}(-2.00)=\frac{1}{1+\exp(2.00)}=0.1192.

For $Y=1$, the predictor is $-2.00+1.20=-0.80$, and

q_1=\operatorname{expit}(-0.80)=\frac{1}{1+\exp(0.80)}=0.3100.

The conditional dropout odds ratio comparing $Y=1$ with $Y=0$ is $\exp(1.20)=3.320$. The marginal dropout probability is not either of the two conditional probabilities; it is

$$ P(D=1)=0.40(0.3100)+0.60(0.1192)=0.1240+0.07152=0.1955. $$

Among retained participants, the probability of $Y=1$ is $\{0.40(1-0.3100)\}/(1-0.1955)=0.2760/0.8045=0.3431$. The example assumes the stated outcome prevalence and selection probabilities apply to the same covariate and history profile. The odds ratio describes a conditional model association; it does not establish that the outcome causes dropout.

Worked Example: A Dropout Record With an Unobserved Current Outcome

Consider a two-visit participant. At visit 1, the model gives $P(Y_{i1}=1)=0.40$, and the participant's dropout probability given $Y_{i1}=1$ is 0.20. The participant is observed to have $Y_{i1}=1$ and remains for visit 2. Conditional on that history, suppose $P(Y_{i2}=1\mid Y_{i1}=1)=0.60$; the visit-2 dropout probabilities are 0.10 for $Y_{i2}=0$ and 0.40 for $Y_{i2}=1$. The participant drops out before the visit-2 outcome is recorded. Find the full-cohort likelihood contribution for this observed record.

The visit-1 outcome is observed, and the participant must also have remained at visit 1. That part of the contribution is $0.40(1-0.20)=0.32$. At visit 2, sum over the unobserved current outcome:

\begin{aligned} P(D_{i2}=1\mid Y_{i1}=1) &=P(Y_{i2}=0\mid Y_{i1}=1)P(D_{i2}=1\mid Y_{i2}=0,Y_{i1}=1)\\ &\quad+P(Y_{i2}=1\mid Y_{i1}=1)P(D_{i2}=1\mid Y_{i2}=1,Y_{i1}=1)\\ &=0.40(0.10)+0.60(0.40)=0.04+0.24=0.28. \end{aligned}

Therefore the observed record's contribution from the original cohort is $0.40(0.80)(0.28)=0.0896$. The factor 0.80 is not optional: the observed first-visit outcome is accompanied by evidence that the participant did not drop out at visit 1. If the analysis were explicitly conditional on being in the visit-2 risk set and on the specified visit-1 history, the visit-2 factor alone would be 0.28; it would not be the unconditional record likelihood.

Justification, Assumptions, and Model Checks

The summation at dropout follows directly from the law of total probability. The observed event $D_{it}=1$ can occur along two disjoint paths: the latent current outcome is zero and dropout occurs, or it is one and dropout occurs. Their probabilities are $P(Y_{it}=0\mid H_{it})q_{it}(0,H_{it})$ and $P(Y_{it}=1\mid H_{it})q_{it}(1,H_{it})$, respectively. Adding them gives the observed probability of dropout. No outcome value is selected or substituted.

This factorization relies on several substantive and statistical assumptions. The outcome is meaningfully defined at the dropout visit even though it is not recorded. Dropout is monotone and its timing is known. The outcome and selection models correctly describe their conditional probabilities given the covariates and history included. The sequence of visits and covariates is correctly represented, and there is adequate information to estimate the model parameters. In particular, the model does not make the missing current outcome empirically observed merely by introducing it as a latent variable.

A nonzero $\gamma_Y$ permits dropout to depend on the current binary outcome after conditioning on modeled history. This is a missing-not-at-random selection mechanism under the stated conditioning set. If $\gamma_Y=0$, dropout is conditionally independent of the current outcome given that history, which is a missing-at-random restriction for this selection step. Because the current outcomes of dropouts are never observed, the amount of current-outcome dependence is generally not identified by the observed data alone without additional restrictions or information. Estimates can therefore be sensitive to the chosen selection model.

Useful diagnostics include checking that fitted outcome and dropout probabilities stay within $[0,1]$, inspecting dropout predictions across visits and observed history groups, and comparing predicted with observed dropout frequencies. Sparse histories, complete or quasi-complete separation, extreme fitted probabilities, unstable estimates, or strong sensitivity to plausible values of $\gamma_Y$ are warnings. Good fit to observed dropout rates does not verify the unobserved outcome distribution among dropouts; that part is a modeling assumption.

How This Formulation Differs From Alternatives

A selection model specifies the distribution of outcomes and then models selection conditional on outcomes. Its central feature here is the joint factor $f_Y(y\mid H)q(y,H)$, summed over $y$ when the current outcome is missing. A pattern-mixture model instead organizes the outcome distribution by missingness pattern. These are alternative factorizations of the observed-data problem; neither removes the need to address assumptions about unobserved outcomes.

The conditional-independence restriction $\gamma_Y=0$ is a simpler alternative when dropout is assumed independent of the current outcome after conditioning on observed history. Complete-case analysis is another simpler approach, but it discards partially observed records and does not in general recover the target outcome distribution when dropout depends on outcome or history. These approaches should not be treated as interchangeable: they encode different assumptions and can target different quantities.

The selection formulation is limited by its dependence on the specified link, covariates, time structure, and outcome-history terms. A single coefficient for current-outcome dependence may be too restrictive across visits or subgroups. It also cannot turn an untestable assumption about missing outcomes into a fact established by the observed data. Conclusions should therefore be reported with the model assumptions made explicit and, where relevant, examined across defensible sensitivity values.

Common Pitfall

Common mistake: For a participant with an observed first-visit outcome who later drops out, use only the probability of that outcome times the summed dropout probability at the later visit. This omits the probability that the participant remained at the earlier visit. The correction is to include $1-q_{i1}(y_{i1},H_{i1})$ for every earlier observed visit, as well as its outcome probability. Alternatively, state clearly that a calculation is conditional on entry into the later visit's risk set.

Check Your Understanding

Answer each question, then compare with the solutions below.

If $P(Y=1)=0.30$, $P(D=1\mid Y=1)=0.50$, and $P(D=1\mid Y=0)=0.10$, what is the marginal dropout probability at this visit?
For the same probabilities, what is $P(Y=1\mid D=0)$?
In the observed-data likelihood, why is a sum over $Y=0,1$ required for someone who drops out before the current outcome is recorded?
A participant has an observed outcome at visit 1 and drops out before visit 2. What retention factor must accompany the visit-1 outcome probability in the full-cohort likelihood?
Does a fitted nonzero current-outcome selection coefficient prove that the outcome causes dropout? Explain.

Solutions. (1) $0.30(0.50)+0.70(0.10)=0.15+0.07=0.22$, so the marginal dropout probability is 0.22. (2) The retained $Y=1$ probability mass is $0.30(0.50)=0.15$, and total retention is $1-0.22=0.78$. Thus $P(Y=1\mid D=0)=0.15/0.78=0.1923$. (3) The current outcome is unobserved, so the observed dropout event can arise under either binary value; their joint probabilities must be added. (4) Include the visit-1 retention probability $P(D_{i1}=0\mid Y_{i1},H_{i1})$, in addition to the observed outcome probability. (5) No. The coefficient is a conditional association in the specified selection model and does not by itself establish a causal effect.