A Worked Biomarker Example With Visit Effects

Why put visit effects in both models?

In a repeated-biomarker trial, the average biomarker can change over scheduled visits, and the chance of dropout can change with visit as well. A selection model makes these features explicit in two related pieces: a model for the biomarker trajectory and a model for dropout conditional on the current biomarker. Keeping the visit effects visible helps distinguish a changing average biomarker from a changing dropout propensity.

The example below is hypothetical. It illustrates a model and calculations; it does not represent a real trial. Let \(A_i=1\) denote assignment to treatment and \(A_i=0\) denote control. Let \(Y_{ij}\) be participant \(i\)'s biomarker at scheduled visit \(j\), where \(j=0,1,2\) corresponds to weeks 0, 4, and 8. Let \(D_{ij}=1\) mean that the participant drops out at visit \(j\), so the scheduled current biomarker \(Y_{ij}\) is not observed because of dropout. The dropout model applies only to participants still under follow-up immediately before that visit.

Specify the trajectory and dropout equations

A useful starting point for the biomarker is a linear trajectory with categorical visit effects. In this example, the model allows a treatment difference at baseline and treatment-by-visit differences at follow-up:

Here \(I(\cdot)\) is an indicator, \(b_i\sim N(0,16)\) is a participant-specific random intercept, and \(\varepsilon_{ij}\sim N(0,9)\) are independent errors, also independent of \(b_i\) and treatment. Thus the random intercept has variance 16 and the visit-specific error variance is 9. The marginal mean trajectory is obtained by averaging over \(b_i\) and \(\varepsilon_{ij}\), both of which have mean zero. The random intercept represents stable between-participant variation; it does not change the fixed-effect mean trajectory.

For the dropout part, use a logistic probability model at the two follow-up visits:

The visit-4 effect is the reference level for dropout, and \(Y_{ij}\) is centered at 100 biomarker units. The equation describes the dropout probability for a participant who has not already dropped out, conditional on their current scheduled biomarker and treatment. It is a sequential dropout probability, not the probability of ever dropping out by week 8.

These equations rely on substantive and statistical assumptions. The trajectory mean and its random-effects distribution must adequately describe biomarker variation; the logit-linear form must adequately describe conditional dropout odds; and the scheduled visit and treatment coding must match the data. The sequential model also assumes dropout is adequately represented by the included current outcome, treatment, and visit terms, conditional on being at risk. In a real analysis, additional pre-specified covariates or history terms may be needed. The current-outcome association is an assumption about the joint model, not something directly observed for participants whose current biomarker is missing.

Calculate conditional dropout risks

To turn a linear predictor into a probability, apply the inverse-logit function. If the model's linear predictor for a specified profile is \(\eta\), the conditional dropout probability is \(e^\eta/(1+e^\eta)\). This probability is conditional on the stated current biomarker and other model covariates, as well as being at risk for dropout at that visit.

What the calculations establish—and what they do not

The inverse-logit calculation follows directly from the definition of the logit. If \(p\) is a probability and \(\eta=\operatorname{logit}(p)=\log\{p/(1-p)\}\), exponentiating gives \(e^\eta=p/(1-p)\). Solving for \(p\) yields \(p=e^\eta/(1+e^\eta)\). A one-unit increase in \(Y_{ij}\) changes \(\eta\) by \(\gamma=0.04\); exponentiating the change shows that the odds are multiplied by \(e^{0.04}\). For a 10-unit increase, the change is \(10(0.04)=0.4\), so the odds ratio is \(e^{0.4}\approx1.49\).

The risks calculated above are conditional on a specified current biomarker. A marginal dropout rate at a visit would require averaging the conditional probabilities over the relevant distribution of current outcomes and covariates among participants still at risk. In this model, that distribution must be combined with the trajectory model. Substituting a single trajectory mean into the logistic equation generally does not give the marginal dropout rate, because the inverse-logit is nonlinear.

The current biomarker for a dropout event may be unobserved. Consequently, observed dropout frequencies by treatment and visit can help assess whether the model reproduces broad response patterns, but they cannot by themselves verify how dropout depends on the missing current biomarker. That dependence is a feature of the specified selection model and needs substantive justification and sensitivity assessment. Also inspect whether the outcome model describes the observed repeated measurements adequately and whether the fitted model has estimation or convergence problems; a plausible calculation for selected profiles does not establish that the full model is adequate.

How this model differs from alternatives

A response model that omits the current outcome makes dropout depend only on the other included variables, such as treatment and visit. That is a simpler specification, but it does not express the current-outcome dependence in this example. A model based on previously observed biomarker history instead conditions dropout on earlier measurements; it answers a different conditional modelling question from one that includes the scheduled current outcome.

A pattern-mixture model organizes the outcome distribution by response or dropout pattern rather than expressing response probabilities conditional on the outcome. The selection formulation used here instead pairs an outcome trajectory with a conditional dropout equation. These approaches make different assumptions about the unobserved data and need not produce the same results. No single model form is guaranteed by the observed measurements alone.

Common pitfalls and limitations

• Confusing conditional and marginal risks. The 14.8% result is conditional on a specified week-8 biomarker, treatment, and being at risk. It is not the overall proportion expected to drop out at week 8.
• Using the wrong visit indicator. Week 4 is the reference for the dropout visit effect in this equation, so its indicator is zero at week 4 and one at week 8. The outcome visit terms have their own specified coding.
• Reading a logit coefficient as a probability difference. The coefficient 0.35 is a change in log odds. Its exponent, about 1.42, is an odds ratio, not a 42-percentage-point increase in risk.
• Treating a hypothetical profile as observed data. A calculation at a specified latent current value illustrates the model implication. It does not mean that the value was measured for a participant who dropped out before measurement.
• Over-interpreting the model. The equations describe an assumed joint structure for biomarker and dropout. They do not, by themselves, show that changing the biomarker would change dropout or that the model's unobserved-outcome assumptions are correct.

The model is also limited by its chosen functional forms. A linear mean trajectory may not capture a sharply curved biomarker course, and a linear current-outcome term on the log-odds scale may not describe the dropout relationship well. With only observed current biomarker values among retained participants, fit to the observed outcomes cannot settle the unobserved current-outcome dependence. These limitations matter when interpreting any selected-profile risk.