Choosing Latent Effects for Both Processes

Why the Choice of Shared Effects Matters

A shared-parameter model represents subject-level heterogeneity in an outcome process and a dropout process using latent effects that enter both parts of the model. Choosing those effects is substantive: a shared random intercept links dropout to a subject’s persistent outcome level, whereas a shared random slope can link dropout to how that subject’s outcome changes over time. A model with both can represent both kinds of heterogeneity, at the cost of additional assumptions and parameters.

The choice should follow the scientific question and the time pattern the model needs to represent. It is not a choice that can be settled just by asking which specification gives the best fit to observed data. In particular, dropout can hide outcome values that would help distinguish subject-level patterns. Model fit to the observed records does not, by itself, validate assumptions about outcomes after dropout.

The earlier material on shared random effects as links between outcome and dropout, and on joint factorization for outcomes and dropout times, provides the setting. Here the focus is which latent effects to share, what their contributions mean, and what can go wrong when the structure is poorly chosen.

Basic Model Components and Units

Let \(Y_i(t)\) denote a continuous outcome for subject \(i\) at time \(t\). As a schematic outcome model, write

Here \(m_i(t)\) is the population-level mean structure, \(b_{0i}\) is a subject-specific random intercept, \(b_{1i}\) is a subject-specific random slope, \(s(t)\) is a prespecified time function, and \(\varepsilon_i(t)\) is residual variation. If \(Y_i(t)\) is measured in outcome points and time in years, then \(b_{0i}\) is in points and \(b_{1i}\) is in points per year when \(s(t)=t\). The random effects describe deviations from the population-level trajectory; they are not the residual errors at individual visits.

For illustration, suppose dropout is represented by a discrete-time hazard among subjects still under observation. A schematic model is

In this expression, \(D_i(t)=1\) denotes dropout during the interval at \(t\), \(\alpha(t)\) is a time-specific baseline term, and \(\boldsymbol{x}_i(t)\) contains observed predictors. The random-effect terms are the shared contributions. The logit, and therefore each term added to it, is dimensionless. With outcome measured in points and \(s(t)=t\) in years, \(\lambda_0\) has units of log-odds per point. The product \(b_{1i}s(t)\) is in points, so \(\lambda_1\) also has units of log-odds per point.

The equations are a template, not a claim that every outcome is continuous or every dropout process is discrete-time. For another outcome type or dropout formulation, the relevant linear predictor changes. The selection question remains: which latent features of a subject’s outcome trajectory should also contribute to their dropout process?

What Each Structure Represents

A random intercept alone is often a reasonable starting structure when persistent level differences are central and there is no strong reason to represent individual trajectory differences. A slope becomes relevant when change over time is substantively important or there is a plausible reason dropout depends on that change. Sharing both effects can express both relationships, but does not guarantee that both are estimable or that the model is correct.

Time coding matters. With \(s(t)=t\), \(b_{0i}\) describes a deviation at time zero, which may be far from the observed period. Centering time at a meaningful reference time changes the interpretation of the intercept. If \(s(t)=t-t_0\), then the intercept describes the subject’s deviation at \(t_0\). This can make the parameters easier to interpret and reduce dependence between intercept and slope, but it does not remove the need to specify their joint distribution. The interpretation of each shared effect and its loading must be updated to match the chosen time function.

Why Multiple Effects Can Change the Link

When both level and slope matter, write the latent vector as \(\boldsymbol{b}_i=(b_{0i},b_{1i})^\mathsf{T}\) and specify its joint distribution. For example, a bivariate normal specification might be used:

This is a modeling assumption, not a requirement inherent in the definition of shared effects. The off-diagonal covariance in \(\boldsymbol{\Sigma}\) allows subjects’ deviations in level and slope to be related. If the two effects are correlated, a dropout association attributed to one can be difficult to distinguish from an association with the other, especially with limited repeated measurements.

A short covariance calculation shows why sharing more than one effect changes the modeled relationship over time. Let the outcome’s random part be \(R_i(t)=b_{0i}+b_{1i}s(t)\) and the dropout predictor’s random part be \(L_i(t)=\lambda_0b_{0i}+\lambda_1b_{1i}s(t)\). Assuming the random effects have finite second moments and mean zero, linearity and bilinearity of covariance give

Thus, with both effects shared, the modeled association between the random outcome component and dropout predictor can vary with time. This calculation concerns their latent linear components; it is not itself a probability of dropout or a causal effect. A random-intercept-only specification removes the slope-related terms, while a random-slope-only specification removes the intercept-related terms.

Choosing and Checking a Structure

A defensible choice starts with the outcome trajectory and the anticipated dropout mechanism, not with a search over every possible random-effects specification. Use a structure that is scientifically interpretable, compatible with the observation schedule, and estimable from the available information. In particular, assess whether there are enough repeated outcome measurements and dropout events to distinguish level and slope variation and their relationships with dropout.

Patterns in observed data can motivate a structure. For example, subjects with different early outcome levels might have different observed dropout rates, suggesting that an intercept link is worth considering. But this observed pattern does not establish the latent relationship: dropout changes which later outcomes are observed, and observed associations can reflect covariates or model misspecification. Likewise, an apparent need for random slopes should be assessed against visit timing and repeated-measurement information, not inferred from a few individual trajectories.

Warning signs include a slope variance estimated near zero, an intercept-slope correlation near its allowable boundary, very large uncertainty in loadings, unstable estimates under small specification changes, and convergence problems. These can indicate weak information, a poorly scaled time variable, or an overcomplicated structure; they are not automatic proof that the structure is false. Centering and rescaling time can improve numerical behavior and interpretation, but cannot create information absent from the data.

Compared with a shared intercept, a shared slope can represent a time-varying link to trajectory differences but requires enough repeated information to characterize those differences. Sharing both permits a richer relationship but adds distributional and association parameters. A model with no shared latent effects is another possible specification, but it does not encode this particular latent link between outcome heterogeneity and dropout. These alternatives express different assumptions; fit criteria alone cannot choose which missing-data assumption is scientifically justified.

Common Mistake and Limitations

Wrong reasoning: “The model with both a random intercept and slope fits best, so dropout must depend on both level and trajectory.” A fit comparison only assesses relative fit under the models and data considered. It cannot demonstrate how unobserved outcomes after dropout relate to dropout, nor prove that a chosen latent-effect distribution is correct.

Correction: Treat the shared-effect structure as a substantive modeling assumption. Justify it using the outcome scale, timing, plausible dropout mechanism, and available information. Compare reasonable alternatives and report how conclusions depend on those choices. A shared random effect is a representation of latent association within the model, not evidence of a causal effect of outcome level or change on dropout.

Finally, the structures above are simplified templates. A linear random slope may not represent nonlinear trajectories; the same random effects may not be suitable for every outcome type or dropout-time scale; and a normal random-effects distribution may be restrictive. More effects can improve representation only when their meaning is clear and the data support their estimation. No choice of latent effects removes the need to state the assumptions that connect observed records to outcomes that are not observed after dropout.