Slope Associations on a Log-Marker Scale

Why Put the Slope on a Log-Marker Scale?

A marker can change by the same number of units per year in two people while changing by very different proportions of their current levels. An increase of 4 units per year is an 8% instantaneous rate of increase when the marker is 50 units, but a 4% rate when it is 100 units. A slope association based on the derivative of the log latent marker captures this relative, or proportional, rate of change. It is therefore different from an association based on the absolute derivative of the marker itself.

The distinction matters when a percentage change is scientifically more comparable across marker levels than a fixed change in marker units. The derivation is simple, but the scale, units, and interpretation must be kept explicit: the log-marker derivative is measured per unit time, not in marker units per unit time.

Definition and Joint-Model Form

The reference value makes the logarithm’s argument dimensionless. Since the reference is constant, it disappears upon differentiation. The identity follows from the chain rule: the derivative of \(\log\{m_i(t)/m_{\mathrm{ref}}\}\) is \(m_i'(t)/m_i(t)\). This derivative requires a positive marker trajectory and a well-defined derivative at the time of interest.

A slope-linked event hazard can be written as

Here \(\lambda_i(t\mid b_i)\) is subject \(i\)’s hazard at time \(t\), conditional on the subject-specific latent trajectory (represented by random effects \(b_i\)) and covariates; \(\lambda_0(t)\) is the baseline hazard; \(w_i\) is a vector of covariates; and \(\gamma\) is its coefficient vector. The parameter \(\alpha_s\) links the instantaneous log-marker slope to the hazard. In this tutorial, the association term is the slope term \(\alpha_s r_i(t)\); no current marker-value term is included.

For \(r_i(t)\) measured in years\(^{-1}\), \(\alpha_s\) has units of years, so their product is dimensionless. At a fixed time and equal covariate values, two subjects’ hazard ratio depends on the difference between their log-marker slopes:

This comparison conditions on the same time, so the shared baseline hazard cancels. A positive \(\alpha_s\) means a greater instantaneous proportional marker growth rate is associated with a higher hazard, conditional on the model. A negative \(\alpha_s\) means it is associated with a lower hazard. Neither interpretation by itself establishes a causal effect.

Deriving the Proportional-Rate Interpretation

The link to proportional change follows directly from the derivative definition. Over a short time increment \(\Delta t\), differentiability gives the first-order approximation

Exponentiating both sides gives \(m_i(t+\Delta t)/m_i(t)\approx \exp\{r_i(t)\Delta t\}\). Thus, for a sufficiently short interval, the relative change is approximately \(r_i(t)\Delta t\), or \(100r_i(t)\Delta t\)% when expressed as a percentage. This is an instantaneous interpretation: if the rate varies over a longer interval, the total proportional change depends on the rate throughout that interval, not just its value at the start.

For comparison, the absolute marker slope is \(m_i'(t)\), with units of marker units per unit time. The two slopes are related by \(m_i'(t)=m_i(t)r_i(t)\). Consequently, the same absolute slope can correspond to different proportional rates at different marker levels, and the same proportional rate can correspond to different absolute slopes.

What the Hazard Ratio Compares

The slope association compares instantaneous log-marker slopes at a specified time. It does not compare marker levels, and it does not generally compare hazards at different times for one subject. For two subjects with the same covariates, a slope difference of \(\Delta r\) gives the same-time hazard ratio \(\exp(\alpha_s\Delta r)\). Because \(\Delta r\) has inverse-time units, the size of the contrast and the units of time used in the model must be reported.

Assumptions, Diagnostics, and Limitations

The mathematical definition and hazard interpretation rely on several conditions. The latent marker must be positive and differentiable at the times being evaluated. The longitudinal model must provide a credible trajectory and derivative; the hazard model must correctly specify the baseline hazard, covariate contribution, and log-linear slope association for the intended interpretation. The hazard ratio formula also assumes a comparison at the same time with the same covariates, so that the baseline hazard and covariate terms cancel.

In a joint model, the association is defined using the latent trajectory, not a raw difference between two noisy observed measurements. Taking differences of observed measurements can amplify measurement error and make a purported slope unstable. Using the fitted latent trajectory addresses that distinction, but it does not remove dependence on the longitudinal model’s assumptions or on how well the available observations determine its derivative.

Useful checks include plotting fitted latent trajectories and their derivatives over the observed time range, checking whether the implied marker values remain positive, and examining whether the fitted slopes are plausible rather than being driven by sparse observations or extrapolation. Investigate whether the log-linear hazard contribution is adequate and whether conclusions change under reasonable alternative trajectory specifications. Signs of trouble include implausible rapid changes between observations, highly uncertain derivatives, strong sensitivity to a few observations, or a systematic mismatch between the assumed trajectory and longitudinal data.

The log-marker slope link is not interchangeable with an absolute-slope link. An absolute-slope link asks whether the hazard is associated with change in marker units per time. The log-marker slope link asks whether it is associated with instantaneous proportional change. A current-value link instead uses marker level; it answers a different question. Which scale is useful depends on the scientific meaning of change and the marker’s measurement scale.

A log transformation is not suitable without care when the latent marker can be zero or negative. Adding an arbitrary constant solely to make values positive changes the transformation and its derivative, so it changes the estimand rather than simply fixing notation. Finally, a slope association describes a conditional relationship in the specified joint model. It does not alone show causation, guarantee a constant effect outside the observed time range, or establish that a particular intervention on the marker would change event risk.

Common Mistake

• Wrong reasoning: “The marker rises by 4 units per year in both people, so their log-marker slopes are equal.”
• Correction: The log-marker slope is \(m_i'(t)/m_i(t)\), not \(m_i'(t)\). With a marker level of 50 units, a 4-unit-per-year slope gives \(0.08\) per year; with a level of 100 units, it gives \(0.04\) per year.
• Wrong reasoning: “A slope of 0.06 per year means the marker increases by exactly 6% in every year.”
• Correction: It is an instantaneous proportional rate. The exact proportional change over an interval depends on how the log-marker slope varies over that interval.
• Wrong reasoning: “The exponential of the slope term alone is always the hazard ratio across two times.”
• Correction: The slope term gives a hazard ratio for a slope contrast at the same time when other covariates are equal. Comparing different times for one subject also involves the baseline hazard and any time-varying covariate contributions.