Interpreting Hazard Ratios for Marker Differences

Why the Comparison Must Be Specified

A hazard ratio for a marker difference is not determined by the two marker values alone unless the comparison also specifies what is held fixed. In a joint model, the event hazard depends on a latent marker trajectory and may also depend on other predictors. The useful question is therefore precise: at the same time, and for the same values of the other survival predictors, how does the conditional hazard change when the latent marker value differs?

This tutorial answers that question for a current-value link. The marker values are latent values from the longitudinal model, not noisy measurements substituted directly into the hazard. The result is a conditional hazard ratio: it compares hazards given the relevant latent trajectory information and covariates. It is not, by itself, a marginal comparison between groups of subjects.

Model and Conditions for the Comparison

Definition: Let $m_i(t;b_i)$ be subject $i$'s latent marker value at time $t$, determined by that subject's longitudinal trajectory and random effects $b_i$. For a current-value link, suppose the conditional hazard is $$ h_i(t\mid b_i,z_i)=h_0(t)\exp\{\gamma^\mathsf{T}z_i+\alpha m_i(t;b_i)\}, $$ where $h_0(t)$ is the common baseline hazard, $z_i$ is the vector of other survival predictors, $\gamma$ is its coefficient vector, and $\alpha$ is the association coefficient for the current latent marker value.

This model states that, conditional on the latent trajectory and the other predictors, the log hazard is the log baseline hazard plus a linear predictor. The marker enters through its value at the time being considered: a marker value from an earlier visit is not substituted for $m_i(t;b_i)$ unless it is in fact the latent value at time $t$. The conditional proportional-hazards form means that the marker contribution is multiplied into the hazard through $\exp\{\alpha m_i(t;b_i)\}$, with $\alpha$ not varying over time in this model.

For a clean marker contrast, compare two conditional states, A and B, at the same time $t$. Hold $z$ fixed, so the other survival predictors have equal values in both states. The common baseline hazard then also has the same value. If an extended model includes an additional direct effect of a random effect or another predictor, that term must also be held fixed for a marker-only contrast. The calculation below concerns the stated model, in which the random effects affect the hazard through the current marker value.

Deriving the Conditional Hazard Ratio

Write $m_A$ and $m_B$ for the two latent marker values at time $t$, and let $z$ be the common value of the other survival predictors. The conditional hazard ratio comparing A with B is the hazard in state A divided by the hazard in state B.

\begin{aligned} \frac{h_A(t\mid z)}{h_B(t\mid z)} &= \frac{h_0(t)\exp\{\gamma^\mathsf{T}z+\alpha m_A\}} {h_0(t)\exp\{\gamma^\mathsf{T}z+\alpha m_B\}}\\ &= \exp\{\alpha(m_A-m_B)\}. \end{aligned}

The common baseline hazard cancels because both hazards are evaluated at the same time. The contribution from $z$ cancels because those predictors are held fixed. Thus the marker contrast depends on the difference $m_A-m_B$, not on either marker value in isolation.

Key result: Under the stated current-value model, the conditional hazard ratio for A versus B at the same time, with other survival predictors held fixed, is $$ \mathrm{HR}_{A:B}(t)=\exp\{\alpha(m_A-m_B)\}. $$ A positive $\alpha$ means a higher current marker value corresponds to a higher conditional hazard; a negative $\alpha$ means it corresponds to a lower conditional hazard. These are model-based conditional associations, not statements of causation.

The order of the comparison matters. Switching from A versus B to B versus A changes the sign of the marker difference and gives the reciprocal hazard ratio: $$ \mathrm{HR}_{B:A}(t)=\exp\{\alpha(m_B-m_A)\} =\frac{1}{\mathrm{HR}_{A:B}(t)}. $$

Worked Interpretations

Worked Example: A Higher Marker Value With Positive Association

Suppose the conditional hazard model contains a treatment indicator $z_i$, and the estimated marker association is $\alpha=0.40$ per marker unit. At a particular time, compare two conditional states with latent marker values $m_A=8.0$ units and $m_B=6.5$ units. Both states have the same treatment value and all other survival predictors are equal.

The marker difference is $8.0-6.5=1.5$ marker units. Therefore, $$ \mathrm{HR}_{A:B}=\exp\{0.40(8.0-6.5)\} =\exp(0.60) \approx 1.822. $$

At that time, state A has an estimated conditional hazard about 1.82 times the hazard in state B, or about 82.2% higher, given the model and the fixed predictors. The comparison concerns the instantaneous event rate among subjects still at risk at that time; it is not a comparison of event probabilities over a follow-up interval.

Worked Example: A Lower Marker Value With Negative Association

Suppose $\alpha=-0.25$ per marker unit. At the same time and with the same values of all other survival predictors, state A has marker value $m_A=12$ units and state B has $m_B=15$ units. Calculate the hazard ratio for A versus B.

The signed marker difference is $12-15=-3$ units. Hence, $$ \mathrm{HR}_{A:B} =\exp\{(-0.25)(12-15)\} =\exp(0.75) \approx 2.117. $$

Although A has the lower marker value, its modeled conditional hazard is about 2.12 times B's. This direction is consistent with a negative $\alpha$: lower marker values correspond to higher hazard under the model. Reversing the comparison gives $\mathrm{HR}_{B:A}=1/2.117\approx0.472$, the reciprocal.

Worked Example: When Another Predictor Is Not Held Fixed

Suppose the model is $h_i(t)=h_0(t)\exp\{0.50z_i+0.20m_i(t)\}$, where $z_i$ is a binary treatment indicator and the marker is measured in its stated units. At a common time, state A has $z_A=1$ and $m_A=9$; state B has $z_B=0$ and $m_B=7$. Calculate the full conditional hazard ratio and separate its two contributions.

The full comparison is $$ \frac{h_A(t)}{h_B(t)} =\exp\{0.50(1-0)+0.20(9-7)\} =\exp(0.90) \approx 2.460. $$

The marker-only contribution, if treatment were held fixed, would be $\exp\{0.20(9-7)\}=\exp(0.40)\approx1.492$. The treatment contribution is $\exp\{0.50(1-0)\}=\exp(0.50)\approx1.649$; multiplying them gives approximately $1.492\times1.649=2.460$. The full ratio is not a marker-only hazard ratio because treatment differs between the states. To answer a question about the marker difference alone, compare states with the same treatment value.

Units, Magnitude, and Uncertainty

The units of $\alpha$ are the inverse of the marker's units, so that $\alpha(m_A-m_B)$ is dimensionless, as required by the exponential. For example, if the marker is measured in milligrams per litre, $\alpha$ is expressed per milligram per litre. Always use marker values in the same units as those used to estimate $\alpha$. Changing the marker's units without changing the coefficient produces an incorrect hazard ratio.

A one-unit increase has conditional hazard ratio $\exp(\alpha)$, but a comparison across $d$ units has ratio $\exp(\alpha d)$. These are not generally the same number. For instance, a per-unit ratio of $\exp(0.20)$ becomes $\exp(0.40)$ over a two-unit difference. The exponentiation also ensures that a hazard ratio is positive; a negative association coefficient does not imply a negative hazard ratio.

If an interval estimate for $\alpha$ is $[L,U]$ and the marker difference $d=m_A-m_B$ is treated as fixed, exponentiating the endpoints gives an interval for $\exp(\alpha d)$. For $d>0$, the endpoints are $[\exp(Ld),\exp(Ud)]$. For $d<0$, their order reverses, so the ordered interval is $[\exp(Ud),\exp(Ld)]$. This transformation does not remove uncertainty in estimated latent marker values; that uncertainty must be accounted for when the marker contrast itself is estimated rather than fixed.

Checks, Alternatives, and Limitations

Before reporting a marker hazard ratio, check that the comparison is at a common time, that the other survival predictors are equal for a marker-only contrast, and that the latent marker values and coefficient use compatible units. Also check the direction explicitly: the numerator state must match the order in $m_A-m_B$. A quick sanity check is that reversing the comparison must produce the reciprocal ratio.

A common mistake is to say, “The marker rose by two units, so the hazard ratio is $\exp(\alpha)$.” That uses the one-unit formula for a two-unit contrast. The correction is to multiply the coefficient by the actual difference: the ratio is $\exp\{2\alpha\}$. Another mistake is to describe this conditional ratio as the population-average hazard ratio. Conditioning on the latent trajectory is not the same as averaging over the distribution of random effects; marginal comparisons can have different behavior, as discussed in the earlier material on conditional and marginal hazards.

The main alternative to a marker-only comparison is a full predictor contrast, which includes differences in treatment or other covariates as in the third example. That answers a different question. A marker-only contrast isolates the model's marker term by holding those predictors fixed; a full contrast describes the combined change in all predictors that differ. Neither calculation alone establishes that changing the marker would cause a change in event hazard.

The formula is only as appropriate as the specified current-value model. It assumes the marker contribution is linear on the log-hazard scale and that $\alpha$ is constant over time. If the model instead uses a different functional form, such as a nonlinear transformation of the marker, this particular difference formula need not apply. The hazard ratio also concerns conditional instantaneous hazards, not cumulative event risk, and it does not quantify how accurately the latent marker values have been estimated.

Key takeaway: For a current-value link, compare two latent marker values at the same time and hold the other survival predictors fixed. The conditional hazard ratio for A versus B is $\exp\{\alpha(m_A-m_B)\}$. State the comparison order, marker units, and conditioning context when interpreting it.

Check Your Understanding

For each question, identify the relevant contrast before interpreting the result. Solutions follow the questions.

If $\alpha=0.30$ per marker unit, what is the conditional hazard ratio for a three-unit higher marker value, with time and other predictors fixed?
If the hazard ratio for A versus B is 0.60, what is the hazard ratio for B versus A?
Two states differ in marker value and treatment. Can their full hazard ratio be called a marker-only hazard ratio? What must be done to obtain a marker-only comparison?
Why must both hazards in a marker-only contrast be evaluated at the same time?

Solutions. (1) The difference is $3$ units, so the ratio is $\exp\{0.30(3)\}=\exp(0.90)\approx2.460$. The higher-marker state has about 2.46 times the conditional hazard. (2) Reverse the comparison: $1/0.60\approx1.667$. (3) No. The full ratio includes the treatment difference; compare states with treatment and all other survival predictors held equal, then use $\exp\{\alpha(m_A-m_B)\}$. (4) The baseline hazard is $h_0(t)$, which cancels only when both hazards use the same time. Comparing different times would also include any change in the baseline hazard.