Why the Association Scale Matters
When a longitudinal marker is binary, ordinal, or a count, its observed values do not lie on an unconstrained continuous biomarker scale. A binary marker records one of two outcomes; an ordinal marker records ordered categories; and a count marker records nonnegative integers. In a joint model, a shared random effect can connect the marker process to the event hazard, but its units are not the units of the observed response. The association must therefore be interpreted on the scale actually used in the model.
For a binary marker, a random effect in a logistic model shifts the log odds of the outcome. For an ordinal marker, it shifts cumulative logits under a cumulative-link model. For a count marker with a log link, it shifts the log expected count. If the event hazard depends on that shared effect, its coefficient describes how a contrast in this model scale relates to a conditional hazard ratio. It does not directly describe the hazard change per additional observed positive result, category, or count.
The choice of \(q_i(t)\) is part of the model. It could be a random intercept, a random slope, or a specified combination of random effects. In some models the hazard instead includes the longitudinal linear predictor \(x_i(t)^T\beta+z_i(t)^Tb_i\). The interpretation then concerns a contrast in that predictor, not automatically a contrast in the random-effects vector alone. State which quantity enters the hazard before interpreting its coefficient.
Conditional Hazard-Ratio Interpretation
Suppose two otherwise identical conditional hazard evaluations differ by \(d\) units in \(q_i(t)\), with the baseline hazard and other covariates held fixed. Under the proportional-hazards form above, the ratio of their hazards is
This is a conditional hazard ratio: it compares instantaneous event rates at time \(t\), conditional on the modeled random effects and covariates. For a one-unit contrast in \(q_i(t)\), the ratio is \(\exp(\alpha)\). A positive \(\alpha\) means a higher value of the specified contrast is associated with a higher conditional hazard; a negative \(\alpha\) means a lower conditional hazard. Neither statement alone establishes a causal effect.
Assumptions Behind the Interpretation
The algebraic hazard-ratio contrast follows from the stated hazard model, but its scientific interpretation relies on the model being suitable for the data. In particular, the marker distribution and link must be appropriate; the random-effects structure and the function \(q_i(t)\) must be specified meaningfully; and the event hazard must have the stated conditional proportional-hazards form. The proportional-hazards assumption here means that, for a fixed contrast \(d\), the conditional hazard ratio \(\exp(\alpha d)\) does not vary with time unless the model explicitly allows a time-varying association.
A standard shared-random-effects interpretation also relies on the joint model’s conditional structure: after conditioning on the shared random effects and modeled covariates, the longitudinal observations and event process are treated as independent. Censoring must be handled under an appropriate assumption, commonly that it is independent of the event time conditional on the modeled information. These are substantive modeling assumptions, not consequences of the hazard-ratio calculation. If important common causes or dependence remain unmodeled, \(\alpha\) may not capture the intended association.
For generalized longitudinal models, the link scale is often useful because it is the scale on which the conditional mean model is linear in its predictors. It is not generally a directly observed biomarker scale. In particular, changing \(b_i\) can change a binary probability or ordinal category probability nonlinearly, and a one-unit change in a log-count predictor multiplies the expected count rather than adding one to it.
Worked Example: Binary Marker and a Logit-Scale Random Intercept
Suppose, at a fixed time and covariate pattern, a binary marker \(Y_i\) has model \(\operatorname{logit}\{P(Y_i=1\mid b_i)\}=b_i\). The hazard is \(\lambda_i(t\mid b_i)=\lambda_0(t)\exp(0.4b_i)\). Compare \(b_i=0\) with \(b_i=1\).
Marker probabilities. At \(b_i=0\), \(P(Y_i=1\mid b_i=0)=1/(1+\exp(0))=0.5000\). At \(b_i=1\), \(P(Y_i=1\mid b_i=1)=1/(1+\exp(-1))=0.7311\). If the signed difference is defined as the probability at \(b_i=0\) minus the probability at \(b_i=1\), then
The negative sign reflects that subtraction order: the probability of \(Y_i=1\) is 0.2311 higher at \(b_i=1\) than at \(b_i=0\), a difference of 23.11 percentage points. This is a response-probability contrast, not the meaning of the hazard coefficient.
Conditional hazard ratio. The random-effect contrast is \(d=1\), so the ratio is \(\exp(0.4\times1)=1.492\). Conditional on the model’s random effects and covariates, the hazard at the higher random intercept is 1.492 times the hazard at the lower random intercept. Equivalently, it is 49.18% higher. The model does not say that observing \(Y_i=1\) rather than \(Y_i=0\) multiplies the hazard by 1.492.
Ordinal Markers: Threshold-Specific Probabilities
For an ordinal marker, a cumulative-link model describes probabilities at category thresholds. Under a proportional-odds cumulative-logit parameterization, one common form is \(\operatorname{logit}\{P(Y_i(t)\leq k\mid b_i)\}=\theta_k-\eta_i(t)\), where \(\theta_k\) is a threshold and \(\eta_i(t)\) includes the shared effect. With this sign convention, increasing \(\eta_i(t)\) decreases the probability of being at or below a threshold and increases the probability of being above it. The sign convention matters: a model written with the opposite sign requires the corresponding interpretation to be reversed.
An ordinal response has several category probabilities, and these probabilities need not change by the same amount. A proportional-odds coefficient describes a common shift in cumulative odds across thresholds; it is not a constant difference in the probability of each category. If the hazard depends on a shared component of \(\eta_i(t)\), its association parameter still yields a conditional hazard ratio for a specified contrast in that component.
Worked Example: An Ordinal Marker With a Shared Predictor
Suppose \(Y_i\) has categories 1, 2, and 3, with \(\operatorname{logit}\{P(Y_i\leq k\mid b_i)\}=\theta_k-b_i\). Let \(\theta_2=0\), so \(P(Y_i>2\mid b_i)=1-P(Y_i\leq2\mid b_i)\). Suppose the event hazard is \(\lambda_i(t\mid b_i)=\lambda_0(t)\exp(0.25b_i)\). Compare \(b_i=0\) and \(b_i=1\).
At \(b_i=0\), \(P(Y_i>2\mid b_i=0)=1-1/(1+\exp(0))=0.5000\). At \(b_i=1\), \(P(Y_i>2\mid b_i=1)=1-1/(1+\exp(1))=0.7311\). Thus, for this coding and threshold, the probability of the highest category increases by \(0.7311-0.5000=0.2311\) as \(b_i\) increases from 0 to 1. The probabilities are determined by the marker model’s sign and threshold; they are not read directly from the event association coefficient.
The conditional hazard ratio for the same one-unit contrast is \(\exp(0.25)=1.284\). Thus the higher value of the specified shared predictor is associated with a conditional hazard 1.284 times as large. The marker probabilities and hazard ratio describe different model components, and one does not numerically determine the other.
Count Markers: A Log-Mean Contrast
For a count marker modeled with a log link, \( \log E\{Y_i(t)\mid b_i\}=x_i(t)^T\beta+z_i(t)^Tb_i\). A one-unit increase in the linear predictor multiplies the conditional expected count by \(\exp(1)\), holding the other terms fixed. It is not an increase of one count. If the hazard uses the same one-dimensional shared contrast, its association parameter translates that contrast into a hazard ratio separately.
Worked Example: Count Marker and Opposite-Signed Associations
At a fixed time and exposure, suppose \(\log E\{Y_i(t)\mid b_i\}=\log(2)+b_i\), and \(\lambda_i(t\mid b_i)=\lambda_0(t)\exp(-0.3b_i)\). Compare \(b_i=0\) and \(b_i=1\).
At \(b_i=0\), the conditional expected count is \(\exp\{\log(2)\}=2.000\). At \(b_i=1\), it is \(\exp\{\log(2)+1\}=2\exp(1)=5.437\). The expected count ratio is \(5.437/2.000=2.718\), while the increase in expected count is \(5.437-2.000=3.437\) counts at the stated measurement occasion. These are model-based expectations, not a claim that each individual’s observed count changes by exactly that amount.
For the event process, the one-unit random-effect contrast gives \(\exp(-0.3)=0.7408\). Conditional on the modeled random effects and covariates, the hazard is 0.7408 times as large at \(b_i=1\) as at \(b_i=0\), or 25.92% lower. This example shows that a predictor can be associated with a higher expected count and a lower event hazard; the signs describe the two model components, not a single direct response-scale effect.
Choosing and Checking the Association
The shared-random-effect association is only one possible way to connect a discrete marker and the event process. The table contrasts it with common alternatives. The relevant choice is scientific and structural: what aspect of the marker process is intended to carry the association, and on what scale is that quantity defined?
| Hazard association target | Interpretation | Main caution |
|---|---|---|
| Specified random-effect contrast \(q_i(t)\) | Conditional hazard ratio for a stated contrast in the shared random effects | The random effect is latent; its units depend on the marker model and coding |
| Longitudinal linear predictor \(\eta_i(t)\) | Conditional hazard ratio for a stated contrast in the link-scale predictor | The predictor can include fixed effects as well as random effects |
| Response-scale conditional mean | Association with a probability, category summary, or expected count | For discrete markers this scale can be nonlinear, bounded, or threshold-dependent |
The response-scale approach can be more immediately interpretable—for example, a change in a predicted probability—but it is not interchangeable with a link-scale shared-effect association. Its hazard-ratio interpretation depends on the precise response-scale function included in the hazard. Conversely, a link-scale coefficient may be simpler and stable to specify but requires careful explanation of the link and contrast.
Diagnostics should assess both parts of the joint specification. Check whether the chosen binary, ordinal, or count model fits the observed marker patterns; assess whether the random-effects structure captures relevant between-individual variation; and investigate whether the conditional proportional-hazards form is plausible. For ordinal proportional odds, examine whether the common slope across thresholds is tenable. For counts, assess overdispersion and excess zeros when relevant. Examine alternative codings and plausible association functions, because an association with a random intercept need not represent an association with a slope or current predictor.
Common Mistakes and Limits
- Calling the random effect an observed biomarker. It is latent and model-defined. State its relationship to the marker’s link predictor rather than describing it as a measured marker value.
- Interpreting \(\exp(\alpha)\) per observed positive result or count. It is the hazard ratio for a one-unit contrast in the exact quantity multiplied by \(\alpha\). That may be a logit-scale random effect, not a one-unit response change.
- Ignoring sign conventions. Cumulative-link equations can use different signs. Derive whether a larger predictor increases or decreases the probability of higher categories from the specified equation.
- Calling a conditional association causal. Shared random effects encode modeled dependence. Without a causal design and additional assumptions, \(\alpha\) is not a causal effect.
- Assuming good fit for one submodel validates the joint model. Marker fit, event-model fit, conditional dependence, and censoring assumptions each require attention.
A shared-effect association also has limits. The hazard association can be sensitive to the random-effects distribution, scale, and chosen function \(q_i(t)\). Different parameterizations may yield coefficients with different units. Further, conditional hazard ratios do not in general equal marginal hazard ratios after averaging over random effects. The interpretation here is deliberately conditional; it should not be extended to population-level comparisons without the appropriate marginal calculation.
Check Your Understanding
Answer each question using the model scale and the stated contrast. Solutions follow the questions.
- A binary marker has logit probability \(b_i\), and the hazard includes \(0.2b_i\). What is the conditional hazard ratio for a one-unit increase in \(b_i\)?
- An ordinal model is written \(\operatorname{logit}\{P(Y_i\leq k\mid b_i)\}=\theta_k-b_i\). Does increasing \(b_i\) increase or decrease the probability of being above threshold \(k\)?
- A count model has \(\log E(Y_i\mid b_i)=\log(3)+b_i\). What is the expected count at \(b_i=1\), to three significant figures, and what is its ratio to the expectation at \(b_i=0\)?
- In a hazard model with association coefficient \(\alpha=-0.4\), what is the conditional hazard ratio for a two-unit increase in the specified shared-effect contrast?
- Why is it incorrect to call a hazard ratio based on a one-unit logit-scale random-effect contrast the hazard ratio for observing a positive rather than negative marker?
Solutions. 1. The ratio is \(\exp(0.2)=1.221\), conditional on the modeled random effects and covariates. 2. It increases: \(\operatorname{logit}P(Y_i\leq k)\) falls as \(b_i\) rises, so \(P(Y_i\leq k)\) falls and \(P(Y_i>k)\) rises. 3. At \(b_i=1\), the expectation is \(3\exp(1)=8.155\); at \(b_i=0\), it is 3, so the ratio is \(\exp(1)=2.718\). 4. The ratio is \(\exp\{(-0.4)(2)\}=\exp(-0.8)=0.4493\). 5. The random effect is a latent logit-scale quantity, not the observed binary outcome; the probability contrast between outcomes depends on the marker model and covariates, while the hazard ratio is conditional on a contrast in the specified latent quantity.