Comparing Association Coefficients on a Common Scale

Why Association Coefficients Need a Common Scale

An association coefficient in a joint model is tied to the quantity that enters the event hazard. A coefficient multiplying a biomarker measured in milligrams per litre has different units from one multiplying an integral measured in milligram-days per litre. Their numerical magnitudes alone do not tell us which association is stronger. A meaningful comparison first states a biomarker change, then calculates the hazard ratio implied by that change under each link.

This tutorial considers conditional hazards: the event hazard is conditional on a subject’s latent biomarker trajectory and any shared random effects in the joint model. The comparison concerns the model-implied hazard contrast at the same event time, with other terms held fixed. It is an association contrast, not by itself a causal effect.

Definition: Let $m_i(t)$ denote the latent biomarker value for subject $i$ at time $t$. Choose a reference value $m_0$ and a positive scale $d$, both expressed in the biomarker’s units, and define the standardized value $$ z_i(t)=\frac{m_i(t)-m_0}{d}. $$ The reference $m_0$ sets the origin; the scale $d$ sets the size of a one-unit change. For example, $d$ may be a standard deviation from a specified reference population or a clinically meaningful change. The choice must be stated.

For a current-value association, a common conditional hazard model is

h_i(t\mid b_i)=h_0(t)\exp\{\gamma^\top x_i+\alpha_m m_i(t)\},

where $h_0(t)$ is the baseline hazard, $x_i$ is a vector of covariates, $b_i$ denotes the shared random effects, and $\alpha_m$ multiplies the biomarker in its original units. The log-linear association assumption means that, conditional on the other terms, a change $\Delta m$ produces a hazard ratio $\exp(\alpha_m\Delta m)$. Thus $\alpha_m$ has units reciprocal to the biomarker’s units.

Rescaling a Current-Value Link

Substitute $m_i(t)=m_0+d z_i(t)$ into the hazard model. The term involving the biomarker becomes $\alpha_m m_0+\alpha_m d z_i(t)$. The coefficient on the standardized value is therefore $\alpha_z=\alpha_m d$. A change of one standardized unit corresponds to a change of $d$ original biomarker units, so its hazard ratio is $\exp(\alpha_z)=\exp(\alpha_m d)$.

\frac{h_i(t\mid z_i+1,b_i)}{h_i(t\mid z_i,b_i)} =\exp(\alpha_z) =\exp(\alpha_m d).

The reference value $m_0$ does not change this coefficient conversion. With a common reference for all subjects, its contribution is constant in the current-value linear predictor and can be incorporated into the baseline hazard. The scale $d$, by contrast, changes the numerical coefficient. Comparisons across models should use the same $m_0$ and $d$, preferably chosen before examining the fitted associations.

Worked Example: Hazard Ratio per Standard Deviation

Suppose a hypothetical analysis measures a biomarker in milligrams per litre. In a specified reference population, its standard deviation is $d=2.5$ mg/L. A fitted current-value coefficient is $\alpha_m=0.18$ per mg/L. Calculate and interpret the hazard ratio for a 2.5 mg/L increase in the current latent biomarker value.

The standardized coefficient is $$ \alpha_z=\alpha_m d=0.18\frac{1}{\text{mg/L}}\times 2.5\text{ mg/L}=0.45. $$

The hazard ratio for a one-standard-deviation increase is $$ \exp(0.45)=1.568. $$

Under the fitted conditional model, at the same time and with the other model terms fixed, a current biomarker value 2.5 mg/L higher is associated with a hazard 1.568 times as large, or about 56.8% higher. This interpretation is conditional and associational; it does not establish that changing the biomarker would cause the hazard to change.

If the coefficient’s hypothetical 95% confidence interval is $[0.08,0.28]$ per mg/L, its standardized interval is $[0.08(2.5),0.28(2.5)]=[0.20,0.70]$. Exponentiating the endpoints gives $[\exp(0.20),\exp(0.70)]=[1.221,2.014]$. This interval estimates the uncertainty in the hazard ratio for a 2.5 mg/L increase; it is not a 95% probability statement about the fixed parameter.

Different Links Require Explicit Change Scenarios

The same principle applies when the hazard link uses a summary of the trajectory rather than the current value. For example, define a cumulative exposure over a fixed window of length $L$ by

A_i(t)=\int_{t-L}^{t}m_i(u)\,du.

If time is measured in days and $m_i(u)$ in mg/L, then $A_i(t)$ is measured in mg-days/L. In a hazard model with term $\alpha_A A_i(t)$, the coefficient has units reciprocal to mg-days/L. It cannot be compared numerically with a current-value coefficient whose units are reciprocal to mg/L.

To compare the models on a biomarker scale, specify a trajectory change. Suppose the biomarker is shifted upward by $d$ throughout the entire window. The cumulative exposure changes by $dL$, not by $d$. Consequently, the model-implied hazard ratio for that scenario is $\exp(\alpha_A dL)$. Equivalently, define the standardized cumulative summary $A_{z,i}(t)=\int_{t-L}^{t}z_i(u)\,du$. Its coefficient is $\alpha_{A,z}=\alpha_A d$, and the specified trajectory shift changes $A_{z,i}(t)$ by $L$, giving the same hazard ratio $\exp(\alpha_{A,z}L)$.

Worked Example: Comparing Current and Cumulative Links

Consider two hypothetical link-sensitivity fits using the same biomarker reference scale, $d=2$ mg/L. One fit uses the current biomarker with coefficient $\alpha_m=0.24$ per mg/L. The other uses cumulative exposure over the previous $L=14$ days, with coefficient $\alpha_A=0.015$ per (mg-day/L). Compare their hazard ratios for a clearly specified change: the biomarker trajectory is 2 mg/L higher throughout the 14-day window, and the current-value contrast is also 2 mg/L higher at the event time.

For the current-value link, the log-hazard contrast is $$ 0.24\frac{1}{\text{mg/L}}\times 2\text{ mg/L}=0.48, $$ so the hazard ratio is $\exp(0.48)=1.616$.

For the cumulative link, the trajectory shift changes cumulative exposure by $$ 2\text{ mg/L}\times14\text{ days}=28\text{ mg-days/L}. $$ The log-hazard contrast is $0.015\times28=0.42$, and the hazard ratio is $\exp(0.42)=1.522$.

For these specified contrasts, the current-value fit implies a conditional hazard ratio of 1.616 and the cumulative fit implies 1.522. The raw coefficients, 0.24 and 0.015, are not the basis for that comparison: they multiply different quantities with different units. Nor do these two contrasts establish that one association structure is true; they express the predictions each fitted model makes for the stated changes.

A Rescaling Check

Changing measurement units must not change a model’s predictions. If $M$ is measured in mg/L and a new unit $U$ is defined by $M=10U$, then $\alpha_M M=\alpha_U U$ requires $\alpha_U=10\alpha_M$. The coefficient’s numerical value changes because the unit changed, but the product of coefficient and covariate change—and hence the hazard ratio—does not.

Worked Example: Converting Units Without Changing the Hazard Ratio

Continue with a hypothetical coefficient of $0.18$ per mg/L and standard deviation $2.5$ mg/L. Define $U=M/10$, so one $U$-unit equals 10 mg/L. Calculate the coefficient and standard deviation in $U$-units, then verify the one-standard-deviation hazard ratio.

Since $M=10U$, the linear predictor term satisfies $0.18M=0.18(10U)=1.8U$. Thus $\alpha_U=1.8$ per $U$-unit. The standard deviation becomes $d_U=2.5/10=0.25$ $U$-units.

The standardized log-hazard contrast is unchanged: $$ \alpha_Ud_U=1.8\times0.25=0.45, \qquad \exp(0.45)=1.568. $$

The raw coefficient changed from 0.18 to 1.8, but the hazard ratio for the same physical biomarker change remains 1.568. A coefficient’s size cannot be interpreted without its units and the change to which it refers.

Procedure, Assumptions, and Checks

A useful comparison requires more than multiplying a coefficient by a convenient number. State the model and the contrast first, then check that the transformation matches the quantity in the hazard link.

Identify the linked quantity.
Write the hazard’s association term and record the units of the linked biomarker summary. For a cumulative integral, include the time unit and window.

Choose a common biomarker scale.
Specify $m_0$ and $d>0$, and use the same reference population and scale when comparing fitted models. A one-standard-deviation change is interpretable only when the population used to calculate that standard deviation is identified.

Define the change being compared.
For a current-value link, state the change at the event time. For an integrated link, state the trajectory change and its duration; the induced change in the integral follows by integration.

Calculate and interpret the hazard ratio.
Multiply the fitted coefficient by the change in its linked quantity and exponentiate. Include the biomarker units, time window if relevant, conditional nature of the contrast, and uncertainty interval when available.

These calculations rely on a log-linear association in the specified link quantity and a valid fitted joint model for the conditional hazard. The hazard ratio compares two conditional hazards at the same time with other terms held fixed; the conditional model does not guarantee that the corresponding marginal hazard ratio after averaging over random effects is identical. The comparison also assumes the same covariate and random-effects conditioning is intended across the models.

Useful checks include verifying units algebraically, confirming that the same biomarker scale was used in all fits, and recalculating the hazard ratio from an equivalent unit conversion. A large raw coefficient may simply reflect a small unit, such as a coefficient per one-thousandth of a biomarker unit. Conversely, standardization does not repair an unsuitable link, a misspecified trajectory, or poor estimation. Examine convergence, uncertainty, and whether the chosen scale is representative; compare model predictions for the stated contrast rather than relying on coefficient magnitudes alone.

Choosing a Scale and Recognizing Its Limits

There are several defensible reporting scales, each answering a different question:

Scale	What it conveys	Main caution
One original biomarker unit	Hazard ratio for a change of one recorded unit	Can be numerically awkward or clinically small
One reference-population standard deviation	Hazard ratio for a change tied to observed biomarker variability	Depends on the population and period used to calculate the standard deviation
A clinically meaningful increment	Hazard ratio for a prespecified change relevant to practice	Requires a clear rationale for the chosen increment
A specified trajectory perturbation	Hazard ratio for a stated change over a stated time pattern or window	Different trajectory patterns can yield different contrasts for cumulative links

Standardized coefficients can help compare associations when the same marker and standardization are used, but they do not make every link’s coefficient directly comparable. A one-unit increase in standardized current value and a one-unit increase in standardized cumulative exposure still represent different changes. For a fair comparison, express each fitted model’s hazard ratio for the same explicitly described biomarker trajectory change whenever that is scientifically meaningful. If a slope or another trajectory summary is used, derive the change in that summary from the stated trajectory perturbation and retain its time units; do not label unlike quantities as equivalent merely because they have been standardized.

A common mistake is to say, “The current-value coefficient is 0.24 and the cumulative coefficient is 0.015, so the current-value association is stronger.” This compares numbers with incompatible units and ignores the amount and duration of biomarker change. The correction is to specify a common biomarker change, calculate the corresponding change in each link quantity, and compare the resulting conditional hazard ratios—with uncertainty and model limitations kept in view.

Key takeaway: A coefficient is interpretable only together with its linked quantity, units, and change. Standardize using a declared reference and scale, then report $\exp(\alpha\Delta q)$ for a stated change $\Delta q$ in the quantity $q$ that enters the hazard.

Check Your Understanding

Answer each question by stating the relevant units and contrast. Brief solutions follow the questions.

A current-value coefficient is $0.12$ per mg/L and the chosen biomarker standard deviation is 3 mg/L. What is the hazard ratio for a one-standard-deviation increase?
A cumulative link integrates biomarker values over 10 days. The biomarker is shifted upward by 1.5 mg/L throughout that window, and the coefficient is $0.02$ per (mg-day/L). What is the implied hazard ratio?
A coefficient changes from 0.4 per unit to 4 per new unit, where one new unit equals 10 old units. Has the model-implied hazard ratio for a fixed physical biomarker change changed? Explain.
Why is it not valid to conclude that a coefficient of 0.3 is a stronger association than a coefficient of 0.02 without further information?

Solutions. (1) The standardized log-hazard contrast is $0.12\times3=0.36$, so the hazard ratio is $\exp(0.36)=1.433$. (2) The cumulative change is $1.5\times10=15$ mg-days/L. The log-hazard contrast is $0.02\times15=0.30$, giving $\exp(0.30)=1.350$. (3) No. If the new-unit value is one tenth of the old-unit value, the coefficient must be ten times larger to preserve the linear predictor. For any fixed physical change, the coefficient times the corresponding change is unchanged. (4) The coefficients may multiply quantities with different units or scales. Their magnitudes alone do not determine comparable hazard contrasts; specify a biomarker change, calculate each linked-quantity change, and exponentiate the resulting log-hazard contrasts.