Treatment-by-Outcome Interactions in Dropout

Why Allow the Outcome Association to Differ by Treatment?

A dropout model may associate the current outcome with the chance of subsequently leaving follow-up. A common-slope model assumes that this association is the same in every treatment group. That restriction can be too strong: for example, an unfavorable biomarker value might be more closely associated with later dropout in one treatment group than another. A treatment-by-outcome interaction represents this difference directly.

The interaction is a feature of the dropout equation, not a claim that treatment causes dropout through the outcome. It describes how the conditional dropout odds vary with the current outcome and treatment assignment, given the variables in the model. Its meaning depends on the chosen link and outcome scale.

A Pooled Dropout Equation With an Interaction

For participant $i$ at visit $t$, let $A_i=0$ denote control and $A_i=1$ denote treatment. Define $D_{it}=1$ if the participant drops out after providing the visit-$t$ outcome, and $D_{it}=0$ otherwise. The model applies to participants still under follow-up and at risk of dropping out at that visit. Thus, the current outcome $Y_{it}$ is recorded before the dropout decision being modelled. Let $z_{it}=(Y_{it}-c)/q$, where $c$ is a prespecified reference value and $q>0$ is a chosen unit, such as 10 biomarker units. Let $X_i$ denote other included covariates.

Definition: A pooled logistic dropout model with a treatment-by-current-outcome interaction is $$ \operatorname{logit}\{P(D_{it}=1\mid \text{at risk at }t,A_i,Y_{it},X_i)\} =\alpha_t+\gamma A_i+\beta z_{it}+\delta A_i z_{it}+\theta^\mathsf{T}X_i. $$ Here $\operatorname{logit}(p)=\log\{p/(1-p)\}$; $\alpha_t$ is a visit-specific intercept; $\gamma$ is the treatment contrast at $z_{it}=0$; $\beta$ is the outcome association in control; and $\delta$ is the difference between the treatment and control outcome associations on the log-odds scale.

The model assumes the specified conditional log odds are linear in the included terms, with the stated visit effects and covariate adjustment. It also assumes the interaction is adequately represented by the product $A_i z_{it}$. The logit link is a modelling choice; the coefficients and their interpretations are specific to it. In repeated-visit data, estimation and uncertainty calculations must account appropriately for the within-participant structure and the dropout process being modelled.

The reference value $c$ matters for interpreting $\gamma$: because $z_{it}=0$ when $Y_{it}=c$, $\gamma$ compares treatment and control at that outcome value, at the same visit and covariate values. Centering at a meaningful value makes this contrast easier to interpret. Rescaling by $q$ changes the numerical slope coefficients, so state the outcome increment whenever interpreting them.

Reading the Interaction

Substituting $A_i=0$ gives the control-group logit, and substituting $A_i=1$ gives the treatment-group logit. These substitutions establish the group-specific outcome slopes without fitting two separately specified equations.

\begin{aligned} A_i=0:\quad & \operatorname{logit}(p_{it})=\alpha_t+\beta z_{it}+\theta^\mathsf{T}X_i,\\ A_i=1:\quad & \operatorname{logit}(p_{it})=\alpha_t+\gamma+(\beta+\delta)z_{it}+\theta^\mathsf{T}X_i. \end{aligned}

Here $p_{it}$ denotes the conditional probability of dropout in the model. For a one-unit increase in $z_{it}$, the dropout odds are multiplied by $\exp(\beta)$ in control and by $\exp(\beta+\delta)$ in treatment, holding visit and covariates fixed. The ratio of those two odds multipliers is $\exp(\delta)$. Therefore, $\delta=0$ means a common outcome slope on the log-odds scale; $\delta\ne0$ means those slopes differ on that scale.

The treatment contrast also depends on the outcome. At a given $z_{it}$, the treatment-to-control conditional dropout odds ratio is $\exp(\gamma+\delta z_{it})$. Thus $\exp(\gamma)$ is the treatment odds ratio only at the reference outcome $z_{it}=0$, not a single treatment odds ratio that applies at every outcome level.

Worked Example: Interpreting the Two Outcome Slopes

Suppose a hypothetical fitted model uses $z=(Y-50)/10$, so one unit represents a 10-unit increase in the biomarker. Suppose the estimated outcome coefficient is $\hat{\beta}=-0.30$ and the interaction coefficient is $\hat{\delta}=0.50$.

Control slope. In control, the log-odds change per 10-unit increase is $\hat{\beta}=-0.30$. The corresponding dropout-odds multiplier is $\exp(-0.30)=0.7408$. Conditional on the model covariates and visit, a 10-unit higher biomarker is associated with about 0.741 times the dropout odds, or about 25.9% lower odds, in control.

Treatment slope. In treatment, the log-odds change is $\hat{\beta}+\hat{\delta}=-0.30+0.50=0.20$. The odds multiplier per 10-unit increase is $\exp(0.20)=1.221$, or about 22.1% higher dropout odds.

Difference between slopes. The ratio of the treatment and control odds multipliers is $\exp(\hat{\delta})=\exp(0.50)=1.649$. The outcome-associated odds multiplier is therefore about 1.65 times as large in treatment as in control for the specified 10-unit increase. This is a comparison of conditional odds multipliers, not a 65% change in dropout probability.

From Log Odds to Conditional Probabilities

An interaction in a logit model does not translate into a constant probability difference. The inverse-logit transformation is nonlinear, so the probability change associated with a fixed outcome increment depends on the starting probability. Similarly, even when $\delta=0$, treatment groups can have different probability-scale outcome associations if their baseline logits differ.

Worked Example: Calculating Probabilities at Two Outcome Levels

Consider a hypothetical model with no additional covariates, $\alpha_t=-2.00$, $\gamma=0.40$, $\beta=-0.30$, and $\delta=0.50$. Use $z=(Y-50)/10$. Assume the model and visit-specific intercept apply to each comparison below.

At the reference outcome $z=0$, control has logit $-2.00$, so its conditional dropout probability is $1/(1+\exp(2.00))=0.1192$, or 11.9%. Treatment has logit $-2.00+0.40=-1.60$, giving $1/(1+\exp(1.60))=0.1680$, or 16.8%.

At $z=1$, corresponding to a biomarker 10 units above 50, control has logit $-2.00-0.30=-2.30$, giving $1/(1+\exp(2.30))=0.0911$, or 9.11%. Treatment has logit $-2.00+0.40-0.30+0.50=-1.40$, giving $1/(1+\exp(1.40))=0.1978$, or 19.8%.

The treatment-to-control dropout odds ratio is $\exp(0.40)=1.492$ at $z=0$, but at $z=1$ it is $\exp(0.40+0.50)=\exp(0.90)=2.460$. The probabilities above are conditional model predictions for the specified visit and outcome values; they are not marginal dropout risks for the treatment groups.

Derivation and Inference for the Interaction

The interaction interpretation follows directly from the difference between the two group-specific logits. Holding visit, outcome, and covariates fixed, subtract the control logit from the treatment logit. The common terms cancel, leaving a treatment contrast that varies with $z_{it}$.

\begin{aligned} \eta_1-\eta_0 &=\{\alpha_t+\gamma+(\beta+\delta)z_{it}+\theta^\mathsf{T}X_i\} -\{\alpha_t+\beta z_{it}+\theta^\mathsf{T}X_i\}\\ &=\gamma+\delta z_{it}. \end{aligned}

Exponentiating this log-odds contrast gives $\exp(\gamma+\delta z_{it})$, the conditional treatment-to-control dropout odds ratio at that outcome value. This derivation also explains why $\delta$ describes the change in the treatment log-odds contrast for a one-unit increase in $z_{it}$.

A usual inferential question is whether the data support a different outcome slope by treatment on the logit scale. The corresponding null hypothesis is $H_0:\delta=0$, against $H_a:\delta\ne0$ for a two-sided assessment. A Wald test uses the estimate divided by its standard error, provided the large-sample approximation and the standard error calculation are appropriate. The confidence interval for $\delta$ is on the log-odds scale; exponentiating its endpoints gives an interval for the ratio $\exp(\delta)$ of outcome-associated dropout-odds multipliers.

Worked Example: Testing for Different Outcome Associations

Suppose a hypothetical analysis estimates $\hat{\delta}=0.45$ with $SE=0.18$. Assume the standard error accounts for the repeated observations as required by the fitted analysis, and use a large-sample normal approximation.

The Wald statistic for $H_0:\delta=0$ is $z=0.45/0.18=2.50$. The two-sided p-value is approximately 0.0124. At $\alpha=0.05$, this is evidence against a common outcome slope on the log-odds scale, subject to the model and inference assumptions.

A 95% Wald confidence interval for $\delta$ is $0.45\mathbin{\pm}1.96(0.18)=0.45\mathbin{\pm}0.3528$, or $(0.0972,0.8028)$. Exponentiating gives $(\exp(0.0972),\exp(0.8028))=(1.102,2.232)$. Under the model, for a one-unit increase in $z$, the treatment-to-control ratio of outcome-associated dropout-odds multipliers is estimated as 1.57, with a 95% confidence interval from 1.10 to 2.23. The interval excludes 1, matching the two-sided test at the 5% level.

Assumptions, Checks, and Alternatives

The interpretation above is conditional on a well-defined risk set and temporal ordering: $Y_{it}$ must be measured before the dropout event represented by $D_{it}$. If dropout occurs before the visit-$t$ outcome is observed, that outcome cannot be treated as an observed predictor of that same dropout decision without additional modelling. The equation also requires a suitable link, adequate covariate adjustment for the intended conditional association, and enough information across treatment groups and outcome values to estimate the interaction.

Useful checks include plotting or tabulating observed dropout frequencies across clinically meaningful outcome ranges separately by treatment group, checking that the fitted relationship is not obviously curved, and examining whether both groups have adequate outcome overlap among participants at risk. These checks can reveal sparse regions or a poor linear-on-the-logit specification; they do not prove the model correct. Implausible fitted probabilities, unstable estimates, or large changes under modest changes in outcome scaling or covariate specification warrant investigation. In particular, extrapolated interaction effects outside the observed outcome ranges may be unreliable.

Specification	What it assumes or represents	Interpretive caution
Common outcome slope, $\delta=0$	The outcome association is shared by treatment groups on the log-odds scale.	Probability-scale associations can still differ when baseline logits differ.
Pooled model with interaction	The outcome slopes differ by the single coefficient $\delta$, while other terms follow the specified shared structure.	It does not allow every model coefficient to vary by treatment.
More flexible treatment-specific equations	Can allow additional parameters, including intercept or covariate effects, to differ across groups.	Requires more parameters and data; it is a distinct, less constrained specification.

The pooled interaction is a focused alternative to assuming one common outcome slope: it adds one parameter and directly tests the specific heterogeneity of interest. More flexible treatment-specific equations can represent broader differences, but may be less precise and require stronger data support. Neither a significant nor a nonsignificant interaction settles whether the chosen outcome scale, link, or overall dropout model is adequate.

Finally, treatment assignment being randomized does not make $\delta$ a causal effect. The equation conditions on a current outcome that may itself be affected by treatment, and it describes a conditional association with subsequent dropout. The coefficient does not establish that changing the outcome would change dropout. It also does not alone describe how dropout changes the distribution of outcomes observed later.

Common Mistake

Common mistake: “The interaction is positive, so treatment raises dropout probability by the same amount for every one-unit increase in outcome.” This is incorrect. A positive $\delta$ means the treatment outcome slope is greater than the control slope on the log-odds scale. The corresponding probability changes depend on the baseline logits, and the association is not automatically causal.

A second frequent error is to interpret $\gamma$ as the treatment contrast at every outcome value. It is the treatment log-odds contrast only at the centering point $z=0$; at other values the contrast is $\gamma+\delta z$. Always state the reference outcome and the unit used to scale the outcome.

Check Your Understanding

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

In a logit dropout model, $\hat{\beta}=-0.20$ and $\hat{\delta}=0.35$. What are the outcome-associated dropout-odds multipliers per unit of $z$ in control and treatment?
If $z=2$, $\gamma=-0.10$, and $\delta=0.25$, what is the treatment-to-control conditional dropout odds ratio?
Does $\delta=0$ imply identical outcome-associated changes in dropout probability across groups? Explain.
Why must the current outcome be measured before the dropout event in this formulation?

Solutions. (1) Control: $\exp(-0.20)=0.819$. Treatment: $\exp(-0.20+0.35)=\exp(0.15)=1.162$. (2) The log-odds contrast is $-0.10+0.25(2)=0.40$, so the odds ratio is $\exp(0.40)=1.492$. (3) No. It implies a common slope on the log-odds scale; probability-scale changes can differ because the inverse-logit is nonlinear and the groups can have different baseline logits. (4) An outcome observed after dropout cannot be an observed predictor of the earlier dropout decision; using it that way would violate temporal ordering and require a different model for unavailable outcomes.

Specification	What it assumes or represents	Interpretive caution
Common outcome slope, \(\delta=0\)	The outcome association is shared by treatment groups on the log-odds scale.	Probability-scale associations can still differ when baseline logits differ.
Pooled model with interaction	The outcome slopes differ by the single coefficient \(\delta\), while other terms follow the specified shared structure.	It does not allow every model coefficient to vary by treatment.
More flexible treatment-specific equations	Can allow additional parameters, including intercept or covariate effects, to differ across groups.	Requires more parameters and data; it is a distinct, less constrained specification.