Time Coding for Biomarker Follow-Up

Why the Time Origin Matters

A biomarker trajectory is often described by its level at a reference time and its rate of change over time. The reference time may be the start of follow-up, a clinical event, or a visit chosen because it makes the intercept easier to interpret. Changing that reference does not inherently change the trajectory. It changes which point on the trajectory is called the intercept.

For example, an intercept coded at time zero means the model’s expected biomarker level at elapsed time zero. If zero is far outside the observed follow-up, that intercept may be an extrapolation. Centering time at a follow-up visit instead makes the intercept the expected level at that visit. The choice is about parameterization and interpretation, not about moving or changing the observed biomarker values.

Model and Assumptions

Consider a linear mixed model with one common linear time trend and a random intercept:

Here \(Y_{ij}\) is the biomarker for subject \(i\) at measurement \(j\); \(\beta_0\) and \(\beta_1\) are fixed coefficients; \(b_{0i}\) is a subject-specific deviation in level; and \(\epsilon_{ij}\) is the residual. For the usual interpretation, assume \(E(b_{0i})=0\), \(E(\epsilon_{ij}\mid b_{0i})=0\), and that the conditional mean is linear in the specified time. The random effects and residuals have the covariance structure specified for the model; likelihood-based normal-theory inference additionally uses the corresponding normality assumptions. These assumptions concern the model, not the choice of time origin.

Replacing elapsed time with \(t^{*}_{ij}=t_{ij}-c\) gives the equivalent expression

Thus the centered-time intercept is \(\beta_0^{*}=\beta_0+c\beta_1\), while the slope remains \(\beta_1^{*}=\beta_1\). The random intercept in this model is unchanged because the random part has no time-varying term. This identity requires the same observations, time units, linear mean form, and random-effects and residual structure in both parameterizations.

Why Fitted Values Are Preserved

The invariance follows by substitution, not by an approximation. Since \(t_{ij}=t^{*}_{ij}+c\), the fixed part of the original model satisfies

The random-intercept and residual terms are identical on both sides. Therefore, for every observed measurement, the conditional fitted value is unchanged when the coefficients are transformed as stated. The population-average fitted value is also unchanged, because \(E(b_{0i})=0\). This is an exact re-expression of the same model, provided the parameterization is transformed rather than independently constrained in a way that changes the model.

The coefficient estimates transform by the same linear rule. If \(\widehat{\beta}_0\) and \(\widehat{\beta}_1\) are the estimates under elapsed time, then \(\widehat{\beta}_0^{*}=\widehat{\beta}_0+c\widehat{\beta}_1\). Its variance is

Consequently, the uncertainty for the intercept depends on the chosen origin, even though predictions at a fixed time do not. A confidence interval for the centered intercept describes the mean biomarker at elapsed time \(c\), not at enrollment.

Centering When Individual Slopes Are Included

Some mixed models allow subject-specific slopes as well as subject-specific levels. The following algebra describes the coding issue without changing the interpretation of the fixed mean slope. Write the subject-specific linear predictor as

After setting \(t^{*}_{ij}=t_{ij}-c\), the same subject-specific predictor is

So the subject-specific intercept deviation also changes: \(b_{0i}^{*}=b_{0i}+cb_{1i}\), and \(b_{1i}^{*}=b_{1i}\). If the random intercept and slope have an unrestricted covariance matrix, the same random-effects distribution can be expressed in the new coordinates by transforming that matrix. If instead a model imposes a zero covariance between the random intercept and slope, that restriction is generally not preserved by the transformation. Refitting under that restriction can therefore change the implied covariance structure and fitted results. Centering is exactly invariant only when the random-effects structure and its constraints are transformed compatibly.

Choosing a Coding and Checking It

Use elapsed time from a clinically meaningful origin when the level at that origin is scientifically important and adequately supported by observations. Center at a well-observed, meaningful time when that makes the intercept more useful or the model easier to interpret. The slope retains its units and interpretation under a simple shift: it is still the expected change in biomarker per month, per week, or other chosen unit.

A change of unit is different from centering. If time is converted from months to years, the numerical slope changes because one unit now represents a different duration. For example, a slope of \(-0.40\) units per month corresponds to \(-4.80\) units per year. Predictions remain unchanged only if the coefficients are converted consistently.

A useful check is to calculate a prediction at the same elapsed time using both codings. If the values differ beyond rounding, check whether \(t^{*}=t-c\) was applied consistently, whether the intercept transformation used \(\beta_0+c\beta_1\), and whether a model constraint changed during refitting. Also inspect whether the chosen origin lies within or near the observed follow-up; an intercept far outside that range may be an extrapolated quantity even if the model fits observed times well.

Centering does not correct a nonlinear mean trajectory, make the observations’ schedule non-informative, or resolve an inappropriate covariance model. It only changes the coordinate used to represent time. For more complicated mean structures, each time-dependent term must be re-expressed using the same substitution; changing only the intercept is not generally sufficient.

Common Mistakes

• Calling the centered intercept the enrollment mean. If time is centered at month six, the intercept represents the modeled mean at month six, not month zero.
• Changing the intercept without transforming the slope contribution. The correct new intercept is \(\beta_0+c\beta_1\), not \(\beta_0\) alone.
• Assuming the coefficient estimates must be identical after recoding. The slope is unchanged by a shift, but the intercept and its standard error generally change.
• Assuming every constrained mixed model is invariant. A random-slope covariance constraint, such as forcing a zero intercept-slope covariance, may describe a different model after centering.
• Treating centering as a remedy for poor fit. A more convenient time origin does not fix nonlinearity, an unsuitable covariance structure, or problems with the observation process.