Baseline-Referenced Current-Value Associations

Why Refer the Marker to Baseline?

A current-value association links the event hazard at time $t$ to the subject’s latent marker level at that time. A baseline-referenced current-value association instead links the hazard to how far that latent level is from the subject’s own baseline. These two choices answer different questions: is the hazard associated with the marker’s absolute level, or with its departure from baseline?

The distinction matters when subjects begin with different marker levels. Two subjects can have the same current value but different changes from baseline; conversely, they can have the same change but different current values. A model using change from baseline assigns its association term according to the first subject-specific comparison, not the second.

Definition: Let $m_i(t)$ denote subject $i$’s latent marker trajectory at time $t$, on the marker scale used in the longitudinal model. Define the baseline-referenced current value as $$ \Delta_i(t)=m_i(t)-m_i(0). $$ A baseline-referenced current-value hazard model has the form $$ h_i(t\mid b_i)=h_0(t)\exp\left\{\gamma^\mathsf{T}w_i+\alpha_{\Delta}\Delta_i(t)\right\}, $$ where $h_0(t)$ is the baseline hazard, $w_i$ is a vector of covariates, $\gamma$ is its coefficient vector, and $b_i$ denotes the subject-specific random effects that determine the latent trajectory. The coefficient $\alpha_{\Delta}$ is the log-hazard association per one-unit greater change from baseline, conditional on the other terms in the model.

The marker in this definition is the latent trajectory, not a single observed measurement. For example, if the longitudinal model is $Y_{ij}=m_i(t_{ij})+\epsilon_{ij}$, then $Y_{ij}$ includes measurement error, whereas $m_i(t_{ij})$ represents the model-based underlying marker level. The baseline value $m_i(0)$ is likewise the trajectory’s value at the defined baseline time; it need not equal a noisy baseline observation.

Hazard-Ratio Interpretation

The model is a proportional-hazards model conditional on its included covariates and latent trajectory: at a fixed time, the ratio of two subjects’ hazards is determined by their covariate and association-term differences. To isolate the baseline-referenced association, compare two subjects at the same time with the same $w_i$. If their changes from baseline differ by $d$ marker units, the hazard ratio due to this association term is $\exp(\alpha_{\Delta}d)$.

\frac{h_a(t\mid b_a)}{h_b(t\mid b_b)} = \exp\left\{\alpha_{\Delta}\left[\Delta_a(t)-\Delta_b(t)\right]\right\} = \exp(\alpha_{\Delta}d), \qquad d=\Delta_a(t)-\Delta_b(t).

In particular, a one-unit greater change from baseline corresponds to a hazard ratio of $\exp(\alpha_{\Delta})$, holding the specified covariates and other model terms fixed. If $\alpha_{\Delta}>0$, a greater change is associated with a higher hazard; if $\alpha_{\Delta}<0$, it is associated with a lower hazard. This is an association in the fitted model, not by itself a causal effect.

At baseline, $\Delta_i(0)=0$ for every subject. Consequently, the baseline-referenced association term contributes zero to the log hazard at $t=0$. This does not imply that all subjects have equal hazards at baseline: covariates or other terms in the hazard model may still differ. Nor does it mean the model has estimated no baseline risk; that risk is represented by the baseline hazard and any other included terms.

How the Baseline-Referenced Link Differs from an Absolute Link

For comparison, an absolute current-value model replaces $\Delta_i(t)$ with $m_i(t)$:

h_i(t\mid b_i)=h_0(t)\exp\left\{\gamma^\mathsf{T}w_i+\alpha_m m_i(t)\right\}.

Here, $\exp(\alpha_m)$ is the hazard ratio for a one-unit higher current marker value, holding the covariates and other model terms fixed. That comparison concerns absolute marker levels. In the baseline-referenced model, $\exp(\alpha_{\Delta})$ concerns a one-unit greater departure from each subject’s own baseline.

The distinction can be made algebraically. The change-referenced term is $\alpha_{\Delta}m_i(t)-\alpha_{\Delta}m_i(0)$. It therefore depends on both the current value and the baseline value. A general model that includes both current and baseline marker values could assign them separate coefficients, but the baseline-referenced model imposes a particular relationship between those coefficients. It is not simply an absolute-value model with a different verbal interpretation.

\alpha_{\Delta}\{m_i(t)-m_i(0)\} = \alpha_{\Delta}m_i(t)-\alpha_{\Delta}m_i(0).

Thus, if two subjects have the same current marker but different baseline markers, the absolute-value association terms are equal, whereas their baseline-referenced terms may differ. If they have the same change from baseline but different current values, the baseline-referenced terms are equal, whereas their absolute-value terms may differ. In an actual fitted analysis, the baseline hazards and association coefficients are estimated under each model; hazard ratios from separately fitted models should not be compared as if every other part of the models were identical.

Worked Example: Same Current Value, Different Baselines

Suppose two hypothetical subjects have the same covariates. At a particular follow-up time, subject A has baseline marker $m_A(0)=5$ and current marker $m_A(t)=8$; subject B has baseline marker $m_B(0)=7$ and current marker $m_B(t)=8$. Consider a baseline-referenced model with $\alpha_{\Delta}=0.18$ per marker unit.

Their changes are $\Delta_A(t)=8-5=3$ units and $\Delta_B(t)=8-7=1$ unit. The difference in changes is $3-1=2$ units, so the hazard ratio due to the association term is

\exp\{0.18(3-1)\} = \exp(0.36) \approx 1.433.

Under this model, subject A’s hazard is about 1.43 times subject B’s hazard at that time, conditional on the same covariates and the model terms shown. This is the comparison implied by the two-unit greater change from baseline. Under an absolute-value link, the subjects’ current values are both 8, so their association terms would be equal and the hazard ratio from that term alone would be 1. The example isolates the association term; other model terms could still distinguish the subjects.

Worked Example: Comparing Different Changes

Consider two hypothetical subjects observed at the same follow-up time, with the same covariates. Subject C has $m_C(0)=4.2$ and $m_C(t)=5.0$, while subject D has $m_D(0)=3.0$ and $m_D(t)=4.0$. Suppose $\alpha_{\Delta}=0.25$ per marker unit.

Subject C’s change is $5.0-4.2=0.8$ units, and subject D’s change is $4.0-3.0=1.0$ unit. D’s change exceeds C’s by $1.0-0.8=0.2$ units. The hazard ratio for D relative to C under the change-referenced link is therefore

\frac{h_D(t)}{h_C(t)} = \exp\{0.25(0.2)\} = \exp(0.05) \approx 1.051.

Conditional on the model terms, D’s hazard is about 1.05 times C’s hazard, or approximately 5.13% higher. For comparison, their absolute current values differ by $4.0-5.0=-1.0$ unit. If an absolute-value model had $\alpha_m=0.25$, its association-term hazard ratio for D relative to C would be $\exp\{0.25(-1)\}=\exp(-0.25)\approx0.7788$. The two links give different comparisons because D has a slightly larger increase from baseline but a lower current marker value.

Model Assumptions and a Short Justification

The hazard-ratio result follows directly from the specified model, but its interpretation depends on the model being appropriate. In particular, the association is linear on the log-hazard scale in $\Delta_i(t)$; the same $\alpha_{\Delta}$ applies over follow-up; and the conditional hazard has the stated proportional form. The marker and its baseline reference must be defined on the same scale and at a meaningful common time origin. The joint model must also represent the longitudinal trajectories and their dependence with event risk adequately. As in standard time-to-event modeling, censoring assumptions must be appropriate for the likelihood used.

To justify the hazard ratio, write the model for two subjects $a$ and $b$ at the same time. If their $w$ vectors are equal, their baseline hazard $h_0(t)$ and covariate contribution $\gamma^\mathsf{T}w$ cancel in the ratio. Subtracting their log hazards leaves only the difference in the association terms:

\begin{aligned} \log h_a(t)-\log h_b(t) &=\alpha_{\Delta}\Delta_a(t)-\alpha_{\Delta}\Delta_b(t)\\ &=\alpha_{\Delta}\{\Delta_a(t)-\Delta_b(t)\}. \end{aligned}

Exponentiating both sides gives $\exp\{\alpha_{\Delta}[\Delta_a(t)-\Delta_b(t)]\}$. For a one-unit difference, this reduces to $\exp(\alpha_{\Delta})$. The cancellation requires the subjects to share the same baseline hazard at that time and the same values of the other covariates being held fixed.

Worked Example: Interpreting a Negative Association

Suppose a hypothetical joint model estimates $\alpha_{\Delta}=-0.12$ per marker unit. Compare two otherwise similar subjects whose changes differ by 2 units, with the first subject having the greater change. The hazard ratio for the first subject relative to the second is

\exp\{-0.12(2)\} = \exp(-0.24) \approx 0.7866.

Under the model, the first subject’s hazard is about 0.787 times the second subject’s hazard, or approximately 21.3% lower, at the same time and with other terms held fixed. This interpretation is about a two-unit difference in change from baseline, not a two-unit difference in current marker level. The negative coefficient indicates an inverse association on the model’s log-hazard scale; it does not establish that increasing the marker would reduce event risk.

Diagnostics, Limitations, and Common Pitfalls

A baseline-referenced link is useful when the scientific question concerns departures from an individual’s starting level. It can also be useful when absolute starting levels are not the intended focus. But subtracting baseline does not automatically remove confounding, correct measurement error, or make the association causal. The baseline value is estimated as part of the longitudinal trajectory, and uncertainty in that trajectory should be propagated through the joint model rather than ignored.

Model assessment should examine whether the chosen longitudinal trajectory describes the marker data over the relevant time range and whether the fitted event model captures the observed event patterns. Investigate the plausibility of a linear log-hazard association in change and a constant association parameter over time. Sparse or noisy baseline measurements can make $m_i(0)$ weakly informed; a poorly estimated baseline reference can then affect every calculated change for that subject. Extrapolation is especially concerning when the trajectory at time zero is far from observed measurement times.

A further limitation is structural: the model uses the difference $m_i(t)-m_i(0)$ with a single coefficient. It does not separately estimate an unrestricted effect of current level and an unrestricted effect of baseline level. If both may be associated with hazard in different ways, the change-only link may be too restrictive. Conversely, including additional terms changes the interpretation and requires sufficient information to estimate them.

Mistake: “A one-unit higher current marker gives a hazard ratio of $\exp(\alpha_{\Delta})$.” This is not generally the interpretation of a change-referenced coefficient. The correct comparison is a one-unit greater change from baseline, conditional on the other model terms.
Mistake: substituting the observed baseline measurement for $m_i(0)$ without accounting for error. An observed value may be noisy. The definition uses the latent trajectory’s baseline value, as estimated under the longitudinal model.
Mistake: treating the difference as a slope. $m_i(t)-m_i(0)$ is a difference in marker units over the interval from baseline to $t$; it is not a rate per unit time.
Mistake: interpreting the association causally. A hazard ratio from this model describes a conditional association under its assumptions. It does not, by itself, describe what would happen if the marker were intervened upon.

The main alternative considered here is the absolute current-value link. Choose between them based on the intended scientific comparison, not on which coefficient appears more convenient. The absolute link asks about higher marker levels at the same time; the baseline-referenced link asks about greater departures from each subject’s own baseline. Neither comparison is universally preferable.

Check Your Understanding

Answer each question using the model’s conditional interpretation. Solutions follow the questions.

If $\alpha_{\Delta}=0.10$, what hazard ratio corresponds to a one-unit greater change from baseline?
Two subjects have the same current marker value but different baseline values. Which link can assign them different association terms: the absolute-value link, the baseline-referenced link, or both?
At baseline, what is $\Delta_i(0)$, and what does that imply for the direct baseline-referenced association term?
Why is a noisy observed baseline measurement not automatically the same as $m_i(0)$?

Solutions. (1) The hazard ratio is $\exp(0.10)\approx1.105$, about 1.11 for a one-unit greater change, with other terms held fixed. (2) The baseline-referenced link can assign different association terms because the changes differ; the absolute-value link gives equal association terms when current values are equal. (3) $\Delta_i(0)=m_i(0)-m_i(0)=0$, so the direct association term is zero at baseline, though other hazard terms may differ. (4) The observed measurement can include measurement error, whereas $m_i(0)$ is the latent trajectory value at the defined baseline time.