Translating Trajectory Choices Into Longitudinal Model Formulas

From a Trajectory to a Formula

A longitudinal formula is more than software syntax: it encodes a claim about how a biomarker changes over time and how subjects may differ in that change. A formula can fit successfully while representing a different trajectory from the one intended. The reliable approach is to write the mathematical mean first, translate each term into the software formula, and then expand the formula back into a subject-specific path.

Let \(y_{ij}\) be the biomarker measurement for subject \(i\) at visit \(j\), and let \(t_{ij}\) be time measured from a stated origin. A useful basic longitudinal model is

The formula specifies the mean trajectory \(m_i(t)\), including its fixed and random effects. In the simple Gaussian version used for illustration, random effects \(b_i\) have mean zero and covariance matrix \(D\), and measurement errors \(\epsilon_{ij}\) have mean zero and variance \(\sigma^2\), independently across measurements and independently of \(b_i\). These distributional assumptions are not guaranteed merely because a formula has been written; they are part of the model and should be assessed in context. Software syntax and supported random-effect structures also vary, so the formula must be checked against the chosen fitting tool.

For example, an intercept represents the expected biomarker level at the time origin when other predictors are at their reference values. A time coefficient represents a rate of change per unit of time only when the trajectory is linear in that time variable. In nonlinear models, the instantaneous rate depends on time.

Linear Trajectories and Formula Expansion

A linear trajectory with a group-level difference in biomarker level, but a common average time slope, can be written as

where \(G_i=0\) denotes the comparison group and \(G_i=1\) the treatment group. Here \(b_{0i}\) is subject \(i\)'s deviation in starting level, and \(b_{1i}\) is the subject's deviation in slope. If the software uses the familiar mixed-model formula syntax, the corresponding formula is y ~ time + group + (1 + time | id). The group variable should be coded or treated as a factor so that its coefficient has the intended contrast interpretation.

A different scientific claim—that groups have different average slopes—requires an interaction. The formula y ~ time * group + (1 + time | id) expands to time, group, and time-by-group fixed effects, plus the stated random effects. Its mean includes an additional term \(\beta_3tG_i\). The group contrast is then \(\beta_2+\beta_3t\), rather than a constant. The formula must match the intended contrast, not just look plausible at a glance.

Nonlinear Trajectories: Curvature and Knots

A quadratic trajectory allows curvature while keeping the model relatively simple. With centered time \(t_c=t-c\), where \(c\) is a chosen reference time, a possible model is

The corresponding formula is y ~ time_c + I(time_c^2) + (1 + time_c | id). In R-style formulas, I(...) requests arithmetic on the expression inside it. Without that protection, some formula systems interpret the caret as an operator rather than as a request to square the variable. This model allows subject-specific intercepts and linear slopes but gives all subjects the same quadratic curvature. Adding a random quadratic term is a stronger and more data-demanding choice.

A piecewise-linear trajectory is useful when a prespecified time point \(k\) marks a change in slope. Define the hinge term \(h(t)=\max(0,t-k)\). A model containing \(t\) and \(h(t)\) is continuous at \(k\): before the knot, the hinge is zero; after it, the hinge increases with time. For subject-specific changes in both the initial slope and the slope change, one possible formula is y ~ time + hinge + (1 + time + hinge | id). This can require substantial data because it estimates several random effects and their covariances.

Time-Varying Terms and Practical Checks

A biomarker trajectory is time-varying by definition: its fitted value changes with \(t\). Adding a time-varying predictor is a separate modeling choice. For example, if treatment status changes across visits, a row-specific treatment variable can enter the mean as a covariate. That does not automatically give treatment a time interaction, nor does it imply that the biomarker's slope differs by treatment. To specify a different slope by a binary group, include the corresponding interaction with time. To specify a nonlinear time pattern, include the chosen nonlinear time basis. Keep these roles distinct when constructing the data table and formula.

After fitting, inspect the formula's implied design: confirm the time variable and units, the reference levels, the included powers or basis functions, and the random effects attached to each subject. Then calculate or plot fixed and subject-specific fitted trajectories across the observed time range. Check that the number and timing of observations can support the chosen complexity, and examine residual patterns against fitted values and time. Curvature left in residuals can indicate an inadequate mean trajectory; implausible subject paths or unstable random-effect estimates can signal an overcomplicated random structure. Sparse observations near a knot, or near one end of the time range, make the shape there especially uncertain.

These checks do not establish that the model is correct. They help detect mismatches between intent and implementation and reveal some failures of fit. A linear model is easy to interpret but may miss curvature. A quadratic model is compact but can behave implausibly outside the observed time range. Piecewise linear models allow a prespecified slope change but depend on the knot choice. Spline bases can represent more flexible smooth patterns, though their basis and boundary behavior need careful checking. More flexibility can improve description while making estimates less stable, especially when subjects have few measurements.

A formula describes the longitudinal mean and the chosen random-effect structure; it does not by itself guarantee that software has fitted that exact model or that the assumptions are adequate. In a joint model, the longitudinal trajectory is one component of a larger specification. Checking the trajectory here is therefore important, but it does not replace checking the other model components or their implementation.