Dropout Dependence on Outcome Level Versus Change

Why Distinguish Outcome Level From Change?

A dropout model can say that participants are more likely to leave when their current symptom level is high. A different model can say that they are more likely to leave when their symptoms have worsened since the preceding visit. These statements can imply quite different dropout probabilities for people with the same current score but different histories. The distinction matters whenever dropout may depend on outcomes that are themselves unobserved after a participant leaves.

The equations below describe a discrete-time dropout hazard. They do not assert that a selection coefficient has a causal interpretation. In particular, an association between a scheduled outcome and dropout is a feature of the specified joint model, not proof that changing the outcome would cause a person to drop out.

Definition: Let $Y_{it}$ be the scheduled outcome for participant $i$ at visit $t$, whether or not it is observed. Let $D_{it}=1$ mean that participant $i$ first drops out just before the scheduled assessment at visit $t$, and let $D_{it}=0$ mean that the participant remains under observation at that point. For a participant observed through visit $t-1$, the dropout hazard is $$ h_{it}=P(D_{it}=1\mid D_{i,t-1}=0,\text{ relevant outcome history}). $$ In a model depending on $Y_{it}$, the outcome at the visit where dropout occurs may be unobserved. The equation describes its relationship with dropout through a joint model for outcomes and dropout.

For clarity, consider logistic hazard equations with a visit-specific intercept $\alpha_t$. Let $\operatorname{logit}(h)=\log\{h/(1-h)\}$. A current-level model and a change model are

\begin{aligned} \text{Level model:}\qquad \operatorname{logit}(h_{it})&=\alpha_t+\gamma_L Y_{it},\\ \text{Change model:}\qquad \operatorname{logit}(h_{it})&=\alpha_t+\gamma_\Delta(Y_{it}-Y_{i,t-1}). \end{aligned}

Here $\gamma_L$ is the change in log odds of dropout per one-unit higher current outcome, while $\gamma_\Delta$ is the change in log odds per one-unit greater increase since the previous visit. Their units are inverse outcome units. If higher outcome values indicate worse health, a positive coefficient means higher modeled dropout odds at higher levels or larger increases, respectively. The direction reverses if lower values indicate worse health.

What the Equations Impose

The level model depends on the current outcome and not explicitly on the previous outcome. Thus, at the same visit and the same current score, it assigns the same hazard to participants regardless of their previous score. The change model instead depends on the difference between current and previous scores. It assigns the same hazard to participants with the same change, even if their current levels differ.

The algebra makes the distinction precise. An unrestricted model containing both current and previous outcomes is

\operatorname{logit}(h_{it})=\alpha_t+\beta_cY_{it}+\beta_pY_{i,t-1}.

Since $Y_{it}-Y_{i,t-1}$ is the change, the change model can be rewritten as

\alpha_t+\gamma_\Delta(Y_{it}-Y_{i,t-1}) =\alpha_t+\gamma_\Delta Y_{it}-\gamma_\Delta Y_{i,t-1}.

Therefore a change-only equation is the unrestricted current-plus-previous equation under the restriction $\beta_c=\gamma_\Delta$ and $\beta_p=-\gamma_\Delta$. Equivalently, it requires $\beta_p=-\beta_c$. A level-only equation is the different restriction $\beta_p=0$. The change model is not simply another name for a model that includes current outcome and adjusts for the previous outcome: it constrains the two coefficients to be equal in magnitude and opposite in sign.

This comparison assumes the same outcome scale and the same time-specific intercept structure. If the outcome is centered, the intercept changes, but the current-versus-change distinction does not. With unequal intervals between visits, a raw difference measures change over the interval, not a rate per unit time. If the scientific question concerns rate, the predictor would need to be defined as change divided by elapsed time, with a corresponding change in coefficient units.

Worked Example: Same Current Score, Different Histories

Suppose a hypothetical 0–12 symptom score is modeled at one follow-up visit. Use the level equation with $\alpha_t=-3$ and $\gamma_L=0.35$ per score point, and the change equation with $\alpha_t=-3$ and $\gamma_\Delta=0.50$ per point of change. Compare two participants whose current score is 8: one had a previous score of 6 and the other a previous score of 10. These are illustrative parameter values, not estimates from a study.

Under the level model, both participants have the same linear predictor:

\eta_L=-3+0.35(8)=-0.20,\qquad h_L=\frac{e^{-0.20}}{1+e^{-0.20}}=0.4502.

The model gives each participant a 0.4502 dropout hazard because their current scores are identical. Their previous scores do not enter this equation.

Under the change model, the first participant’s change is $8-6=2$, while the second participant’s change is $8-10=-2$. Their hazards are

\begin{aligned} \eta_{\Delta,1}&=-3+0.50(2)=-2, &h_{\Delta,1}&=\frac{e^{-2}}{1+e^{-2}}=0.1192,\\ \eta_{\Delta,2}&=-3+0.50(-2)=-4, &h_{\Delta,2}&=\frac{e^{-4}}{1+e^{-4}}=0.0180. \end{aligned}

With these illustrative equations, the change model assigns a higher hazard to the participant whose score increased, even though both current scores are 8. The numerical probabilities should not be compared as if the two equations had been fitted and calibrated to data; the calculation demonstrates how their predictors respond differently to history.

Worked Example: Same Increase, Different Current Levels

Now compare two hypothetical score histories: a change from 5 to 7 and a change from 10 to 12. Both participants increase by 2 points. Use the same illustrative equations as above.

For the level model, the two current scores produce different predictors and hazards:

\begin{aligned} Y_{it}=7:\quad &\eta_L=-3+0.35(7)=-0.55, &h_L&=\frac{e^{-0.55}}{1+e^{-0.55}}=0.3659,\\ Y_{it}=12:\quad &\eta_L=-3+0.35(12)=1.20, &h_L&=\frac{e^{1.20}}{1+e^{1.20}}=0.7685. \end{aligned}

The change model uses the same change, $2$, for both histories:

\eta_\Delta=-3+0.50(2)=-2,\qquad h_\Delta=\frac{e^{-2}}{1+e^{-2}}=0.1192.

Thus, this change model assigns equal hazards to the two histories, while the level model distinguishes them by current score. The example also shows why the scientific meaning of the outcome scale matters: a 2-point increase may be treated as the same change at different starting levels, even when the practical implications of that increase differ.

Coefficient Restrictions in a Combined Model

A useful way to express the contrast is to begin with the combined current-and-previous model. Its two coefficients can represent separate associations with current level and previous level, conditional on the other. The level-only and change-only equations are special cases with different restrictions, as shown here.

Specification	Current-outcome coefficient	Previous-outcome coefficient	Implication
Current level only	$\beta_c=\gamma_L$	$\beta_p=0$	At a fixed current score, previous score adds no term.
Change only	$\beta_c=\gamma_\Delta$	$\beta_p=-\gamma_\Delta$	Equal and opposite coefficients make the predictor depend on the difference.
Current plus previous	$\beta_c$ unrestricted	$\beta_p$ unrestricted	Allows separate associations without imposing the change restriction.

Worked Example: Translating a Change Equation

Suppose a hypothetical change model for a symptom score has $\alpha_t=-4.2$ and $\gamma_\Delta=0.30$ per point. For a participant whose score changes from 6 to 8, the change is 2 points. The model gives

\eta_\Delta=-4.2+0.30(8-6)=-3.6,\qquad h_\Delta=\frac{e^{-3.6}}{1+e^{-3.6}}=0.0266.

Expanding the equation gives $-4.2+0.30Y_{it}-0.30Y_{i,t-1}$. In the combined model’s notation, this is $\beta_c=0.30$ and $\beta_p=-0.30$. For comparison, the current-level restriction with $\beta_c=0.30$, $\beta_p=0$, and the same illustrative intercept gives

\eta_L=-4.2+0.30(8)=-1.8,\qquad h_L=\frac{e^{-1.8}}{1+e^{-1.8}}=0.1419.

The different hazards follow from the different coefficient restrictions, not from an arithmetic discrepancy. If these were alternative fitted models, their intercepts and coefficients would generally be estimated separately; holding them fixed here isolates the role of the previous-outcome term.

Assumptions, Diagnostics, and Limitations

A level or change equation is meaningful only alongside a defensible outcome model and a clear dropout-time definition. In particular, the scheduled current outcome must be defined even for participants whose value is not observed because they drop out at that visit. The selection equation is then part of a joint model for outcomes and missingness. It is not justified to substitute a participant’s last observed value for the unobserved current outcome and treat that replacement as the value in the equation.

The logistic forms also make specific assumptions. Conditional on the stated outcome term and visit, the log odds are linear in that term; $\alpha_t$ permits baseline dropout odds to differ by visit. A constant $\gamma_L$ or $\gamma_\Delta$ assumes the same per-unit association across visits unless further terms are included. The model must also respect the outcome’s scale and plausible range. For a bounded score, a linear predictor may imply extreme hazards near the boundaries; the implied probabilities should be inspected over plausible outcome histories.

Diagnostics cannot directly verify the dependence on an outcome that is missing precisely when dropout occurs. Useful checks include whether the model produces plausible hazards across the joint range of current and previous scores, whether its implications fit observed dropout patterns across visits and observed histories, and whether conclusions change under reasonable alternative selection equations. Strong correlation between current and previous outcomes can also make their separate coefficients difficult to estimate in a combined model. A change predictor may have limited variation or may be measured noisily, especially when scores are close to their scale limits.

The main alternatives answer different questions. A current-level equation targets dependence on absolute current score; a change-only equation targets dependence on the difference; and the combined current-plus-previous equation relaxes the equal-and-opposite restriction. A model based only on observed history makes a different missingness assumption: it does not posit additional dependence on the unobserved current outcome once that history is given. These specifications should be understood as alternative assumptions about the dropout process, not as interchangeable parameterizations with automatically identical implications.

Because the current outcome is unavailable for dropouts, observed data alone generally cannot establish which dependence is correct without additional assumptions. A good observed-history fit does not prove that dropout is independent of the missing current outcome. The selection coefficients therefore require careful substantive interpretation and sensitivity analysis; they describe conditional model associations, not causal effects or directly observed relationships for those who dropped out.

Common Mistake

Common mistake: “A change model is just a current-outcome model with the previous score controlled for.” That is incorrect. A change model includes the previous score with a coefficient constrained to be the negative of the current-score coefficient. A combined model that estimates both coefficients freely does not impose this restriction. To check the distinction, expand $\gamma_\Delta(Y_{it}-Y_{i,t-1})$ and identify both coefficients explicitly.

Check Your Understanding

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

In the combined model, what restriction on $\beta_c$ and $\beta_p$ gives a change-only equation?
With a change-model intercept of $-2$ and coefficient $0.4$, calculate the hazard for a change of $-1$ point.
Two participants have the same current score but different previous scores. Which of the two basic specifications necessarily assigns them the same hazard, at a fixed visit?
Why is a current-outcome selection equation not directly checked by comparing predicted dropout with observed current outcomes among participants who dropped out at that visit?

Solutions.

1. Expanding the change term gives $\gamma_\Delta Y_{it}-\gamma_\Delta Y_{i,t-1}$. Therefore $\beta_c=\gamma_\Delta$, $\beta_p=-\gamma_\Delta$, or equivalently $\beta_p=-\beta_c$.

2. The linear predictor is $\eta=-2+0.4(-1)=-2.4$. Thus $h=e^{-2.4}/(1+e^{-2.4})=0.08317$, or about an 8.32% modeled hazard at that visit, conditional on remaining under observation through the previous visit.

3. The level-only specification does: at a fixed visit it depends on the current score, not the previous score. The change model can assign different hazards because the changes differ.

4. The current outcomes for those participants are missing by the dropout definition. Their unobserved values cannot be used as directly observed validation data; evaluating the equation requires the specified joint model and assumptions about the missing outcomes.

Specification	Current-outcome coefficient	Previous-outcome coefficient	Implication
Current level only	\(\beta_c=\gamma_L\)	\(\beta_p=0\)	At a fixed current score, previous score adds no term.
Change only	\(\beta_c=\gamma_\Delta\)	\(\beta_p=-\gamma_\Delta\)	Equal and opposite coefficients make the predictor depend on the difference.
Current plus previous	\(\beta_c\) unrestricted	\(\beta_p\) unrestricted	Allows separate associations without imposing the change restriction.