Biomarker Units and Association-Scale Changes

Why Biomarker Units Matter

An association coefficient in a joint model is expressed per unit of the biomarker used in the analysis. Changing that unit can therefore change the coefficient’s numerical value, sometimes substantially, even though it does not change the risk contrast for the same underlying biomarker difference. A coefficient per milligram per deciliter and a coefficient per millimole per liter are not directly comparable as bare numbers.

The key distinction is between a change of scale and a change in substantive conclusions. A scale conversion changes how a biomarker value is represented. It does not, by itself, change the fitted relationship between the underlying biomarker trajectory and the event hazard. This tutorial derives that result for current-value and slope links, including conversions with an additive offset.

Model and Unit-Conversion Rule

Let \(Y_{ij}\) be the observed biomarker for individual \(i\) at visit time \(t_{ij}\), and let \(m_i(t)\) denote that individual’s latent mean biomarker trajectory. Consider a joint model whose conditional event hazard is

h_i(t)=h_0(t)\exp\left\{\gamma^\mathsf{T}w_i+\alpha_L m_i(t)+\alpha_S m_i'(t)\right\}.

Here \(h_0(t)\) is the baseline hazard, \(w_i\) is a vector of event-model covariates, \(\gamma\) is its coefficient vector, \(\alpha_L\) is the current-value association coefficient, and \(\alpha_S\) is the slope association coefficient. A current-value-only model sets \(\alpha_S=0\); a slope-only model sets \(\alpha_L=0\). The derivative \(m_i'(t)\) is the instantaneous rate of change in the biomarker per unit of time.

For the algebra below, assume the same individuals, event times, covariates, time units, and latent trajectories are represented before and after conversion. The hazard is conditional on the specified covariates and latent trajectory, and the link is the log-linear form displayed above. These conditions establish the equivalence of the two parameterizations; they do not establish that the link or trajectory model is scientifically adequate.

Definition: An affine conversion of biomarker units is \(Z=a+cY\), where \(a\) is a fixed offset and \(c>0\) is a fixed scale factor. The transformed latent trajectory is \(m_{Zi}(t)=a+c\,m_i(t)\), so its slope is \(m_{Zi}'(t)=c\,m_i'(t)\). The positive factor \(c\) covers ordinary unit conversions that preserve direction.

The slope formula follows because the offset is constant in time: differentiating \(a+c\,m_i(t)\) gives \(c\,m_i'(t)\). Thus an additive offset changes the level but not the slope. If the time unit also changes, the slope must be converted for that change as well; the formulas here hold time units fixed.

Deriving the Coefficient Transformation

First consider the original link contribution to the log hazard. Since \(m_i(t)=(m_{Zi}(t)-a)/c\) and \(m_i'(t)=m_{Zi}'(t)/c\), substitution gives

\alpha_Lm_i(t)+\alpha_Sm_i'(t) = \frac{\alpha_L}{c}m_{Zi}(t) +\frac{\alpha_S}{c}m_{Zi}'(t) -\frac{\alpha_La}{c}.

The last term is constant across individuals and time. It can be absorbed into the baseline hazard, giving an equivalent model with transformed coefficients

\alpha_{L,Z}=\frac{\alpha_L}{c}, \qquad \alpha_{S,Z}=\frac{\alpha_S}{c}, \qquad h_{0,Z}(t)=h_0(t)\exp\left(-\frac{\alpha_La}{c}\right).

When there is no current-value term, \(\alpha_L=0\), so an offset does not require a baseline-hazard adjustment. When \(a=0\), the baseline hazard is unchanged. The formulas show that the association coefficients are divided by the factor that multiplies the biomarker; the slope coefficient is also divided by that factor because the slope itself is multiplied by it.

Theorem: Under the stated log-linear hazard model, an affine change \(Z=a+cY\) with \(c>0\) preserves the same conditional hazard function when the current-value and slope coefficients are divided by \(c\), and the baseline hazard is multiplied by \(\exp(-\alpha_La/c)\). A scale conversion alone changes coefficient units, not the hazard contrast for a fixed underlying biomarker change.

For a current-value difference \(\Delta m\), the hazard ratio is \(\exp(\alpha_L\Delta m)\). On the converted scale the same difference is \(c\Delta m\), so its hazard ratio is

\exp\left(\alpha_{L,Z}c\Delta m\right) = \exp\left(\frac{\alpha_L}{c}c\Delta m\right) = \exp(\alpha_L\Delta m).

The same reasoning applies to a slope difference \(\Delta m'\): its converted value is \(c\Delta m'\), and \(\exp(\alpha_{S,Z}c\Delta m')=\exp(\alpha_S\Delta m')\). These are conditional hazard ratios for the specified contrast, holding the other model terms fixed. They describe an association in the model, not a causal effect.

Worked Example: Converting a Current-Value Link

Suppose a hypothetical current-value model uses glucose in milligrams per deciliter and estimates \(\alpha_L=0.0120\) per mg/dL. For illustration, use the approximate conversion \(Z=Y/18.0\), where \(Z\) is in mmol/L. Find the converted coefficient and compare the hazard ratio for an 18.0 mg/dL increase with that for a 1.00 mmol/L increase.

Here \(a=0\) and \(c=1/18.0\). Therefore

\alpha_{L,Z}=\frac{0.0120}{1/18.0} =0.0120(18.0) =0.216\ \text{per mmol/L}.

An increase of 18.0 mg/dL corresponds to an increase of 1.00 mmol/L. On the original scale, the modeled hazard ratio is \(\exp(0.0120\times18.0)=\exp(0.216)=1.241\). On the converted scale it is \(\exp(0.216\times1.00)=1.241\). The numerical coefficient is 18 times as large on the mmol/L scale, but the hazard ratio for this same biomarker change is identical.

If the original 95% confidence interval for \(\alpha_L\) were 0.00900 to 0.0150 per mg/dL, the converted interval would be \(18.0(0.00900)\) to \(18.0(0.0150)\), or 0.162 to 0.270 per mmol/L. The corresponding hazard-ratio interval for a 1.00 mmol/L increase is \(\exp(0.162)\) to \(\exp(0.270)\), approximately 1.176 to 1.310. The interval concerns the hazard ratio for that specified increase, not for every possible contrast.

Slope Links and Offset Conversions

A slope coefficient is reported per unit of biomarker change per unit of time. Keeping time units fixed, changing the biomarker unit by \(c\) divides the slope coefficient by \(c\), just as for a current-value coefficient. The interpretation should name both units: for example, a coefficient per (cm/year), not merely “per centimeter.”

Worked Example: Converting a Slope Link

Suppose a hypothetical slope-only model uses a biomarker in millimeters and time in years. Its estimated slope coefficient is \(\alpha_S=0.0800\) per (mm/year). Convert the biomarker to centimeters using \(Z=Y/10\), then calculate the hazard ratio associated with a slope difference of 2.00 mm/year.

The scale factor is \(c=0.100\), so the transformed slope coefficient is

\alpha_{S,Z}=\frac{0.0800}{0.100} =0.800\ \text{per (cm/year)}.

A difference of 2.00 mm/year is 0.200 cm/year. On the original scale, the hazard ratio is \(\exp(0.0800\times2.00)=\exp(0.160)=1.174\). On the converted scale it is \(\exp(0.800\times0.200)=\exp(0.160)=1.174\). Both calculations describe the same modeled contrast in rate of change.

For an offset conversion, the transformed current-value coefficient still follows the same division rule, but the baseline hazard adjustment matters to preserve absolute hazard predictions. The adjustment is a constant multiplier, not a change in the relative hazard between individuals that arises from a particular biomarker contrast.

Worked Example: A Shifted Scale in a Level-and-Slope Link

Consider a hypothetical model with \(\alpha_L=0.300\) per original biomarker unit and \(\alpha_S=0.0400\) per original unit/year. Suppose a new assay scale is \(Z=5+2Y\), with time still measured in years. For an individual with \(m_i(t)=4.00\) original units and \(m_i'(t)=1.50\) original units/year, show how the link contribution is preserved.

Here \(a=5\) and \(c=2\). The transformed coefficients are \(\alpha_{L,Z}=0.300/2=0.150\) per new unit and \(\alpha_{S,Z}=0.0400/2=0.0200\) per new unit/year. The converted level is \(5+2(4.00)=13.0\) new units, and the converted slope is \(2(1.50)=3.00\) new units/year. The original link contribution is

$$ 0.300(4.00)+0.0400(1.50)=1.26. $$

The transformed link contribution is \(0.150(13.0)+0.0200(3.00)=2.01\). Because the level was shifted, the transformed baseline hazard is \(h_{0,Z}(t)=h_0(t)\exp[-(0.300)(5)/2]=h_0(t)\exp(-0.750)\). Combining the baseline adjustment with the transformed link gives \(\exp(-0.750)\exp(2.01)=\exp(1.26)\), exactly the original multiplier. Omitting the baseline adjustment would alter absolute fitted hazards.

Interpreting Estimates and Checking an Analysis

If an estimate and its standard error are both transformed consistently, the standard error is divided by \(c\), just like the coefficient. For \(c>0\), confidence-interval endpoints are divided by \(c\) and retain their order. Consequently, the coefficient-to-standard-error ratio is unchanged, as are the corresponding Wald test and its p-value under this exact reparameterization. These statements concern the same fitted model expressed on different scales; they do not imply that separately specified models, priors, or constraints will always produce identical results.

A useful reporting strategy is to give the coefficient with its units and also report a hazard ratio for a prespecified, clinically interpretable contrast. For example, report the association per 10 units or per 1 standard deviation only if that contrast is clearly defined. To compare estimates across unit systems, first convert them to a common scale or compare their hazard ratios for the same underlying change.

The longitudinal model must also represent the conversion coherently. If observed biomarker values are multiplied by \(c\), the modeled trajectory and its uncertainty must be expressed on the converted scale. In a linear mixed model, the fixed-effect trajectory terms and random-effect contributions must together imply \(m_{Zi}(t)=a+c\,m_i(t)\); otherwise the transformed hazard coefficient alone cannot ensure equivalent fitted trajectories. This is a consistency check, not a claim that fitting on every scale will be numerically identical under every estimation setup.

When diagnosing an apparent discrepancy, check that the conversion factor is oriented correctly, that the coefficient is labelled with the correct units, that time units were not also changed, and that the same clinical contrast is being compared. Also check whether an offset was absorbed into the baseline hazard. A fixed baseline hazard or other parameter constraint that cannot accommodate the required multiplier may prevent exact equivalence. Rounding can cause small displayed differences even when unrounded results agree.

Unit conversion is different from changing the association structure. A current-value link and a slope link use different trajectory features and answer different scientific questions; changing units does not turn one into the other. Likewise, standardizing a biomarker is a re-expression of scale, whereas changing the time window or functional form of the link changes the model specification. Interpret apparent sensitivity only after separating those changes.

Wrong reasoning: “The coefficient became much larger after conversion, so the biomarker has a stronger association with the event.”
Correction: A coefficient’s numerical magnitude depends on the unit. Compare hazard ratios for the same biomarker contrast, or convert coefficients to a common unit before judging their scale.
Wrong reasoning: “The slope coefficient should not change because the direction and shape of the trajectory are unchanged.”
Correction: The slope’s numerical unit changes with the biomarker unit. Transform the slope coefficient and the slope contrast together.

Key takeaway: Under \(Z=a+cY\), divide current-value and slope coefficients by \(c\). For a nonzero offset, adjust the baseline hazard when a current-value term is present. Interpret and compare fitted associations using the same biomarker contrast and the same units.

Check Your Understanding

Answer each question, then check the solutions below. All scenarios are hypothetical, and time units are held fixed unless stated otherwise.

A current-value coefficient is 0.050 per unit. The biomarker is rescaled as \(Z=2Y\). What is the transformed coefficient, and what hazard ratio corresponds to a 3-unit increase on the original scale?
A biomarker is converted by \(Z=10+5Y\) in a current-value model with \(\alpha_L=0.20\). What is the transformed coefficient, and what baseline-hazard multiplier preserves the same hazard?
A slope coefficient is 0.12 per (unit/year). The biomarker is converted as \(Z=Y/4\). What is the transformed slope coefficient, and what converted slope difference corresponds to 2 units/year originally?
After converting units, the coefficient and its standard error are both multiplied by 10. What happens to their ratio, and what should be compared to assess the same clinical association?

Solutions. 1. Here \(c=2\), so \(\alpha_{L,Z}=0.050/2=0.0250\) per new unit. A 3-unit original increase is a 6-unit converted increase; the hazard ratio is \(\exp(0.050\times3)=\exp(0.150)=1.162\), also \(\exp(0.0250\times6)=1.162\). 2. Here \(a=10\) and \(c=5\), so \(\alpha_{L,Z}=0.20/5=0.0400\). The baseline multiplier is \(\exp(-\alpha_La/c)=\exp[-0.20(10)/5]=\exp(-0.400)=0.6703\). 3. Since \(c=1/4\), \(\alpha_{S,Z}=0.12/(1/4)=0.480\) per (new unit/year). A 2-unit/year original difference becomes \(0.500\) new unit/year. 4. The ratio is unchanged because both numerator and denominator have the same factor. Compare hazard ratios for the same specified biomarker change, rather than comparing bare coefficients reported in different units.