Time Trends in Repeated-Measure Means

Why the Time Scale Matters

Repeated measurements are often taken at different intervals: a baseline assessment, a follow-up after two months, and another a year later. A model that uses visit numbers \(1,2,3\) treats each step as the same change in time, even though the elapsed intervals differ. That may be appropriate if the scientific question concerns differences between scheduled visits, but it is not automatically appropriate for a question about change per month or year.

A time effect is part of the marginal linear predictor: it describes how the modeled population-average mean changes with time, for specified values of other covariates. The key decisions are whether time should be represented by actual elapsed time, by visit categories, or by a flexible function of elapsed time, and whether the interpretation is on the link scale or the response scale.

The measurements within a subject may be correlated; the time-effect specification describes the mean pattern, not that dependence. For interpretation, hold any other covariates fixed at stated values. If the time pattern itself differs across covariate values, the model needs a corresponding interaction; here the focus is the basic time pattern.

Three Ways to Represent Time

With continuous linear time, the predictor contains an elapsed-time term such as \(\beta_1t_{ij}\). This assumes a constant change per unit time on the link scale. With categorical time, each scheduled occasion has its own mean, without requiring equal intervals or a straight-line trend. With a nonlinear time function, elapsed time enters through a curve, such as a polynomial or a chosen set of basis functions.

These are different assumptions, not merely different coding conventions. A linear effect uses the distances between measurement times; categorical time distinguishes occasions but does not use their elapsed-time distances to constrain the means. A nonlinear function uses elapsed time while allowing a specified degree of curvature or flexibility.

Proof. At the reference occasion all nonreference indicators equal zero, so the model mean is \(\alpha\). Setting \(\alpha=\mu_1\) gives the desired reference mean. At occasion \(k\), its indicator equals one and all other occasion indicators equal zero, so its model mean is \(\alpha+\delta_k\). Setting \(\delta_k=\mu_k-\mu_1\) yields \(\alpha+\delta_k=\mu_1+(\mu_k-\mu_1)=\mu_k\). Thus every specified vector of occasion means is represented. Conversely, the model has one coefficient for the reference mean and one difference for each other occasion, so it does not impose an equality or linear-trend constraint among those means. \(\square\)

Linear Time and the Meaning of a Slope

For a continuous linear time effect, write \(\eta(t)=a+bt\), holding other covariates fixed. The parameter \(b\) is the change in the linear predictor for a one-unit increase in elapsed time. The unit matters: if time is measured in years, \(b\) is a per-year link-scale change. For an identity link, the response mean itself changes linearly by \(b\) per unit time. For a nonlinear inverse link, the response-scale change generally depends on the starting time and the interval length.

Derivation. Subtract the two linear predictors:

Applying \(h\) gives the two response means, whose difference is \(h(a+bt_2)-h(a+bt_1)\). Dividing by the elapsed interval gives the average response-scale rate. If \(h\) is differentiable, the instantaneous response-scale rate follows from the chain rule:

This derivative generally varies with \(t\), even though the link-scale slope \(b\) is constant. Thus, a linear time effect on the link scale should not be described as a constant response-scale change unless the link is the identity.

Nonlinear Time Effects

A nonlinear time effect allows the rate of change to vary with elapsed time. One transparent option is a quadratic predictor, \(\eta(t)=\beta_0+\beta_1t+\beta_2t^2\). A nonzero \(\beta_2\) represents curvature on the link scale. More flexible specifications can use prespecified basis functions, for example \(\eta(t)=\beta_0+\sum_{r=1}^{R}\gamma_r B_r(t)\), where each \(B_r(t)\) is a known function of elapsed time. The coefficients describe the combined curve; in general, an individual basis coefficient is not itself a simple per-year mean change.

Common Mistakes and Practical Choices

Choose a time representation to match the scientific question and measurement design. A categorical effect is useful when each scheduled visit is substantively distinct or when no simple time curve is justified. Continuous linear time is compact and directly interpretable when a constant link-scale rate is reasonable. A nonlinear function can capture curvature while retaining the information in elapsed-time distances.

• Using visit number as elapsed time. Coding visits as 1, 2, and 3 imposes equal predictor steps. If the actual times are 0, 2, and 5 years, use those elapsed times for a rate-per-year interpretation.
• Calling a link-scale coefficient a response-scale rate. With a nonlinear link, the response mean is transformed. Report fitted response means or an interval-specific response-scale change when that is the quantity of interest.
• Reporting a change without its interval. A difference of 2 units over one year is not the same rate as 2 units over four years. State the time endpoints and divide by their actual separation if giving an average rate.
• Assuming categorical time interpolates. Visit-specific fitted means do not specify a curve between visits. Do not infer an intermediate-time mean without an explicit model for it.
• Ignoring the time origin and unit. A slope per month differs numerically from a slope per year. State the origin and units so coefficients and predictions are reproducible.