Why the Marker Scale Matters
A current-value association links the event hazard at time \(t\) to the longitudinal marker’s latent current value \(m_i(t)\). Choosing the marker’s original scale or its log scale changes the meaning of the association parameter: the first describes additive differences in marker units, while the second describes ratios between marker values. As a result, the same numerical change in the marker need not imply the same hazard ratio under both models.
The comparison here is conditional on the modeled subject-specific trajectory and on the same baseline covariates. It concerns the hazard at a given time, not a cumulative event probability, and an association in the model is not by itself a causal effect.
Two Current-Value Links
Let \(h_i(t\mid b_i)\) denote subject \(i\)’s hazard at time \(t\), conditional on the random effects \(b_i\) that determine the latent marker trajectory. Let \(h_0(t)\) be the baseline hazard, \(w_i\) a vector of baseline covariates, and \(\gamma\) its coefficient vector. The following models differ only in how the current marker enters the hazard:
Log-scale current-value link: \(h_i(t\mid b_i)=h_0(t)\exp\{\gamma^\mathsf{T}w_i+\alpha_{\log}\log(m_i(t)/m_{\mathrm{ref}})\}\). This requires \(m_i(t)>0\). Here \(m_{\mathrm{ref}}>0\) is a fixed reference value in the same units as the marker, and \(\alpha_{\log}\) describes the association with the log of the marker relative to that reference.
The reference value makes the logarithm’s argument dimensionless. Changing \(m_{\mathrm{ref}}\) adds a constant to the log term; that constant can be absorbed into the baseline hazard. It does not change hazard ratios comparing two marker values at the same time. If the marker is recorded in different units, the numerical value of \(m_{\mathrm{ref}}\) should be converted as well.
For both links, the hazard ratio below compares two possible current marker values at the same time for the same covariates, with the same baseline hazard. It is a conditional contrast. The model assumes that the association is linear in the specified marker scale on the log-hazard scale, and that the other terms are held fixed.
Deriving the Marker-Contrast Hazard Ratios
Suppose the marker changes from \(m_A\) to \(m_B\), with both values in the domain of the chosen link. Dividing the hazard at \(m_B\) by the hazard at \(m_A\) cancels the shared baseline hazard and covariate terms. For the original-scale link, subtracting the two log hazards gives \(\alpha_{\mathrm{orig}}(m_B-m_A)\). For the log-scale link, it gives \(\alpha_{\log}\log(m_B/m_A)\). Exponentiating yields:
These expressions reveal the central distinction. If \(m_B=m_A+1\), the original-scale hazard ratio is \(\exp(\alpha_{\mathrm{orig}})\), regardless of the starting value. Under the log-scale link, it is \(\{(m_A+1)/m_A\}^{\alpha_{\log}}\), which depends on \(m_A\). If \(m_B=c\,m_A\), where \(c>0\), the log-scale hazard ratio is \(c^{\alpha_{\log}}\), regardless of the starting value. Under the original-scale link, it is \(\exp\{\alpha_{\mathrm{orig}}(c-1)m_A\}\), which generally depends on \(m_A\).
The coefficient units also differ. If the marker is measured in, for example, mg/L, then \(\alpha_{\mathrm{orig}}\) is measured per mg/L. The log-scale coefficient is dimensionless when the logarithm is taken of a ratio of marker values. A positive coefficient means higher marker values are associated with a higher conditional hazard under either link; a negative coefficient means they are associated with a lower conditional hazard.
Worked Example: Equal One-Unit Changes on the Original Scale
Suppose a hypothetical original-scale model uses a marker measured in mg/L and has \(\alpha_{\mathrm{orig}}=0.20\) per mg/L. Compare the conditional hazards at marker values of 4 and 5 mg/L, and at 10 and 11 mg/L.
For either one-unit increase, the marker difference is \(1\) mg/L. Thus
At each comparison, the model implies a hazard about 1.221 times as high at the larger marker value, or about 22.1% higher, conditional on the model terms. The equal hazard ratios follow from the equal additive differences, even though the starting marker values differ.
For contrast, doubling the marker from 4 to 8 mg/L is a difference of 4 mg/L, not 1 mg/L:
Thus this model does not assign one fixed hazard ratio to every doubling; the implied ratio depends on the size of the additive difference.
Worked Example: One-Unit and Proportional Changes on the Log Scale
Suppose a hypothetical log-scale model has \(\alpha_{\log}=0.75\). The marker is positive and measured in mg/L. Compare 4 with 6 mg/L, then calculate the hazard ratios for one-unit increases from 4 to 5 and from 10 to 11 mg/L.
For the change from 4 to 6 mg/L, the marker is multiplied by \(6/4=1.5\). The reference value cancels from the contrast:
The conditional hazard at 6 mg/L is therefore about 1.355 times the hazard at 4 mg/L, or about 35.5% higher. This is the model’s implication for a 50% increase, wherever that proportional change starts.
A one-unit increase is not the same proportional change at every starting value. From 4 to 5 mg/L:
From 10 to 11 mg/L:
Each increase is 1 mg/L, but the first is a 25% increase and the second is a 10% increase. The model consequently implies different hazard ratios. The calculations are conditional hazard contrasts, not statements that changing a marker would cause the hazard to change.
How the Models Compare
The choice of scale determines which kinds of contrasts are represented simply. An original-scale link is natural when an additive difference has a stable interpretation: one unit higher is associated with the same hazard ratio at every starting value. A log-scale link is natural when a proportional difference has a stable interpretation: a doubling, for example, has the same hazard ratio at every positive starting value.
The local change in log hazard makes the distinction precise. In the original-scale model, the derivative of log hazard with respect to the marker is \(\alpha_{\mathrm{orig}}\), a constant per marker unit. In the log-scale model it is \(\alpha_{\log}/m\), so a one-unit increase has a smaller relative implication at larger marker values. Equivalently, the derivative of log hazard with respect to log marker is \(\alpha_{\log}\). This last quantity is an elasticity: a small proportional change in the marker corresponds approximately to \(\alpha_{\log}\) times that proportional change in hazard.
The coefficients are on different scales and should not be compared by their numerical magnitudes alone. A value of 0.20 per mg/L and a value of 0.20 on the log scale do not encode the same marker contrast. To compare fitted models or communicate an effect, specify concrete marker values and calculate the corresponding hazard ratio.
Worked Example: The Same Proportional Change at Different Starting Values
Consider two hypothetical fitted models for a positive marker in mg/L: an original-scale model with \(\alpha_{\mathrm{orig}}=0.08\) per mg/L and a log-scale model with \(\alpha_{\log}=0.60\). Compare a doubling from 5 to 10 mg/L with a doubling from 10 to 20 mg/L under each model.
For the first doubling, the original-scale difference is 5 mg/L:
For the second doubling, the difference is 10 mg/L:
Under the log-scale model, both comparisons are doublings, so both have the same ratio:
The models give different implications because their coefficients were specified on different scales. The example illustrates the structural contrast: under the original-scale link, the larger additive difference for the second doubling yields a larger hazard ratio; under the log-scale link, equal proportional changes yield equal hazard ratios.
Assumptions, Diagnostics, and Limitations
These interpretations require the model to represent the conditional hazard using the stated link and to compare marker values while holding the other model terms fixed. The original-scale link assumes a linear contribution of \(m_i(t)\) to log hazard. The log-scale link assumes a linear contribution of \(\log(m_i(t)/m_{\mathrm{ref}})\) to log hazard and requires the latent marker to remain positive wherever the link is used. Neither link automatically guarantees that its assumed functional form is appropriate.
When checking an analysis, examine whether the scale makes sense for the marker and whether the model’s implied contrasts are plausible over the observed range. For the original-scale model, ask whether equal additive changes should really imply equal hazard ratios at low and high marker values. For the log-scale model, check that the marker is positive over the relevant follow-up and consider whether the implied effect of a proportional change is plausible across its range. A log link can be especially sensitive when values are close to zero, because a fixed additive change then corresponds to a very large proportional change.
Patterns in residual or calibration diagnostics, and systematic disagreement between observed outcomes and model-based predictions across marker levels, can signal a poor functional form. A useful comparison is to calculate and inspect hazard ratios for clinically interpretable contrasts under both candidate scales. Model fit measures may also help, but better fit alone does not make an association causal or ensure that predictions remain reliable outside the data’s marker range.
The log link is not defined for zero or negative marker values. Adding a constant before taking logs changes the transformation and therefore changes the model; it is not an automatic repair. The original-scale link can accommodate such values algebraically, but its linear log-hazard assumption may still be unsuitable. Both models can extrapolate implausibly beyond the marker values supported by the data, and neither contrast describes a marginal hazard obtained by averaging over random effects.
Common Pitfall
Correction: That constant one-unit multiplier applies to the original-scale link. Under a log-scale link, a change from \(m\) to \(m+1\) has hazard ratio \(\{(m+1)/m\}^{\alpha_{\log}}\), which depends on the starting value. The log-scale coefficient instead gives the ratio \(c^{\alpha_{\log}}\) for a proportional change by factor \(c\).
A related mistake is to read a hazard ratio as a change in event probability. A hazard ratio compares instantaneous event rates conditional on the model terms at a time; it is not a ratio of cumulative probabilities over follow-up.
Check Your Understanding
Use the stated model and show which marker contrast determines each hazard ratio.
- Under an original-scale link with \(\alpha_{\mathrm{orig}}=0.10\) per unit, what is the hazard ratio for an increase from 3 to 5 marker units?
- Under a log-scale link with \(\alpha_{\log}=0.50\), what is the hazard ratio for doubling a positive marker?
- Under that same log-scale link, compare the hazard ratios for 4 to 5 and 8 to 9 marker units. Why are they different?
- What positivity condition is required for the log-scale link, and what does its coefficient describe for a proportional change by factor \(c\)?
Solutions. (1) The difference is \(5-3=2\), so \(\mathrm{HR}=\exp\{0.10(2)\}=\exp(0.20)\approx1.221\). The hazard is about 22.1% higher at 5 units, conditional on the model terms. (2) A doubling has \(c=2\), so \(\mathrm{HR}=2^{0.50}=\sqrt{2}\approx1.414\). (3) From 4 to 5, \(\mathrm{HR}=(5/4)^{0.50}\approx1.118\). From 8 to 9, \(\mathrm{HR}=(9/8)^{0.50}=1.061\). The same one-unit increase is a larger proportional change from 4 than from 8. (4) The current marker must be positive wherever its logarithm is used. A change from \(m\) to \(cm\) has \(\mathrm{HR}=c^{\alpha_{\log}}\), so the coefficient describes the hazard association with proportional, rather than additive, marker changes.