Logistic Dependence on the Current Outcome

Why Model Retention as a Function of the Current Outcome?

In a longitudinal study with monotone dropout, a participant who leaves the study stops providing outcomes at later visits. A selection model describes the response process conditional on outcomes in the full data. In particular, the Diggle–Kenward logistic model can let the probability of remaining observed at a visit depend on the outcome at that same visit. This makes the model capable of representing a form of missing-not-at-random selection: people with different current outcomes can have different response probabilities, even after conditioning on being observed at the preceding visit.

The conditional statement is important. It concerns the response probability among participants still at risk of dropout, not the probability of being observed among everyone enrolled at baseline. Also, the current outcome may be unobserved for a participant who drops out at that visit. The model is therefore a specification involving the full-data outcome, not merely a regression that can always be fitted using recorded outcomes.

Definition: Let \(Y_{it}\) be the outcome for participant \(i\) at visit \(t\), and let \(R_{it}=1\) if that outcome is observed and \(R_{it}=0\) otherwise. Under monotone dropout, \(R_{it}=0\) implies \(R_{i,t+1}=0\). Set \(R_{i0}=1\). For a participant observed through visit \(t-1\), define the conditional retention probability \(q_t(y)=P(R_{it}=1\mid R_{i,t-1}=1,Y_{it}=y)\). The current-outcome logistic selection model is \(\operatorname{logit}\{q_t(y)\}=\alpha_t+\delta y\), where \(\operatorname{logit}(q)=\log\{q/(1-q)\}\). Here \(\alpha_t\) is a visit-specific intercept and \(\delta\) is a current-outcome coefficient.

What the Equation Assumes

The equation specifies the response mechanism among participants who reached the visit while still observed. It makes several substantive restrictions. Dropout is monotone; the conditional retention probability has a logistic form; the intercept may vary by visit; and the current outcome enters linearly on the logit scale with a common coefficient \(\delta\). In this basic version, no other variable or earlier outcome appears in the response equation. Thus, conditional on being at risk and on \(Y_{it}\), retention does not depend on additional predictors in the specified model.

These are modeling assumptions, not consequences of the observed data. If relevant baseline covariates or earlier outcomes also affect retention, omitting them can make the current-outcome term absorb patterns it should not represent. A richer equation could include such predictors, but the model here is deliberately restricted to current outcome and visit-specific intercepts.

The equation is understood as a conditional distribution involving the complete-data value \(Y_{it}\), whether or not that value is recorded. It does not, by itself, settle the substantive question of when a dropout decision is made relative to measurement of the outcome. Nor does a plausible fit to observed data establish that the assumed dependence on an unobserved outcome is correct.

From Logit to Retention and Dropout Probabilities

The logistic equation is often easiest to use after solving for \(q_t(y)\). Write \(\eta_t(y)=\alpha_t+\delta y\). By the definition of the logit,

\log\!\left(\frac{q_t(y)}{1-q_t(y)}\right)=\eta_t(y) \quad\Longrightarrow\quad \frac{q_t(y)}{1-q_t(y)}=\exp\{\eta_t(y)\}.

Multiplying by \(1-q_t(y)\), then collecting the terms containing \(q_t(y)\), gives the probability form:

q_t(y)=\frac{\exp\{\alpha_t+\delta y\}} {1+\exp\{\alpha_t+\delta y\}}, \qquad P(R_{it}=0\mid R_{i,t-1}=1,Y_{it}=y) =\frac{1}{1+\exp\{\alpha_t+\delta y\}}.

The second expression is the conditional probability of dropout at visit \(t\), given that the participant was observed through visit \(t-1\). These probabilities add to one. If \(\delta=0\), retention in this model does not vary with the current outcome. If \(\delta<0\), higher values of the outcome correspond to lower conditional retention probabilities; if \(\delta>0\), they correspond to higher conditional retention probabilities. These directions follow because the logistic function is increasing in its input.

Worked Example: Calculating Retention at Two Outcome Values

Suppose, hypothetically, that at visit 2 the model has \(\alpha_2=-1.20\) and \(\delta=-0.50\). Compare conditional retention for participants with \(Y_{i2}=0\) and \(Y_{i2}=2\). Assume both are observed through visit 1, as required for the risk set.

For \(Y_{i2}=0\), the linear predictor is \(-1.20+(-0.50)(0)=-1.20\). Therefore, \(q_2(0)=\exp(-1.20)/\{1+\exp(-1.20)\} =0.3011942/1.3011942\approx0.2315\). The corresponding conditional dropout probability is \(1-0.2314752\approx0.7685\).

For \(Y_{i2}=2\), the linear predictor is \(-1.20+(-0.50)(2)=-2.20\). Thus, \(q_2(2)=\exp(-2.20)/\{1+\exp(-2.20)\} =0.1108032/1.1108032\approx0.09975\). The conditional dropout probability is \(1-0.0997505\approx0.9002\).

Under these hypothetical model parameters, the conditional retention probability is lower at outcome value 2 than at outcome value 0. These are model-implied probabilities, not estimates from a real study.

Worked Examples Across Visits

Worked Example: Comparing Two Current Outcomes

Consider a hypothetical visit with \(\alpha_1=0.30\) and \(\delta=0.20\). Calculate retention for outcome values \(Y_{i1}=1\) and \(Y_{i1}=-1\), among participants observed at baseline.

At \(Y_{i1}=1\), the linear predictor is \(0.30+(0.20)(1)=0.50\). Hence \(q_1(1)=\exp(0.50)/\{1+\exp(0.50)\} =1.648721/2.648721\approx0.6225\). The conditional dropout probability is \(1-0.6224593\approx0.3775\).

At \(Y_{i1}=-1\), the linear predictor is \(0.30+(0.20)(-1)=0.10\). Hence \(q_1(-1)=\exp(0.10)/\{1+\exp(0.10)\} =1.105171/2.105171\approx0.5250\). The conditional dropout probability is \(1-0.5249792\approx0.4750\).

Since \(\delta\) is positive, the model assigns greater retention to the higher of these two outcomes. The comparison is conditional on the same visit and risk set; it does not compare overall observation rates from baseline.

Worked Example: Allowing the Baseline Retention Level to Change by Visit

Suppose a hypothetical two-visit model has \(\alpha_1=-0.40\), \(\alpha_2=0.20\), and a common current-outcome coefficient \(\delta=-0.30\). Calculate conditional retention at both visits for a current outcome value of \(1.5\). For visit 1, the risk set consists of participants observed at baseline; for visit 2, it consists of participants observed through visit 1.

At visit 1, the linear predictor is \(-0.40+(-0.30)(1.5)=-0.85\). Therefore \(q_1(1.5)=\exp(-0.85)/\{1+\exp(-0.85)\} =0.4274149/1.4274149\approx0.2994\).

At visit 2, the linear predictor is \(0.20+(-0.30)(1.5)=-0.25\). Therefore \(q_2(1.5)=\exp(-0.25)/\{1+\exp(-0.25)\} =0.7788008/1.7788008\approx0.4378\).

The two probabilities differ because the visit-specific intercepts differ. They are conditional probabilities for their respective risk sets, not cumulative probabilities of being observed at each visit. The negative current-outcome coefficient gives the same direction of outcome dependence at both visits in this model.

Why the Specification Matters

An outcome-independent response model is obtained by setting \(\delta=0\). It allows retention to vary by visit through \(\alpha_t\), but not by the current outcome within a visit. The current-outcome model is more flexible in this particular respect, but it requires a dependence on potentially unobserved values. A pattern-mixture representation describes distributions conditional on missingness patterns instead; it is a different factorization of the joint distribution, not evidence that the selection equation is true. The earlier material on equivalent selection and pattern-mixture representations explains why distinct representations can describe the same joint distribution.

The current-outcome term matters because it encodes how response is allowed to depend on an outcome that may be missing precisely when the response indicator is zero. The observed data generally cannot directly reveal the response probability at each unobserved value. As emphasized in the earlier material on identification and MNAR extrapolation, a well-fitting model for the observed data cannot by itself validate this untestable part of the specification. Interpretation therefore depends on the assumed outcome model and response model together, as well as on the available identifying information or sensitivity assumptions.

Diagnostics, Failure Modes, and Limitations

Some features of the specification can be checked against observed data, while the central MNAR dependence cannot usually be verified from those data alone. Useful checks include confirming the monotone dropout pattern, examining observed retention rates by visit, checking whether the chosen outcome scale makes a linear logit dependence plausible over its observed range, and comparing observed response patterns across observed predictors. Such checks can reveal contradictions or poor fit in observable margins; they cannot establish the model's behavior at outcome values missing because of dropout.

Nonmonotone response histories: If participants can return after missing a visit, the monotone-dropout setup does not describe the response process as stated.
Omitted predictors: Strong associations of retention with recorded baseline variables or earlier outcomes may indicate that the basic equation is too restrictive.
Implausible extrapolation: The logistic equation gives probabilities between zero and one, but that mathematical property does not ensure plausible probabilities at unobserved outcome values.
Unstable conclusions: If substantive results change substantially across reasonable values of \(\delta\), that sensitivity is important to report rather than concealing it behind a single fitted specification.

A common mistake is to treat \(\delta\) as a directly observed association because the equation resembles an ordinary logistic regression. That reasoning overlooks that \(Y_{it}\) is unavailable for some participants in the relevant risk set. The correction is to regard the equation as a model for the full-data response mechanism and to distinguish observable fit checks from assumptions about missing outcomes. Another mistake is to interpret \(q_t(y)\) as the probability of being observed from study entry through visit \(t\); it is conditional on being observed through visit \(t-1\). Overall observation through a visit involves the sequence of conditional retention probabilities, not just the single \(q_t(y)\).

Key takeaway: The basic current-outcome Diggle–Kenward specification is \(\operatorname{logit}\{P(R_{it}=1\mid R_{i,t-1}=1,Y_{it}=y)\}=\alpha_t+\delta y\). It defines a conditional retention model, makes restrictive functional-form assumptions, and includes a dependence that cannot generally be checked using observed outcomes alone.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

Write the conditional retention probability when \(\alpha_t=0\), \(\delta=-0.4\), and \(Y_{it}=1\). What is the corresponding conditional dropout probability?
What does \(\delta=0\) imply about current-outcome dependence in this model? Does it imply that retention is constant across visits?
For \(\alpha_t=-0.5\), \(\delta=0.2\), and \(Y_{it}=2\), calculate the linear predictor and approximate retention probability.
Why can an observed-data fit check not by itself verify the model's dependence on a current outcome that is unobserved for dropouts?

Solutions. (1) The linear predictor is \(0+(-0.4)(1)=-0.4\), so retention is \(\exp(-0.4)/\{1+\exp(-0.4)\}=0.4013\), and conditional dropout is \(1-0.4013=0.5987\). (2) It removes current-outcome dependence, but the visit-specific intercepts \(\alpha_t\) can still make retention differ by visit. (3) The predictor is \(-0.5+(0.2)(2)=-0.1\), so retention is \(\exp(-0.1)/\{1+\exp(-0.1)\}\approx0.4750\). (4) For dropouts, the current outcome needed to assess that conditional dependence is missing; observed-data checks can assess observable implications but cannot directly verify the response probability at those unobserved values.