Centering and Scaling Current-Value Associations

Why Re-express a Current-Value Marker?

A current-value association is often easier to interpret when the marker is centered at a meaningful value or scaled to a familiar increment. For example, an association per one standard deviation may be more interpretable than an association per one unit when one unit is very small. Centering can also make the hazard intercept refer to a subject whose marker is at a clinically relevant value. These changes of reference and units alter numerical coefficients, but—when performed consistently—they do not alter the fitted hazard contrasts.

Let \(m_i(t)\) denote subject \(i\)'s latent marker value at time \(t\), and consider a shared-random-effects model with a current-value term:

Here \(h_i(t)\) is the subject-specific hazard conditional on the latent trajectory and shared random effects, \(h_0(t)\) is the baseline hazard, \(w_i\) is a vector of other covariates, and \(\alpha\) is the current-value association coefficient. The coefficient is on the log-hazard scale per one marker unit. The model links the latent marker value, not a noisy observed measurement at a particular visit, to the hazard.

Coefficient and Intercept Transformation

Since \(m_i(t)=c+s m_i^*(t)\), substitution into the hazard model gives

The transformed association coefficient is therefore \(\alpha^*=s\alpha\). The constant term \(\alpha c\) can be absorbed into the baseline hazard by defining \(h_0^*(t)=h_0(t)\exp(\alpha c)\). Equivalently, if the log-hazard has a constant intercept \(\gamma_0\), the transformed intercept is \(\gamma_0^*=\gamma_0+\alpha c\). The transformed model is then

This is an algebraic reparameterization: the transformed marker is a one-to-one affine function of the original marker when \(s>0\), and the parameter changes compensate exactly for that re-expression. The other covariate coefficients remain the same. If the baseline hazard is left in its original form rather than absorbing the constant, the factor \(\exp(\alpha c)\) must remain explicitly in the linear predictor.

Why Hazard Contrasts Are Preserved

At a fixed time, compare two subjects with the same values of \(w_i\), but latent marker values \(m_1(t)\) and \(m_2(t)\). In the original parameterization, their hazard ratio is

The shared baseline hazard and the constant centering shift cancel in this comparison. In the transformed parameterization, \(m_1^*(t)-m_2^*(t)=(m_1(t)-m_2(t))/s\), so the contrast is

Thus, \(\exp(\alpha)\) is the hazard ratio for a one-unit marker increase in the original scale, while \(\exp(\alpha^*)=\exp(s\alpha)\) is the hazard ratio for an increase of \(s\) original units. These are different increments, not conflicting estimates of the same increment.

Centering Changes the Reference, Not the Slope

With centering alone, \(s=1\). The coefficient remains \(\alpha\), because a one-unit difference in the centered marker is still a one-unit difference in the original marker. The intercept or baseline hazard changes so that its reference now corresponds to the chosen center. This can improve interpretation: the hazard intercept is associated with marker value \(c\), rather than marker value zero. It does not make the association stronger or weaker.

This change of reference is distinct from a baseline-referenced current-value association. In that model, the hazard association involves a difference between the current latent value and a baseline latent value. Merely centering the marker subtracts a fixed constant from every value; it does not subtract each subject’s own baseline value.

Changing Units and Reporting Uncertainty

A marker can also be rescaled simply by changing its units. For example, if a new numerical marker value is one hundredth of the old value, then one new unit represents one hundred old units. The coefficient must adjust inversely to the size of the new unit. More generally, if \(x^*=a+bx\), where \(b>0\), the coefficient on \(x^*\) is \(\alpha/b\). This is the same rule as above with \(a=-c/s\) and \(b=1/s\).

For a fixed positive \(s\), uncertainty transforms along with the coefficient: \(SE(\alpha^*)=sSE(\alpha)\), and a confidence interval with endpoints \(L\) and \(U\) for \(\alpha\) becomes \((sL,sU)\) for \(\alpha^*\). A test of \(H_0:\alpha=0\) has the same test statistic and p-value after this positive rescaling, since both estimate and standard error are multiplied by \(s\). If the intercept is also transformed, its uncertainty must account for its covariance with \(\alpha\): for fixed \(c\), \(\gamma_0^*=\gamma_0+c\alpha\), so \(\operatorname{Var}(\gamma_0^*)=\operatorname{Var}(\gamma_0)+c^2\operatorname{Var}(\alpha)+2c\operatorname{Cov}(\gamma_0,\alpha)\).

Conditions, Diagnostics, and Limitations

The exact invariance depends on a consistent affine transformation of the latent marker and its longitudinal model. In particular, the same \(c\) and \(s\) must be used at every time and for every subject; the transformed trajectory must represent \((m_i(t)-c)/s\); and the hazard model must include the corresponding transformed coefficient and intercept or baseline-hazard factor. Under these conditions, each subject’s linear predictor, and therefore fitted hazard, is unchanged.

If \(c\) and \(s\) are calculated from the observed data, the algebraic identities still hold for the resulting values. But when reporting inference, make clear whether they are treated as fixed reference constants or whether their estimation uncertainty is part of the inferential target. Reparameterizing a likelihood does not by itself ensure that every estimation procedure is invariant: constraints, priors, or penalties that are not transformed to match the new parameterization can lead to different estimates.

Useful checks include verifying that the transformed marker equals \((m_i(t)-c)/s\) in the longitudinal model, that the transformed coefficient equals \(s\alpha\), and that the transformed intercept includes \(\alpha c\). For selected marker values, calculate the linear predictor both before and after transformation. A mismatch signals an omitted intercept shift, an inconsistent unit conversion, or a different model rather than a harmless change of scale.

The main alternatives are to retain the original scale, center only, or scale to a convenient increment such as one standard deviation. Retaining the original scale can make units directly interpretable; centering can make the intercept meaningful; scaling can make coefficient magnitudes easier to compare across marker units. None is universally preferable. The choice changes presentation, not the underlying hazard contrasts, when the transformation is handled correctly. It does not correct a misspecified trajectory or establish that the marker causes the event.

Common Mistake

A common error is to multiply the coefficient by \(s\) and then report \(\exp(s\alpha)\) as the hazard ratio for a one-unit increase in the original marker. That interpretation is wrong: the transformed coefficient is per one unit of \(m_i^*\), which equals \(s\) original units. The original one-unit hazard ratio remains \(\exp(\alpha)\); the transformed one-unit hazard ratio is \(\exp(s\alpha)\). Another error is to center the marker but leave the intercept unchanged. That omission shifts every fitted log hazard by \(\alpha c\), so the resulting model is no longer equivalent.