Why the target of an ordinal model matters
An ordinal outcome has categories with a meaningful order, such as mild, moderate, and severe symptoms. A model for repeated ordinal responses can describe how covariates relate to outcomes within a cluster at fixed random effects, or how they relate to outcomes averaged over a population of clusters. These are distinct questions. A conditional model is useful when the comparison concerns units sharing the same modeled cluster effect; a marginal model describes average outcome probabilities across the target population.
The link also determines the reported effect scale. In a cumulative-logit model, a coefficient is a log cumulative-odds contrast. In a cumulative-probit model, it is a contrast on the probit scale. Neither coefficient is, without further calculation, a change in the probability of any one category. The earlier material on GEE for repeated count outcomes introduced population-averaged mean models; here the mean structure is an ordered set of cumulative probabilities.
Conditional and marginal cumulative-logit models
A common conditional model is the proportional-odds cumulative-logit random-intercept model
Here \(\operatorname{logit}(p)=\log\{p/(1-p)\}\), \(\alpha_k\) is a cutpoint, \(\beta\) is a vector of covariate coefficients, and \(b_i\) is a cluster random intercept. The minus signs are a convention: a positive coefficient shifts probability toward higher categories. The cutpoints must satisfy \(\alpha_1<\cdots<\alpha_{K-1}\), so the cumulative probabilities increase with \(k\). The proportional-odds assumption means that the same \(\beta\) applies at every cutpoint.
For a one-unit increase in a covariate with coefficient \(\beta_r\), holding the other covariates and \(b_i\) fixed, the odds of being above rather than at or below any cutpoint are multiplied by \(\exp(\beta_r)\). This is a conditional odds ratio. The comparison is not a probability ratio, and it is not an odds ratio averaged across clusters.
A population-averaged cumulative-logit model instead has the form
The superscript \(M\) marks a marginal coefficient. In a GEE analysis, this model describes the marginal cumulative probabilities while a working association structure accounts for within-cluster dependence in estimation. Its coefficient is a population-averaged cumulative-odds contrast under the specified marginal proportional-odds model. It is not defined by conditioning on a particular cluster effect.
These formulations rely on different assumptions. A conditional GLMM specifies an ordinal response distribution given \(b_i\), a distribution for the random effects, and typically conditional independence of responses within a cluster given those effects. The random effects are usually assumed independent across clusters and, for the simple interpretation above, independent of the modeled covariates. A marginal GEE requires independent clusters and a correctly specified marginal mean structure for the target interpretation; its working correlation may be imperfect, with robust variance estimation providing protection for standard errors under suitable large numbers of independent clusters. Robust standard errors do not repair a misspecified marginal mean.
Worked Example: Reading a Conditional Proportional-Odds Coefficient
Suppose a hypothetical symptom outcome has categories 1 = mild, 2 = moderate, and 3 = severe. A fitted conditional model is
where \(x_{ij}=1\) denotes a treatment condition and \(x_{ij}=0\) a comparison condition. For illustration, take \(\alpha_1=-1\), \(\alpha_2=1\), and \(b_i=0\). The cutpoints are ordered, as required. At \(x=0\), the cumulative probabilities are \(\operatorname{logistic}(-1)=0.268941\) and \(\operatorname{logistic}(1)=0.731059\). Thus the category probabilities, calculated from the unrounded cumulative probabilities, are approximately \(0.2689\), \(0.4621\), and \(0.2689\).
At \(x=1\), the cumulative probabilities are \(\operatorname{logistic}(-1.8)=0.141851\) and \(\operatorname{logistic}(0.2)=0.549834\). The category probabilities are therefore approximately \(0.1419\), \(0.4080\), and \(0.4502\). Each set sums to 1 apart from rounding.
The coefficient interpretation is separate from these category probabilities. At either cutpoint, the conditional odds of being above the cutpoint rather than at or below it are multiplied by \(\exp(0.8)=2.226\) for \(x=1\) versus \(x=0\), holding \(b_i\) fixed. The larger severe-category probability in this illustration is compatible with that direction, but the odds ratio itself is not a multiplier for the severe-category probability.
Why conditional and marginal effects can differ
For a random-intercept model, the marginal cumulative probability is obtained by integrating the conditional probability over the random-effects distribution:
This integral averages probabilities, not linear predictors. Because the inverse-logit function is nonlinear, in general the result is not the inverse-logit evaluated at the average random effect. Even when \(E(b_i)=0\), it does not follow that the marginal cumulative-logit coefficient equals the conditional coefficient. This difference is not, by itself, evidence that either model is incorrectly fitted: the models target different summaries.
Worked Example: Averaging Over a Cluster Effect
Continue with a hypothetical three-category outcome and consider only the first cutpoint. Let \(\alpha_1=-1\), \(\beta=0.8\), and suppose the random intercept takes values \(-1\) and \(1\), each with probability \(1/2\). This deliberately simple distribution has mean zero. For \(x=0\), the conditional cumulative probabilities at the two random-effect values are \(\operatorname{logistic}(0)=0.500000\) and \(\operatorname{logistic}(-2)=0.119203\). Their average is
For \(x=1\), the corresponding probabilities are \(\operatorname{logistic}(-0.8)=0.310026\) and \(\operatorname{logistic}(-2.8)=0.057324\), giving
The marginal odds of being above category 1 are \(0.690399/0.309601=2.230\) for \(x=0\) and \(0.816325/0.183675=4.444\) for \(x=1\). The marginal odds ratio at this cutpoint is therefore approximately \(4.444/2.230=1.993\), not the conditional odds ratio \(\exp(0.8)=2.226\). This calculation averages over a stated random-effects distribution; it is not a claim that the two effects must differ by this amount in other models or populations.
The link determines the reported effect scale
The cumulative logit is not the only possible link. For example, a cumulative-probit model replaces the logit with the inverse standard normal cumulative distribution function \(\Phi^{-1}\):
A coefficient in this model is a change on the probit scale per unit change in a covariate, conditional on the random effects. It is not an odds ratio. In the usual latent-variable interpretation, the probit scale is expressed in standard-normal units; translating a coefficient into category probabilities requires the cutpoints and covariate values.
Worked Example: Interpreting a Probit-Scale Contrast
Consider a hypothetical three-category outcome with a conditional cumulative-probit model, \(b_i=0\), cutpoints \(\alpha_1=-1\) and \(\alpha_2=1\), and treatment coefficient \(\beta=0.5\). For \(x=0\), the cumulative probabilities are \(\Phi(-1)=0.158655\) and \(\Phi(1)=0.841345\). The category probabilities are thus \(0.1587\), \(0.6827\), and \(0.1587\).
For \(x=1\), the cumulative probabilities become \(\Phi(-1.5)=0.066807\) and \(\Phi(0.5)=0.691462\). The category probabilities are \(0.0668\), \(0.6247\), and \(0.3085\). The coefficient \(0.5\) is a half-unit shift on the probit scale for a one-unit treatment contrast, at fixed \(b_i\). It is not a 50% increase in odds. The computed probabilities show how the same scale-specific effect is translated into category probabilities at these cutpoints.
| Model or link | What the coefficient describes | What it does not directly describe |
|---|---|---|
| Conditional cumulative logit | Conditional log cumulative-odds contrast, holding random effects fixed | A population-averaged odds ratio or a category-probability difference |
| Marginal cumulative logit | Population-averaged log cumulative-odds contrast under the marginal model | A contrast conditional on a particular cluster effect |
| Cumulative probit | Contrast on the probit link scale; conditional or marginal according to the model | A cumulative odds ratio |
Diagnostics, alternatives, and limitations
The main alternative to a conditional ordinal GLMM is a marginal ordinal GEE when the scientific target is population-averaged and there are enough independent clusters for reliable large-sample inference. A GLMM makes the random-effects distribution and conditional response model explicit; a GEE specifies the marginal mean and uses a working association structure. These are not merely two computational routes to the same coefficient. Their coefficients can differ because their targets and, in nonlinear models, their scales of averaging differ.
The proportional-odds restriction is also a substantive modeling choice. Before relying on a single slope across cutpoints, examine whether fitted cumulative probabilities and residual patterns are compatible with that restriction. Check that estimated cutpoints are ordered, fitted cumulative probabilities increase with \(k\), and implied category probabilities are nonnegative. For a GLMM, assess whether the random-effects distribution and conditional model are plausible; for a GEE, inspect the marginal mean fit and dependence assumptions. Poor fit, sparse categories, few clusters, or strong sensitivity to modeling choices can make estimates and inference unreliable.
A frequent mistake is to call the conditional GLMM coefficient a population-averaged effect because the data came from a population sample. That reasoning confuses the sampling population with the model’s conditioning statement. The correction is to read the probability in the model: if it is conditioned on \(b_i\), the coefficient is conditional; if it describes probabilities averaged over clusters, it is marginal. Another mistake is to compare a logit coefficient directly with a probit coefficient as though both were in the same units. State the link and the conditioning target, and translate effects to probabilities at specified covariate values when those are the quantities of interest.
No link removes the need to define the outcome ordering, target population, covariates, and cluster structure. A marginal model may be easier to interpret for population-wide averages, but it does not by itself describe cluster-specific variation. A conditional model represents that variation through its random effects, but a conditional odds ratio does not directly give the average odds ratio across clusters. Report the model’s target and effect scale explicitly.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions.
- In a conditional cumulative-logit model with the form shown above, what does \(\exp(\beta_r)\) compare?
- Why can a mean-zero random intercept still produce marginal probabilities different from probabilities evaluated at \(b_i=0\)?
- A cumulative-probit coefficient is 0.4. Is it a cumulative odds ratio? State its scale.
- Name one probability check for fitted ordinal models and one assumption to examine in a conditional GLMM.
- Why does a marginal GEE coefficient not automatically describe a comparison between two observations with the same cluster effect?
Solutions. (1) It is the conditional odds ratio for being above rather than at or below any cutpoint, per one-unit covariate increase, holding other covariates and random effects fixed. (2) Averaging a nonlinear inverse link over a random-effects distribution is generally different from evaluating that inverse link at the mean random effect. (3) No; it is a 0.4-unit contrast on the probit scale, with the interpretation conditional or marginal according to the model. (4) Cumulative probabilities should increase across cutpoints and implied category probabilities should be nonnegative; for a GLMM, examine the assumed random-effects distribution and conditional response model. (5) The marginal model describes probabilities averaged over the target population rather than conditioning on a specified cluster effect.