Scaling the Marker Before Linking It to Hazard

Why Change the Marker’s Scale?

A marker may be measured in units that are inconvenient for computation or interpretation. A coefficient per one unit can be very small when the marker’s values are in the hundreds, while a coefficient per one standard deviation may be easier to compare or communicate. Centering can also give the marker’s zero value a useful reference meaning. These changes are re-expressions of a model, not changes to the underlying subject-specific marker trajectories or their fitted hazards—provided all affected parameters are transformed consistently.

The key distinction is that centering alone does not change the coefficient of a linear marker term in the log hazard. It changes the reference point and therefore the baseline-hazard parameterization. Rescaling changes the numerical coefficient because one unit on the new scale represents a different marker difference.

Model and Affine Re-expression

Let \(m_i(t)\) be subject \(i\)’s latent marker value at time \(t\), and let \(h_i(t\mid b_i)\) be that subject’s hazard conditional on the shared random effects \(b_i\). Consider the conditional hazard model

where \(h_0(t)\) is the baseline hazard, \(w_i\) is a vector of baseline covariates, \(\gamma\) is its coefficient vector, and \(\alpha\) is the marker-association coefficient. This tutorial concerns a linear marker contribution \(\alpha m_i(t)\) to the log hazard. The longitudinal submodel supplies \(m_i(t)\); the hazard uses that latent value, not a noisy observed measurement substituted for it.

Substitute \(m_i(t)=c+s\,m_i^*(t)\) into the hazard model:

Thus an exactly equivalent parameterization is

The transformed baseline hazard is still a baseline hazard: it is multiplied by a positive constant that does not depend on subject or time. The transformed and original conditional hazards are therefore equal at every time for every subject and random-effects value. This equivalence assumes the same fixed \(c\) and \(s\) for all subjects and times, the linear marker term shown above, and a baseline-hazard parameterization that can accommodate the constant multiplier. With a fixed finite-dimensional baseline-hazard parameterization, the multiplier must also be representable in that parameterization.

What Centering and Rescaling Do

If the marker is only centered, \(s=1\). Then \(\alpha^*=\alpha\): the coefficient per marker unit is unchanged. The baseline hazard becomes \(h_0^*(t)=h_0(t)\exp(\alpha c)\), so the zero point of the transformed marker corresponds to the original marker value \(c\). If the marker is also rescaled, the coefficient becomes \(s\alpha\). Its units change accordingly: if the original coefficient is per milligram per liter and \(s=5\) milligrams per liter, then \(\alpha^*\) is the log-hazard change per five milligrams per liter.

The marker’s longitudinal model must also be handled coherently. If the marker is merely re-expressed inside the hazard, the longitudinal model can remain on its original scale and the identity \(m_i(t)=c+s m_i^*(t)\) is used in the hazard. If the observed marker and its longitudinal submodel are instead fitted on the transformed scale, all marker measurements and the longitudinal mean and error components must be transformed too. For example, a measurement error with variance \(\sigma^2\) on the original scale has variance \(\sigma^2/s^2\) on the transformed scale. The transformed latent trajectory must represent \((m_i(t)-c)/s\), including its fixed and random components.

Derivation and Invariance of the Association Contrast

The preservation of the hazard follows directly from substitution. For any two latent marker values \(m_a\) and \(m_b\) at the same time, with the same covariates and other model quantities, the conditional hazard ratio is

Under the affine transformation, \(m_b-m_a=s(m_b^*-m_a^*)\). Consequently,

The marker contrast and its interpretation are therefore invariant when expressed in corresponding units. A one-unit contrast on the transformed scale represents an \(s\)-unit contrast on the original scale; its hazard ratio is \(\exp(\alpha^*)=\exp(s\alpha)\). A center does not appear in a difference, which is another way to see why centering alone does not change the association coefficient.

For fixed \(s\), standard errors and confidence limits for the association coefficient transform by the same positive factor: \(SE(\hat{\alpha}^*)=s\,SE(\hat{\alpha})\), and a confidence interval for \(\alpha\) maps to one for \(\alpha^*\) by multiplying both endpoints by \(s\). The Wald statistic for testing \(H_0:\alpha=0\) is unchanged because both the estimate and its standard error are multiplied by \(s\). This statement treats \(s\) as fixed for the reparameterization; if a data-estimated scale is treated as uncertain in a separate inferential target, that uncertainty must be accounted for.

Practical Checks, Alternatives, and Limitations

Raw units, centered units, and standardized units are alternative coordinate systems for the same linear marker association when the transformation and model are consistent. Raw units preserve direct measurement units. Centering makes the zero marker value a chosen reference and may improve the meaning of the baseline-hazard reference. Scaling can make a coefficient numerically easier to read or improve numerical conditioning when marker values are very large or very small. These are not substitutes for changing the association model; a linear link remains linear after an affine transformation.

Useful checks during fitting include confirming the formulas for \(\alpha^*\) and \(h_0^*(t)\), comparing conditional hazards at several marker values, and checking that fitted linear predictors agree after the transformation. If an equivalent model fitted with different coordinates gives materially different fitted hazards, investigate whether the marker transformation was applied consistently, whether the baseline-hazard parameterization can absorb the multiplier, and whether optimization has converged adequately. Small numerical discrepancies can occur from rounding or numerical optimization; systematic discrepancies are a warning.

Equivalence need not hold for a subject-specific center \(c_i\) or a time-varying center \(c(t)\): the resulting factor \(\exp\{\alpha c_i\}\) or \(\exp\{\alpha c(t)\}\) cannot generally be absorbed into one common baseline hazard. Nor does affine rescaling repair a misspecified longitudinal trajectory, an unsuitable hazard association, or problems caused by weak information about the latent marker. A nonlinear transformation of the marker is not just a change of units in this linear model; it changes the form of the hazard association unless the model is transformed accordingly.

• Common mistake: “Centering changes the association coefficient.” With a linear marker term, centering alone has \(s=1\), so \(\alpha^*=\alpha\). It changes the reference point and rescales the baseline hazard.
• Common mistake: changing the marker but not its coefficient. If one transformed unit represents \(s\) original units, the coefficient must become \(s\alpha\); otherwise the marker contribution to the log hazard changes.
• Common mistake: comparing baseline hazards as if their reference scales matched. The transformed baseline hazard includes the factor \(\exp(\alpha c)\). A difference in its numerical value does not by itself indicate different fitted subject-specific hazards.
• Common mistake: treating a standardized coefficient as a per-original-unit coefficient. State the scale explicitly and translate contrasts using \(m_i(t)=c+s\,m_i^*(t)\).