Time Centering in Slope-Based Association Links

Why the Time Origin Matters

A longitudinal trajectory’s intercept is its expected value at time zero. If time zero is changed—for example, from enrollment to a later clinical landmark—the intercept changes its meaning and its numerical value. The slope of a linear trajectory, however, describes change per unit time and does not depend on where time zero is placed. This distinction matters in a joint model when the event hazard is linked to the longitudinal slope: a changed intercept parameterization need not imply a changed hazard association.

The key is to distinguish changing the coordinate used to describe a trajectory from changing the trajectory itself. Centering is only a re-expression of the same model if the fixed effects, random effects, and their covariance are all transformed consistently. A comparison that changes the time origin but leaves some of those parameters fixed may instead compare different models.

Model and Time-Centering Transformation

Consider a linear mixed model for subject \(i\)'s longitudinal measurements \(Y_{ij}\) at physical times \(t_{ij}\):

Here \(\beta_0\) and \(\beta_1\) are the population intercept and slope, \(b_{0i}\) and \(b_{1i}\) are subject-specific random intercept and slope, and \(\varepsilon_{ij}\) is measurement error. Assume the random-effects vector \(b_i=(b_{0i},b_{1i})^\mathsf{T}\) has mean zero and covariance matrix \(G\), and that the errors have mean zero and are independent of the random effects. These assumptions define the part of the longitudinal model needed for the reparameterization below.

Let \(c\) be a fixed time, measured in the same units as \(t\), and define the centered time \(\tau=t-c\). The new time origin is therefore physical time \(t=c\). The same trajectory can be written

The transformation also changes the random-effects covariance. In matrix form, \(b_i^*=A b_i\), where

If \(G\) has entries \(G_{00}=\operatorname{Var}(b_{0i})\), \(G_{01}=\operatorname{Cov}(b_{0i},b_{1i})\), and \(G_{11}=\operatorname{Var}(b_{1i})\), then

The random-intercept variance and its covariance with the random slope can therefore change, even though the population slope, random slope, and individual trajectories do not. Since \(A\) is invertible, this is a one-to-one change of coordinates, not a restriction on the possible trajectories.

Why a Linear Slope Link Is Invariant

Suppose the event hazard, conditional on the random effects, is modeled as

Here \(\lambda_0(t)\) is the baseline hazard, \(w_i\) is a vector of event-model covariates, \(\gamma\) is its coefficient vector, and \(\alpha_s\) is the slope-association parameter. The model assumes that the event hazard depends on the current derivative of the latent linear trajectory through this term. It also assumes that the longitudinal and event processes are conditionally linked through the shared random effects; the same physical time \(t\) is used in both processes.

Because \(\beta_1^*=\beta_1\) and \(b_{1i}^*=b_{1i}\), the derivative under centered time is

Thus, at any given physical time, a slope-only association term is unchanged by a shift of the time origin. If the baseline hazard, covariates, and association parameter are also kept on the same physical scale, the conditional hazard is unchanged. This is a parameterization invariance, not evidence that all ways of changing the time scale leave a model unchanged.

When Other Link Parameters Change

A slope-only link is especially simple because the slope coordinate is unchanged by linear centering. A link that directly uses random-effect coordinates must be transformed with them. For example, suppose a contribution to the log hazard is \(\alpha_0 b_{0i}+\alpha_1 b_{1i}\). Since \(b_{0i}=b_{0i}^*-cb_{1i}^*\) and \(b_{1i}=b_{1i}^*\), the identical contribution in centered coordinates is

The numerical coefficient on the centered random slope is therefore not generally the same as the coefficient on the original random slope in this joint level-and-slope expression. The two expressions describe the same association only when the coefficients are transformed consistently.

Justification, Diagnostics, and Limitations

The invariance can be seen directly from the algebra: the centered trajectory equals the original trajectory at every physical time, and its derivative with respect to centered time equals the original derivative with respect to physical time. Consequently, the longitudinal fitted values and the slope term in the conditional hazard are identical for each subject. Because the transformation \(b_i^*=A b_i\) is invertible and \(G^*=AGA^\mathsf{T}\), the random-effects distribution is also the same distribution written in new coordinates. A fully reparameterized model therefore represents the same joint model.

In practice, centering can make the intercept scientifically interpretable at a meaningful landmark and may improve numerical behavior when measurements are clustered far from the original zero. It does not guarantee improved estimation: the transformed intercept and slope can remain strongly correlated, and sparse data may still provide weak information about subject-specific slopes.

When two centered and uncentered fits appear to give different slope-link conclusions, check whether the comparison has actually preserved the model. Useful checks include verifying the time conversion \(t=\tau+c\), transforming the fixed and random intercepts, using \(G^*=AGA^\mathsf{T}\), and evaluating both fitted trajectories at the same physical times. Also check that the hazard baseline is indexed by the same physical event time, that \(\alpha_s\) uses the same biomarker and time units, and that the link is based on the derivative rather than a coordinate-specific intercept or slope-like coefficient.

For a polynomial trajectory, the derivative at a given physical time is still invariant if the polynomial is transformed exactly and evaluated at the corresponding centered time. But the coefficient attached to a polynomial basis term need not itself equal the instantaneous slope. A slope link must use the actual derivative intended by the model. More generally, centering does not protect against a misspecified trajectory, an incorrectly transformed covariance, or a change in the time scale.

The main alternative is to retain the original time origin. This leaves the original intercept interpretation intact but may place time zero outside the observed range. Centering at a clinically meaningful landmark changes the intercept interpretation and covariance representation, while preserving the linear trajectory and correctly transformed slope link. Neither choice is universally superior; the essential requirement is to keep the parameterization and interpretation consistent.

Common Pitfall