Separating Current and Lagged Outcome Effects

Why Separate Current and Lagged Outcomes?

A dropout model can allow the chance of leaving a study to depend on more than one outcome value. The current outcome and the most recent previous outcome play different roles: one represents the outcome at the visit where response is missing, while the other represents an already observed part of the participant’s history. If both appear in the model, each coefficient describes a conditional association, holding the other outcome fixed. That distinction matters when interpreting dropout odds or comparing participants with different histories.

The model below describes dropout before a scheduled outcome measurement. This timing makes the current outcome $Y_t$ unavailable for someone who drops out at visit $t$, even though it appears in the selection equation. By contrast, $Y_{t-1}$ is observed for someone at risk at $t$. Treating these two variables as interchangeable, or assuming that both are observed for every participant, changes the model being described.

Definition: Let $R_t=1$ mean that the outcome at visit $t$ is observed, and $R_t=0$ mean that it is missing because of dropout. For a participant observed at $t-1$, the dropout hazard at $t$ is $P(R_t=0\mid R_{t-1}=1,\ldots)$. A logistic current-and-lagged-outcome selection model is $$ \operatorname{logit}P(R_t=0\mid R_{t-1}=1,Y_t,Y_{t-1}) =\alpha_t+\gamma Y_t+\delta Y_{t-1}, $$ where $\operatorname{logit}(p)=\log\{p/(1-p)\}$, $\alpha_t$ is the visit-specific intercept, $\gamma$ is the current-outcome coefficient, and $\delta$ is the lagged-outcome coefficient. The equation applies to visits at which participants can be at risk, ordinarily after an observed previous visit.

What Each Coefficient Means

The equation is conditional: it compares dropout odds at the same visit and at the same value of the other outcome. If $Y_t$ increases by $c$ outcome units while $Y_{t-1}$ stays fixed, the dropout log-odds change by $\gamma c$, and the dropout odds are multiplied by $\exp(\gamma c)$. If instead $Y_{t-1}$ increases by $d$ units while $Y_t$ stays fixed, the log-odds change by $\delta d$, and the odds are multiplied by $\exp(\delta d)$.

\frac{\operatorname{odds}(R_t=0\mid Y_t+c,Y_{t-1})} {\operatorname{odds}(R_t=0\mid Y_t,Y_{t-1})} =\exp(\gamma c), \qquad \frac{\operatorname{odds}(R_t=0\mid Y_t,Y_{t-1}+d)} {\operatorname{odds}(R_t=0\mid Y_t,Y_{t-1})} =\exp(\delta d).

Here dropout odds are $p/(1-p)$, not dropout probability $p$. A positive coefficient means higher values of that outcome are associated with higher conditional dropout odds; a negative coefficient means lower conditional odds. Neither coefficient alone gives a marginal dropout risk, and neither establishes a causal effect. The interpretation also depends on the units used for $Y_t$ and $Y_{t-1}$.

When both outcomes change, their contributions add on the log-odds scale. For a contrast from $(Y_t,Y_{t-1})$ to $(Y_t+c,Y_{t-1}+d)$, the odds ratio is $\exp(\gamma c+\delta d)$. Consequently, different histories can have the same predicted odds if their current- and lagged-outcome contributions offset one another. A coefficient is not an unconditional description of what happens when an entire outcome history changes.

Worked Example: Isolating the Current-Outcome Effect

Suppose, for illustration, the dropout model at a particular visit is $\operatorname{logit}(p)=-2+0.50Y_t-0.25Y_{t-1}$, with both outcomes measured in the same stated units. Compare histories with $Y_{t-1}=4$ and current outcomes $Y_t=6$ and $Y_t=4$. Treat the coefficients as stipulated hypothetical values, not estimates from a real study.

For $Y_t=6$, the linear predictor is $\eta_1=-2+(0.50)(6)-(0.25)(4)=-2+3-1=0$. Thus $p_1=\exp(0)/(1+\exp(0))=0.500$.

For $Y_t=4$, the predictor is $\eta_2=-2+(0.50)(4)-(0.25)(4)=-2+2-1=-1$. Thus $p_2=\exp(-1)/(1+\exp(-1))\approx0.2689$.

The current outcome is 2 units higher in the first history, with the lagged outcome held fixed. Its conditional odds ratio is $\exp\{(0.50)(6-4)\}=\exp(1)\approx2.718$. The first history therefore has about 2.718 times the dropout odds of the second under this model. The corresponding probabilities are 0.500 and 0.2689; their ratio is not 2.718, because the coefficient specifies an odds ratio rather than a probability ratio.

Why the Current Outcome Is a Selection Variable

In this timing convention, dropout at $t$ means $Y_t$ is not recorded. Therefore, although the dropout equation conditions on $Y_t$, that value is latent for participants who drop out at $t$. The lagged outcome $Y_{t-1}$, by contrast, is known for those participants because they were observed at the preceding visit. This is the essential distinction between the two terms in the selection equation.

A likelihood for the observed data must respect that distinction. For a participant who drops out at $t$, the contribution involving the missing current outcome generally combines its outcome density with the dropout probability and integrates over possible values of $Y_t$. Schematically, if $f_t(y\mid\text{past})$ is the specified conditional outcome density, the contribution has the form

\int f_t(y\mid\text{past})\, P(R_t=0\mid R_{t-1}=1,Y_t=y,Y_{t-1})\,dy.

This expression is schematic: the full likelihood also accounts for observed history and the model’s other response-pattern contributions. It shows why an ordinary logistic regression using the unobserved current outcome as if it were recorded cannot directly estimate the selection equation for dropouts. The outcome model and selection model must be considered together, as in the earlier material on observed-data likelihood contributions by dropout time and integrating out outcomes after dropout.

The conditional odds calculations remain valid as interpretations of the specified model. Estimating the coefficients from incomplete data, however, requires a likelihood or other estimation procedure that handles the latent current outcomes and states the outcome-model assumptions. The selection relationship is not identified from the observed dropout indicators alone without such modeling assumptions.

Worked Example: Isolating the Lagged-Outcome Effect

Consider the hypothetical model $\operatorname{logit}(p)=-1.20+0.40Y_t+0.30Y_{t-1}$. Compare two histories with the same current outcome, $Y_t=5$, but lagged outcomes $Y_{t-1}=2$ and $Y_{t-1}=4$.

For the first history, $\eta_1=-1.20+(0.40)(5)+(0.30)(2)=-1.20+2+0.60=1.40$. Thus $p_1=\exp(1.40)/(1+\exp(1.40))\approx0.8022$.

For the second history, $\eta_2=-1.20+(0.40)(5)+(0.30)(4)=-1.20+2+1.20=2.00$. Thus $p_2=\exp(2)/(1+\exp(2))\approx0.8808$.

The lagged outcome is 2 units higher, while the current outcome is held fixed. The conditional dropout odds ratio is $\exp\{(0.30)(4-2)\}=\exp(0.60)\approx1.822$. Under this model, the second history has about 1.822 times the dropout odds. The probabilities differ by about 0.0786, or 7.86 percentage points; that probability difference is specific to these histories and is not the coefficient’s general interpretation.

Contrasting Histories and Conditional Comparisons

To compare two arbitrary histories at the same visit, subtract their linear predictors. The intercept cancels, leaving a weighted sum of the differences in current and lagged outcomes. This calculation is a direct way to check whether a verbal comparison really isolates one coefficient or combines both effects.

\eta_A-\eta_B =\gamma(Y_{t,A}-Y_{t,B}) +\delta(Y_{t-1,A}-Y_{t-1,B}), \qquad \frac{\operatorname{odds}_A}{\operatorname{odds}_B} =\exp(\eta_A-\eta_B).

For example, return to the model $-2+0.50Y_t-0.25Y_{t-1}$. Compare history A, $(Y_t,Y_{t-1})=(6,4)$, with history B, $(4,0)$. The current-outcome contribution to $\eta_A-\eta_B$ is $0.50(6-4)=1$; the lagged contribution is $-0.25(4-0)=-1$. Their sum is zero, so the odds ratio is $\exp(0)=1$. Both predictors are zero, and both predicted probabilities are 0.500. This equality does not mean the coefficients are zero: the two nonzero contributions cancel for this particular contrast.

Worked Example: Comparing Two Histories Together

Use the hypothetical model $\operatorname{logit}(p)=-2+0.50Y_t-0.25Y_{t-1}$. Compare history A, $(Y_t,Y_{t-1})=(6,4)$, with history B, $(4,8)$.

For A, $\eta_A=-2+(0.50)(6)-(0.25)(4)=0$, so $p_A=0.500$. For B, $\eta_B=-2+(0.50)(4)-(0.25)(8)=-2+2-2=-2$, so $p_B=\exp(-2)/(1+\exp(-2))\approx0.1192$.

The log-odds difference is $0.50(6-4)-0.25(4-8)=1+1=2$. Therefore the odds ratio for A versus B is $\exp(2)\approx7.389$. Both the current and lagged differences contribute in the same direction here. This contrast cannot be attributed to the current coefficient alone or the lagged coefficient alone.

Assumptions, Diagnostics, and Limitations

Interpretation of the equation relies on explicit modeling assumptions. The logistic link and the additive linear predictor are assumed adequate on the log-odds scale; the current and lagged outcomes enter linearly unless the model specifies otherwise. The visit-specific intercept must reflect the intended time structure, and the definition and timing of dropout must be consistent across visits. For likelihood-based estimation, a suitable outcome model must also be specified so that missing current outcomes can be integrated out. These assumptions are not verified merely by fitting the equation.

Useful checks include verifying that $R_t=0$ denotes dropout at the intended time, confirming which outcome values are observed for each dropout pattern, and checking that each predictor is measured and scaled consistently. Examine whether the fitted model reproduces observed dropout frequencies across visits and across strata of observed lagged outcomes. Since the current outcome is missing for those dropping out at $t$, observed-data checks cannot directly verify the assumed dependence on that latent value. Sensitivity to plausible alternative current-outcome coefficients or outcome-model specifications is therefore important.

Strong association between $Y_t$ and $Y_{t-1}$ can make their separate coefficients imprecise: the model may contain little information to distinguish their conditional contributions. Wide uncertainty, unstable estimates, or large changes under small specification changes are warnings against overinterpreting either coefficient. A linear term can also be inadequate if dropout odds change nonlinearly with outcome level; residual patterns and comparisons with substantively plausible alternatives can help diagnose this, although missing current outcomes limit direct assessment.

Selection equation	What it allows	Main interpretive caution
Current outcome only	Dropout odds depend on $Y_t$	Does not represent a separate conditional association with $Y_{t-1}$
Lagged outcome only	Dropout odds depend on observed $Y_{t-1}$	Does not represent dependence on the missing current outcome
Current and lagged outcomes	Each outcome has its own coefficient, conditional on the other	Separate effects can be weakly determined when outcomes are highly related

The model is a conditional selection model, not a causal explanation of why a person leaves a study. Its coefficients are conditional odds effects, not marginal risk differences. Nor does including a lagged outcome automatically solve selection bias: the current outcome may still be unobserved precisely when dropout occurs, and inference relies on the joint modeling assumptions. Comparisons with a current-only or lagged-only equation can clarify what dependence is being assumed, but should be treated as model comparisons rather than proof of a particular mechanism.

Common Pitfall

Wrong reasoning: “The current-outcome coefficient is positive, so every participant with a high current outcome has a higher dropout probability than every participant with a low one.” This ignores the lagged outcome, the visit intercept, and the conditional nature of the comparison. It also turns an odds statement into an absolute probability statement.

Correction: At the same visit and the same lagged outcome, increasing $Y_t$ by $c$ multiplies the modeled dropout odds by $\exp(\gamma c)$. For participants with different lagged outcomes, compare the full linear predictors. In addition, $Y_t$ is latent for a participant who drops out at $t$, so the coefficient’s estimation depends on the joint likelihood and its assumptions.

Key takeaway: The current coefficient $\gamma$ describes a conditional association with the outcome at the dropout visit; the lagged coefficient $\delta$ describes a conditional association with the previous outcome. Their contributions add on the log-odds scale, and the current outcome’s missingness must be handled in estimation.

Check Your Understanding

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

In the model $\operatorname{logit}(p)=\alpha_t+0.20Y_t-0.10Y_{t-1}$, what is the conditional odds ratio for a 3-unit increase in $Y_t$, holding $Y_{t-1}$ fixed?
For the same model, what is the conditional odds ratio for a 2-unit increase in $Y_{t-1}$, holding $Y_t$ fixed?
Why can’t the current outcome term simply be entered as an observed predictor for participants who drop out at visit $t$?
If two outcome histories have the same linear predictor, must their current and lagged outcomes be equal? Explain.
Name one warning sign that the separate current and lagged coefficients may be difficult to interpret.

Solutions. (1) The odds ratio is $\exp(0.20\times3)=\exp(0.60)\approx1.822$. (2) It is $\exp(-0.10\times2)=\exp(-0.20)\approx0.8187$; the higher lagged outcome corresponds to lower conditional dropout odds in this model. (3) $Y_t$ is missing for someone who drops out before that scheduled measurement, so estimation must account for its latent value, typically through a joint likelihood. (4) No. Different current and lagged values can yield the same weighted sum $\gamma Y_t+\delta Y_{t-1}$, including when their contributions offset. (5) Strong correlation between the two outcomes can produce imprecise or unstable conditional coefficients, because the data may not clearly distinguish their separate contributions.

Selection equation	What it allows	Main interpretive caution
Current outcome only	Dropout odds depend on \(Y_t\)	Does not represent a separate conditional association with \(Y_{t-1}\)
Lagged outcome only	Dropout odds depend on observed \(Y_{t-1}\)	Does not represent dependence on the missing current outcome
Current and lagged outcomes	Each outcome has its own coefficient, conditional on the other	Separate effects can be weakly determined when outcomes are highly related