Random Effects for Repeated Ordinal Responses

Why Repeated Ordinal Responses Need a Subject Effect

A pharmacodynamic outcome may be recorded at several visits as an ordered category, such as absent, mild, moderate, or severe. Measurements from the same subject are not generally independent: a subject with a tendency toward more severe responses may remain more likely to have severe responses across visits, even after accounting for measured exposure and time. A cumulative-logit mixed model represents that persistent, unobserved subject tendency with a shared random effect.

The key distinction is between the response model conditional on a subject’s random effect and probabilities averaged across subjects. The fixed-effect coefficients describe the conditional model. Marginal probabilities require integrating over the random-effect distribution, and repeated outcomes become dependent after that shared effect is integrated out.

Model Definition and Assumptions

Definition: Let $Y_{ij}$ be the ordinal pharmacodynamic response for subject $i$ at occasion $j$, with ordered categories $1,\ldots,K$. A cumulative-logit random-intercept model is $$ \operatorname{logit}\{P(Y_{ij}\leq k\mid b_i)\} =\alpha_k-\beta C_{ij}-b_i,\qquad k=1,\ldots,K-1, $$ where $\alpha_1<\cdots<\alpha_{K-1}$ are ordered thresholds, $C_{ij}$ is exposure, $\beta$ is its fixed effect, and the subject random intercept satisfies $b_i\sim N(0,\sigma_b^2)$. Conditional on $b_i$ and the covariates, the responses from a subject are assumed independent. Random effects are independent across subjects and independent of the covariates.

The minus signs establish the direction of the model. Increasing exposure by one unit multiplies the conditional odds of being at or below any threshold by $\exp(-\beta)$. Equivalently, it multiplies the conditional odds of being above that threshold by $\exp(\beta)$. Thus, with $\beta>0$, greater exposure shifts response toward higher categories. A positive $b_i$ also shifts response toward higher categories, because it lowers each cumulative probability.

The thresholds must be ordered so that cumulative probabilities increase with $k$. The proportional-odds assumption is that the same $\beta$ and the same random intercept apply at every threshold; only the threshold $\alpha_k$ changes. This is a conditional assumption, given $b_i$. The earlier material on partial proportional-odds models covers threshold-specific exposure effects; adding a random intercept does not by itself remove the proportional-odds restriction.

For a model with multiple measured predictors, the term $\beta C_{ij}$ can be replaced by a linear predictor such as $\boldsymbol{\beta}^{T}\mathbf{x}_{ij}$. The essential repeated-measures feature is that the same $b_i$ enters every occasion for subject $i$, rather than a newly drawn effect being used at each visit.

From Cumulative Probabilities to Category Probabilities

Write $F_{ijk}=P(Y_{ij}\leq k\mid b_i)$. The probability of category 1 is $F_{ij1}$; an intermediate category $k$ has probability $F_{ijk}-F_{ij,k-1}$; and category $K$ has probability $1-F_{ij,K-1}$. These differences are nonnegative when the thresholds are ordered. The category probabilities sum to one by cancellation.

Worked Example: Interpreting a Conditional Response

Suppose a hypothetical three-category response has thresholds $\alpha_1=-0.5$ and $\alpha_2=0.8$, exposure effect $\beta=0.7$ per exposure unit, and a subject with $b_i=0.4$ measured at $C_{ij}=0.5$. Calculate the conditional category probabilities and interpret the exposure effect.

For the first threshold, the linear predictor is $-0.5-(0.7)(0.5)-0.4=-1.25$, so $F_{ij1}=\operatorname{logit}^{-1}(-1.25)\approx0.2227$. For the second threshold, it is $0.8-(0.7)(0.5)-0.4=0.05$, giving $F_{ij2}=\operatorname{logit}^{-1}(0.05)\approx0.5125$.

Therefore the category probabilities are $P(Y_{ij}=1\mid b_i)\approx0.2227$, $P(Y_{ij}=2\mid b_i)\approx0.5125-0.2227=0.2898$, and $P(Y_{ij}=3\mid b_i)\approx1-0.5125=0.4875$. They sum to 1.0000, subject to rounding.

At either threshold, a one-unit exposure increase multiplies the conditional odds of being at or below the threshold by $\exp(-0.7)\approx0.4966$; the corresponding odds of being above it are multiplied by $\exp(0.7)\approx2.014$. These are conditional odds ratios for subjects with the same random effect, not marginal odds ratios across the population.

Why a Shared Random Intercept Creates Dependence

Given $b_i$, the model assumes that two responses from subject $i$ are independent after conditioning on their covariates. But both responses depend on the same unknown $b_i$. If that effect is averaged over, knowing one response gives information about the subject’s latent tendency and therefore about another response. Conditional independence does not imply marginal independence.

This mechanism can be shown for the indicators $A_{ij}=I(Y_{ij}\leq k)$ and $A_{i\ell}=I(Y_{i\ell}\leq k)$, at two occasions with the same covariates and threshold. Let $q(b)=P(Y\leq k\mid b)$. Conditional independence gives $E(A_{ij}A_{i\ell}\mid b_i)=q(b_i)^2$, so

\operatorname{Cov}(A_{ij},A_{i\ell}) =E\{q(b_i)^2\}-[E\{q(b_i)\}]^2 =\operatorname{Var}\{q(b_i)\}.

Unless $q(b_i)$ is constant, that variance is positive. Under the stated model, $q(b)$ decreases as $b$ increases, so subjects with larger random intercepts are less likely to be below the threshold on both occasions. This shared ordering of responses generates positive covariance for these same-threshold indicators. It does not mean every pair of category indicators must have positive covariance: indicators for different, mutually exclusive categories can behave differently.

Worked Example: Marginal Probabilities and Within-Subject Dependence

Use a hypothetical model with $\alpha_1=-0.5$, $\alpha_2=0.8$, $\beta=0.7$, and $\sigma_b=0.6$. At exposure $C=0$, approximate marginal probabilities and dependence using three-point quadrature for a normal random effect. For $b\sim N(0,\sigma_b^2)$, use nodes $0,\pm\sqrt{3}\sigma_b$, with weights $2/3,1/6,1/6$. This is a numerical approximation to the normal integral, not a claim that the random effect has only three possible values.

The nodes are approximately $0,1.039,-1.039$. At the first threshold, the conditional probabilities $q(b)=P(Y\leq1\mid b)$ at these nodes are approximately $0.3775,0.1766,0.6316$, respectively. The quadrature-weighted marginal probability is

E\{q(b)\}\approx(2/3)(0.3775)+(1/6)(0.1766)+(1/6)(0.6316)=0.3864.

At the second threshold, the conditional probabilities at the same nodes are approximately $0.6900,0.4405,0.8629$. Their weighted mean is $(2/3)(0.6900)+(1/6)(0.4405)+(1/6)(0.8629)\approx0.6772$. Thus the three marginal category probabilities are approximately $0.3864$, $0.6772-0.3864=0.2908$, and $1-0.6772=0.3228$.

For two occasions with the same covariates, the covariance of their indicators for $Y\leq1$ is approximately the weighted variance of the three $q$ values:

\operatorname{Cov}(A_{ij},A_{i\ell}) \approx\{(2/3)(0.3775^2)+(1/6)(0.1766^2)+(1/6)(0.6316^2)\}-(0.3864)^2 \approx0.0174.

The marginal variance of either indicator is approximately $0.3864(1-0.3864)=0.2371$, so their marginal correlation is approximately $0.0174/0.2371=0.0734$. These numbers illustrate the dependence mechanism; quadrature accuracy depends on the number and placement of integration points.

Likelihood and a Marginal-versus-Conditional Comparison

For a subject, the conditional likelihood is the product of that subject’s category probabilities across occasions, given $b_i$. The subject’s marginal likelihood contribution is obtained by integrating that product over the random-effect distribution:

L_i=\int\left\{\prod_j P(Y_{ij}=y_{ij}\mid b)\right\} \frac{1}{\sigma_b}\phi\!\left(\frac{b}{\sigma_b}\right)\,db,

where $\phi$ is the standard normal density. The order matters: the shared $b$ is integrated after multiplying the within-subject conditional contributions. Integrating each observation separately would discard the shared subject effect and its induced dependence. In practice, software approximates these integrals using numerical methods such as Gaussian quadrature or Laplace approximations.

Worked Example: Conditional and Marginal Exposure Odds Ratios

Continue with the hypothetical model in the previous example and compare the odds of $Y\leq1$ at exposure 0 and exposure 1. The conditional cumulative odds ratio per exposure unit is $\exp(-0.7)\approx0.4966$. To obtain approximate marginal odds, integrate the cumulative probabilities over the same three quadrature nodes.

At exposure 0, the marginal probability was approximately $0.3864$, giving marginal odds $0.3864/(1-0.3864)\approx0.6297$. At exposure 1, the three conditional logits for $Y\leq1$ are $-1.2$, $-2.239$, and $-0.161$, at nodes $0,1.039,-1.039$. The corresponding probabilities are approximately $0.2315,0.0962,0.4598$, so the marginal probability is $(2/3)(0.2315)+(1/6)(0.0962)+(1/6)(0.4598)\approx0.2470$.

The exposure-1 marginal odds are $0.2470/(1-0.2470)\approx0.3280$. The marginal odds ratio comparing exposure 1 with exposure 0 is therefore approximately $0.3280/0.6297=0.5209$, rather than the conditional value $0.4966$. Integration over subject heterogeneity changes the odds ratio; a conditional coefficient should not be reported as a marginal odds ratio.

Model Checks, Alternatives, and Limitations

A random intercept is useful when repeated ordinal responses show persistent subject-level heterogeneity, but it does not guarantee a good model. Check whether the fitted model converges reliably, whether threshold estimates remain ordered, and whether the estimated $\sigma_b^2$ is plausible. A variance near zero suggests little support for this particular source of heterogeneity; it is not proof that all within-subject structure is absent. Very few subjects can make the random-effect variance difficult to estimate, even when each subject has several observations.

Also examine whether one exposure coefficient is adequate across thresholds and whether the model captures the observed pattern of responses across visits. Large discrepancies in category frequencies or repeated-response patterns can indicate an inadequate proportional-odds structure, random-effect distribution, covariate specification, or conditional-independence assumption. A random intercept captures stable subject heterogeneity, but it does not automatically model serial dependence that remains after conditioning on that intercept, time-varying subject effects, or threshold-specific exposure effects.

A population-averaged alternative, such as a marginal ordinal model fitted with a working correlation structure, targets average response relationships rather than subject-conditional effects. A random-intercept model instead gives a conditional interpretation and specifies a distribution for subject heterogeneity; its marginal probabilities are derived by integration. These approaches answer different questions and are not interchangeable by simply re-labelling a coefficient. A model without random effects may be appropriate if the repeated observations are conditionally independent under the chosen structure, but treating correlated repeated outcomes as independent can misstate uncertainty and fit.

Common Pitfall

Common mistake: “The conditional model assumes independence, so repeated responses are marginally independent.” The correction is that independence is conditional on the shared $b_i$. After averaging over $b_i$, outcomes share the same latent subject tendency, and their joint probability generally does not factor into the product of their marginal probabilities.

A second frequent error is interpreting $\beta$ as a population-averaged log odds ratio. In this model, $\beta$ describes the exposure association conditional on $b_i$. Marginal odds ratios generally differ because odds ratios are non-collapsible and because the random effect is integrated out.

Key takeaway: A cumulative-logit random-intercept model describes ordered repeated outcomes conditional on a shared subject effect. The shared effect both represents persistent heterogeneity and induces within-subject dependence after it is integrated out.

Check Your Understanding

Answer each question, then compare with the brief solutions.

In the stated model, what does a positive $\beta$ imply about the conditional odds of being above a threshold as exposure increases by one unit?
Why must the same $b_i$ appear at every occasion for subject $i$ to represent a shared subject tendency?
For a three-category response, write the category-2 probability using cumulative probabilities.
What operation gives the marginal subject likelihood: integrate each occasion separately, or integrate the product of conditional occasion likelihoods?
Why can the marginal exposure odds ratio differ from $\exp(-\beta)$?

Solutions. (1) The conditional odds of being above the threshold are multiplied by $\exp(\beta)$. (2) Reusing the effect makes all of that subject’s responses depend on the same latent tendency; drawing a new effect each occasion would not encode a persistent subject effect. (3) $P(Y=2\mid b_i)=P(Y\leq2\mid b_i)-P(Y\leq1\mid b_i)$. (4) Integrate the product over the shared random-effect distribution. (5) Integration over heterogeneity changes marginal probabilities and odds, so the marginal odds ratio generally is not the conditional odds ratio.