Why Scale Errors Matter
In a joint model, the longitudinal submodel describes a marker over time, while the event submodel relates event hazard to covariates and often to a feature of the marker trajectory. For a Gaussian marker, a linear predictor and conditional mean may coincide. For a binary or count marker, they generally do not. Mixing those quantities can change predicted values, coefficient interpretations, and the scientific meaning of the association with event risk.
The key discipline is to label each quantity before using it: the observed response, the conditional mean on the response scale, the linear predictor on the link scale, and the particular feature placed in the event model. These are related, but they are not interchangeable.
For a binary response with a logit link, \(\mu_i(t)\) is a conditional probability and \(\eta_i(t)\) is its log-odds. For a Poisson count model with a log link, \(\mu_i(t)\) is a conditional expected count (or rate, depending on the model and exposure), and \(\eta_i(t)\) is its log. The realized \(Y_i(t)\) remains a zero-or-one value or a nonnegative integer; it is not the same thing as the conditional mean.
Keep the Longitudinal Scales Separate
A common source of error is to treat a generalized linear predictor as though it were a fitted response. The inverse link is essential. For example, if a binary marker has \(\eta_i(t)=0.8\), the predicted probability is not 0.8. Under the logit link it is \(1/\{1+\exp(-0.8)\}\), approximately 0.6900.
Worked Example: A Binary Marker’s Predictor Is Not Its Probability
Suppose a hypothetical binary biomarker is modeled conditionally on random effects with a logit link. At a particular time for a particular subject, the fitted predictor is \(\eta_i(t)=-0.4\). Find the conditional probability of a positive marker, and explain the relevant scales.
The logit model gives \(\eta_i(t)=\log\{\mu_i(t)/(1-\mu_i(t))\}\). Applying the inverse-logit function, \(\mu_i(t)=1/\{1+\exp[-\eta_i(t)]\}\).
Substituting the fitted predictor, \(\mu_i(t)=1/\{1+\exp(0.4)\}=1/(1+1.4918)=0.4013\), to four significant figures. The model therefore assigns a conditional probability of about 0.4013 to \(Y_i(t)=1\). A future observed value is still either 0 or 1; it is not 0.4013. The number \(-0.4\) is the log-odds, not the probability.
As a check, the odds corresponding to probability 0.4013 are \(0.4013/(1-0.4013)\approx0.6703\), and \(\log(0.6703)\approx-0.4000\). This verifies the link-scale calculation.
The distinction also affects changes over time. With a logit link, the change in \(\eta_i(t)\) is a change in log-odds, not a constant change in probability. Differentiating \(\mu_i(t)=\{1+\exp[-\eta_i(t)]\}^{-1}\) gives
The multiplier \(\mu_i(t)\{1-\mu_i(t)\}\) varies with the probability. Consequently, the same link-scale slope can correspond to different probability-scale slopes at different times or for different subjects. A slope on the logit scale should not be described as a fixed percentage-point change per unit time.
Worked Example: A Count Mean Is Not a Log Mean
Consider a hypothetical Poisson longitudinal submodel with log link and conditional mean \(\mu_i(t)=\exp\{\eta_i(t)\}\). At two times, the fitted predictors are 0.7 and 1.2. Find the corresponding conditional means and their ratio.
At the first time, \(\mu_i(t)=\exp(0.7)=2.0138\), or approximately 2.014 expected counts per observation under the model. At the second time, \(\mu_i(t)=\exp(1.2)=3.3201\), or approximately 3.320 expected counts.
The predictor difference is \(1.2-0.7=0.5\). The ratio of conditional means is \(\exp(1.2)/\exp(0.7)=\exp(0.5)=1.6487\). Thus the second conditional mean is about 1.649 times the first, a relative increase of about 64.87%, not an increase of 0.5 counts. The observed count at either time remains an integer; a conditional mean need not be an integer.
This calculation assumes the specified log-link mean model is appropriate and that both predictors refer to the same exposure scale. If exposure differs, its offset or rate definition must also be handled consistently before interpreting a mean or mean ratio.
Check the Event-Association Feature
A typical conditional event model can be written
where \(h_i(t\mid b_i)\) is the subject-specific hazard conditional on the random effects, \(h_0(t)\) is the baseline hazard, \(w_i\) is a vector of event covariates, and \(f_i(t)\) is the chosen longitudinal feature. The association coefficient \(\alpha\) is interpreted per one-unit increase in that feature, holding the other terms fixed. The choice of \(f_i(t)\) is a modeling decision, not an automatic consequence of having a longitudinal submodel.
For example, an event model might use the response-scale conditional mean \(f_i(t)=\mu_i(t)\), the link-scale predictor \(f_i(t)=\eta_i(t)\), or another defined feature. With \(f_i(t)=\mu_i(t)\), the hazard ratio for a contrast from \(\mu_a\) to \(\mu_b\) is \(\exp\{\alpha(\mu_b-\mu_a)\}\). With \(f_i(t)=\eta_i(t)\), it is \(\exp\{\alpha(\eta_b-\eta_a)\}\). The features, units, and resulting interpretations differ. In either case, the model ordinarily uses the specified latent or fitted trajectory feature, not a noisy realized binary or count observation substituted without justification.
Worked Example: Response-Scale and Link-Scale Event Associations
Suppose a hypothetical binary marker has conditional probabilities 0.1813 and 0.3297 for two marker profiles at the same time, with corresponding logit predictors \(-1.508\) and \(-0.709\) (rounded). An event model has association coefficient \(\alpha=0.4\). Compare the hazard ratios if the event feature is the probability or the logit predictor.
For the response-scale feature \(f_i(t)=\mu_i(t)\), the contrast is \(0.3297-0.1813=0.1484\). The hazard ratio is \(\exp\{0.4(0.1484)\}=\exp(0.05936)=1.0612\). Under this specification, the contrast corresponds to a conditional hazard about 1.061 times as large, with the same baseline and covariates.
For the link-scale feature \(f_i(t)=\eta_i(t)\), the rounded predictor contrast is \(-0.709-(-1.508)=0.799\). The hazard ratio is \(\exp\{0.4(0.799)\}=\exp(0.3196)\approx1.3766\). The different answer is not a computational inconsistency: it follows from using a different feature and therefore a different coefficient interpretation. The comparison assumes the stated conditional hazard model and does not establish a causal effect.
The rounded probabilities and predictors are used only to illustrate the scale contrast. In an actual analysis, calculate the contrast from unrounded fitted values and state explicitly which feature enters the event model.
Hazard Ratios Are Not Risk Ratios
A second scale mistake is to call a hazard ratio a risk ratio. A hazard is an instantaneous event rate among those still event-free, whereas a risk is the probability of experiencing the event over a specified interval. Even under a constant hazard and no competing events, the two ratios generally differ because risk is a nonlinear function of cumulative hazard.
Worked Example: Constant Hazards and Two-Year Risks
Suppose, hypothetically, that two groups have constant event hazards of 0.10 and 0.20 per year over two years, with no competing risks. Under a constant hazard \(h\), event risk by time \(\tau\) is \(1-\exp(-h\tau)\). Calculate both ratios.
The hazard ratio is \(0.20/0.10=2.00\). The two-year risk for the first group is \(1-\exp\{-(0.10)(2)\}=1-\exp(-0.2)=0.1813\). For the second group it is \(1-\exp\{-(0.20)(2)\}=1-\exp(-0.4)=0.3297\).
Using the displayed risks, the risk ratio is \(0.3297/0.1813=1.8185\), which rounds to 1.819. Using the unrounded risks gives approximately 1.8187, also 1.819 to four significant figures. Thus the hazard ratio is 2.00, while the two-year risk ratio is about 1.819. The result depends on the follow-up horizon; do not substitute one measure for the other.
Assumptions, Diagnostics, and Alternatives
Scale interpretation is meaningful only within a coherent model specification. In a standard shared-random-effects joint model, the longitudinal response distribution and link must be suitable for the observed outcome; the random-effects structure must represent relevant subject-level heterogeneity; and the event hazard must be correctly specified conditional on the random effects and covariates. The chosen association feature must be defined from the trajectory and used consistently. Inference also depends on appropriate handling of censoring and on the likelihood or estimation method being valid for the assumed model. These assumptions are not verified merely by obtaining converged estimates.
Useful checks include comparing fitted conditional means with the observed response range and distribution, inspecting residual or simulation-based diagnostics appropriate to the response family, checking the exposure and offset used for counts, and examining whether predicted binary probabilities are plausible across time and subjects. For the event component, check the baseline hazard and covariate specification, and ensure the stated association feature matches the quantity actually passed to the hazard model. A coefficient whose reported unit does not match its feature is a warning sign.
The principal scale alternatives are not interchangeable shortcuts. A response-scale association may be attractive when risk is hypothesized to track a probability or expected count directly. A link-scale association may be useful when multiplicative changes in the mean or changes in log-odds are scientifically relevant. The choice should follow the scientific question and be stated in the model. A model using the observed \(Y_i(t)\) directly is yet another specification; because the observed response is noisy and discrete, it is not automatically equivalent to using \(\mu_i(t)\) or \(\eta_i(t)\).
Sensitivity analysis can compare defensible choices of \(f_i(t)\), but it does not make their coefficients directly comparable: units differ, and predictions may differ. Such comparisons should focus on model fit, calibration, and scientifically relevant predictions, while clearly identifying the feature and scale for each model. Limitations remain: a chosen link or association structure can be misspecified, sparse measurements may weakly identify a latent trajectory, and estimated associations are conditional model relationships rather than proof of causation.
A Practical Scale Audit
State whether the recorded marker is binary, a count, or another response, and identify its conditional distribution.
Record \(g\{\mu_i(t)\}=\eta_i(t)\) and use the inverse link before calling a quantity a probability, mean, or rate.
Write \(f_i(t)\) explicitly and interpret \(\alpha\) per unit of that feature, not per unit of an unmentioned scale.
Distinguish a hazard ratio at a time from an event-risk ratio over a specified follow-up interval.
Check Your Understanding
Answer each question, then compare with the brief solutions below.
- A binary marker has logit predictor \(\eta=0\). What is its conditional probability, and what values can the observed response take?
- For a Poisson log-link model, a predictor increases by 0.3. By what factor does the conditional mean change?
- An event model uses \(f_i(t)=\mu_i(t)\). What is the hazard ratio for a 0.10 increase in the conditional mean if \(\alpha=0.5\)?
- Why is a hazard ratio of 2 not generally a two-year risk ratio of 2?
- Name two checks that can reveal scale confusion in a non-Gaussian joint model.
Solutions. (1) The inverse logit of 0 is \(1/(1+\exp(0))=0.5\); the observed response is either 0 or 1. (2) The mean ratio is \(\exp(0.3)=1.350\), approximately, not an additive increase of 0.3 counts. (3) The hazard ratio is \(\exp\{0.5(0.10)\}=\exp(0.05)=1.051\), approximately, for that contrast with other terms held fixed. (4) Risk over a fixed interval is a nonlinear function of cumulative hazard, so the risk ratio depends on baseline hazard and follow-up duration. (5) Examples include checking fitted probabilities against the binary response scale, checking count means and exposure offsets, and verifying that the event feature in the implementation matches the stated response or link scale.