Nonlinear Outcome Trajectories Under Selection

Why Combine a Nonlinear Trajectory With Selection?

Repeated outcomes need not change at a constant rate. A biomarker might rise quickly and then level off; a symptom score might decline and later rebound. At the same time, the chance that a participant drops out may depend on the outcome at the current visit. A useful model must therefore represent both the shape of the outcome trajectory over time and the relationship between the current outcome and dropout.

These are separate modeling choices. A quadratic or spline term allows the expected outcome to vary nonlinearly with visit time. A current-outcome term in the dropout equation allows dropout risk to vary with the outcome at that visit. Neither choice implies the other.

The model below treats the current outcome as preceding the dropout decision at that visit. If dropout occurs before \(Y_{ij}\) is recorded, \(Y_{ij}\) is still a latent outcome in the model, not an observed value. This timing convention is essential: the selection equation can depend on an outcome that is missing precisely because dropout occurred.

Modeling the Outcome Trajectory and Dropout Hazard

A general mean trajectory can be written using a vector of time-basis functions \(B(t)\). For example, a quadratic trajectory uses \(B(t)=(1,t,t^2)^\mathsf{T}\), while a spline uses basis functions joined at selected knots. A simple outcome model is

where \(X_i\) denotes baseline covariates. This equation specifies the conditional mean, not the full outcome distribution. For example, a Gaussian model additionally specifies a covariance structure for repeated measurements. The mean model and covariance assumptions should be chosen to reflect the outcome and the study design.

One common selection model uses a logistic dropout hazard:

Here \(C(t)\) can include visit-time terms, \(c\) is a centering value, and \(s>0\) is a scale chosen for the outcome. The precise history being conditioned on must be stated as part of the model. In this version it includes the current outcome, previous observed outcomes as specified, and covariates; the model does not add future outcomes to the visit-\(j\) hazard. The term involving \(\gamma_y\) is a linear effect of the scaled current outcome on the log odds of dropout, conditional on the other terms.

The outcome trajectory can be nonlinear in time while the hazard remains linear in the scaled outcome on the logit scale. A quadratic term in the outcome mean does not mean that the hazard has a quadratic outcome effect. If such an effect is scientifically justified, it must be included separately and assessed as an additional model choice.

Assumptions and Sequential Structure

A model with a current-outcome dropout hazard requires assumptions about both the outcome process and the selection process. In particular, it assumes a correctly specified outcome distribution (including its time-varying mean and dependence across visits) and a correctly specified hazard link and covariate structure. It also assumes that dropout is monotone as defined above, unless the model is extended to allow intermittent missingness.

The hazard is defined among individuals still observed at the start of a visit. A sequential model can generate the complete outcome trajectory from its outcome distribution and then generate each visit's dropout decision, in order, using the current outcome and the stated history. The resulting joint model is built from the sequential outcome and dropout factors. This does not justify replacing those factors with a product of hazards while conditioning on an entire future outcome history: future outcomes are not part of the visit-specific conditioning set unless the model explicitly says so. Being precise about conditioning avoids confusing sequential survival probabilities with probabilities conditional on a full trajectory.

Because \(Y_{ij}\) may be missing when \(R_{ij}=0\), fitting the joint model requires accounting for that unobserved current outcome under the specified outcome distribution. A dropout hazard cannot be calculated from an unobserved value as though it were recorded. The model supplies a probability distribution for the missing outcome; estimation must account for uncertainty about it. The details of likelihood-based estimation and numerical integration depend on the chosen outcome model.

Identification also matters. The observed data alone generally do not determine how dropout would depend on outcomes that are unobserved after dropout. Conclusions about \(\gamma_y\) and the unobserved trajectory therefore rely on the joint-model assumptions and may be sensitive to them. A selection coefficient should not be presented as a directly observed empirical quantity in the same way as a complete-case regression coefficient.

Worked Model Calculations

Why the Basis and Link Choices Matter

For a quadratic mean, the derivative with respect to time is \(\beta_1+2\beta_2t\), so the rate of change varies with time. For a spline mean, the basis functions allow the rate of change to differ across time regions. These forms can represent curvature more flexibly than a single straight line, but their interpretation depends on the time scale, centering, and chosen knots.

The dropout link is a separate choice. With the logistic link, \(\gamma_y\) has a conditional log-odds interpretation. A complementary log-log link instead models a transformed hazard and has a different coefficient interpretation. Do not transfer the logistic odds-ratio interpretation to a different link. Whatever the link, state the outcome scale: changing from raw units to five-unit increments rescales the coefficient. If \(z=(Y-c)/s\), then the coefficient is per \(s\) outcome units, not per one raw unit.

Diagnostics, Alternatives, and Limitations

A flexible time trend can fit observed outcomes better than a straight line while still being unsuitable. Plot observed outcomes against time by relevant groups and compare them with fitted means; inspect residual patterns and the plausibility of extrapolated curves. For splines, check whether the number and placement of knots create implausible bends, especially near the boundaries where data are sparse. For the selection equation, examine whether fitted dropout probabilities are plausible over the observed outcome range and whether a few extreme outcomes dominate the fitted relationship. These checks assess the model's compatibility with observed information; they cannot verify the assumed dependence on outcomes missing after dropout.

Common warning signs include unstable curvature when a quadratic term is added, large changes in the selection coefficient after modest rescaling, fitted probabilities near zero or one across much of the relevant range, and poor overlap in observed outcomes between people who do and do not drop out. Such patterns may reflect sparse data, an over-flexible basis, an unsuitable hazard form, or weak information about the selection mechanism. They call for model revision and sensitivity analysis, not automatic confidence in a more complicated fit.

A linear outcome model with a simpler dropout equation is easier to interpret but can miss genuine curvature. Pattern-mixture models parameterize outcome distributions by missingness pattern rather than placing outcome dependence directly in a selection equation. Models assuming missing at random are another alternative, but they do not represent residual dependence of dropout on the current unobserved outcome in the same way. These approaches answer related questions under different assumptions; one is not a purely data-driven correction for another.

The principal limitation is that the current-outcome selection relationship after dropout is not directly observed. Results can depend on the assumed outcome distribution, trajectory basis, link, and selection coefficient. A nonlinear time basis cannot remove that uncertainty. Report the time scale, basis and knots, outcome scaling, hazard conditioning set, and sensitivity of conclusions to plausible alternative assumptions.

Common Mistake

A related error is to calculate a dropout hazard using a missing current outcome as if its value had been observed. The model defines a hazard conditional on that latent value; when it is missing, its uncertainty must be handled through the joint model rather than by silently substituting a single value.