Cumulative Link Models for Ordinal Responses

Why Model an Ordinal Response?

Many pharmacometric outcomes are recorded in ordered categories rather than on a continuous scale: symptom severity might be none, mild, moderate, or severe; a clinician’s global assessment might range from poor to excellent. The order carries information, but the gaps between categories are not usually known to be equal. Treating category codes as numerical measurements can therefore impose an unjustified scale. A cumulative link model uses the ordering directly by modeling the probability of being at or below each possible category boundary.

In a mixed-effects setting, repeated responses from the same individual may be related because that individual has a persistent tendency toward higher or lower categories. A subject-specific random effect can represent this tendency. The model then describes category probabilities conditional on the individual’s random effect; population-level probabilities require averaging over its distribution.

Model and Assumptions

Definition: Let $Y_{ij}$ be an ordinal response for individual $i$ at observation $j$, with ordered categories $1,\ldots,K$. A cumulative-logit mixed-effects model is $$ \operatorname{logit}\{P(Y_{ij}\leq k\mid b_i)\} =\alpha_k-\eta_{ij},\qquad k=1,\ldots,K-1, $$ where $\operatorname{logit}(p)=\log\{p/(1-p)\}$, the thresholds satisfy $\alpha_1<\cdots<\alpha_{K-1}$, and $\eta_{ij}=x_{ij}^{T}\beta+z_{ij}^{T}b_i$. Here $x_{ij}$ and $z_{ij}$ are covariate vectors, $\beta$ is a vector of fixed effects, and $b_i$ is an individual-level random effect, commonly modeled as $b_i\sim N(0,D)$. The category order is meaningful, and the conditional cumulative probabilities follow the specified logistic form.

This parameterization uses the minus sign before $\eta_{ij}$: a larger linear predictor lowers the cumulative probability $P(Y_{ij}\leq k\mid b_i)$, and thus shifts probability toward higher categories. The same $\eta_{ij}$ appears at every threshold, so a predictor has the same slope on each cumulative-logit scale. The implications and assessment of that restriction are the subject of the proportional-odds material that follows.

The thresholds define the baseline positions of the category boundaries on the latent cumulative-logit scale. A model with an unrestricted fixed intercept in $\eta_{ij}$ would confound that intercept with a shift in every threshold. A common identifiable parameterization therefore omits a separate fixed intercept from $\eta_{ij}$, allowing the thresholds to locate the categories. Random effects can still enter $\eta_{ij}$. The distribution of $b_i$, including its mean-zero convention, must be specified; a normal distribution is common, not automatic.

For the usual likelihood, responses are conditionally independent given the random effects and covariates, unless additional dependence is explicitly modeled. The model also assumes that the recorded categories are genuinely ordered and that the covariates and random-effects structure adequately represent systematic differences relevant to the response. These are modeling assumptions, not consequences of having categorical data.

From Cumulative Probabilities to Category Probabilities

Write $F(u)=\operatorname{expit}(u)=1/(1+\exp(-u))$, the logistic cumulative distribution function. At a given observation and random effect, define $C_k=P(Y_{ij}\leq k\mid b_i)$. Then $C_k=F(\alpha_k-\eta_{ij})$. The probability of one category is obtained by subtracting adjacent cumulative probabilities, not by treating a cumulative probability as the probability of a single category.

\begin{aligned} P(Y_{ij}=1\mid b_i)&=C_1,\\ P(Y_{ij}=k\mid b_i)&=C_k-C_{k-1},\qquad 2\leq k\leq K-1,\\ P(Y_{ij}=K\mid b_i)&=1-C_{K-1}. \end{aligned}

This construction gives valid category probabilities when the thresholds are ordered. Since $F$ is increasing and $\alpha_k>\alpha_{k-1}$, $C_k>C_{k-1}$, so each interior category probability is positive. The probabilities sum to one because the differences telescope:

C_1+\sum_{k=2}^{K-1}(C_k-C_{k-1})+(1-C_{K-1})=1.

For an individual-level likelihood contribution, an observed category $y_{ij}$ contributes its conditional probability $P(Y_{ij}=y_{ij}\mid b_i)$. When the random effect is unobserved, the individual’s likelihood contribution integrates the product of such conditional probabilities over the distribution of $b_i$. The conditional probabilities describe an individual with a specified random effect; they are not generally the same as probabilities averaged over the population distribution of random effects.

Worked Example: Calculating Three Category Probabilities

Suppose a hypothetical response has three ordered categories, with $\alpha_1=-0.5$, $\alpha_2=1.0$, and $\eta_{ij}=0$ for the individual and observation being considered. The thresholds are ordered because $-0.5<1.0$.

The cumulative probabilities are $C_1=F(-0.5)=0.3775407$ and $C_2=F(1.0)=0.7310586$. Therefore,

\begin{aligned} P(Y_{ij}=1\mid b_i)&=0.3775407,\\ P(Y_{ij}=2\mid b_i)&=0.7310586-0.3775407=0.3535179,\\ P(Y_{ij}=3\mid b_i)&=1-0.7310586=0.2689414. \end{aligned}

Rounded to four decimal places, the probabilities are $0.3775$, $0.3535$, and $0.2689$. They sum to one before rounding; the displayed rounded values sum to $0.9999$ because of rounding. The middle probability was calculated from the unrounded cumulative probabilities, rather than by subtracting rounded values.

Interpreting Thresholds and Predictor Effects

A threshold is a cumulative-logit intercept at the covariate and random-effect reference point $\eta_{ij}=0$. For example, $\alpha_2=0.5$ means that at this reference point, the log odds of $Y_{ij}\leq2$ are $0.5$, and the corresponding odds are $\exp(0.5)$. A threshold is not the mean response, nor does it directly give the probability of category $k$ alone.

For a predictor $x$ with coefficient $\beta$, increasing $x$ by one unit while holding other terms fixed changes each cumulative logit by $-\beta$. Equivalently, the odds of being above boundary $k$, $P(Y_{ij}>k\mid b_i)/P(Y_{ij}\leq k\mid b_i)$, are $\exp(\eta_{ij}-\alpha_k)$. A one-unit increase in $x$ multiplies these upper-category odds by $\exp(\beta)$, conditional on the random effect and other covariates. The units and coding of $x$ therefore matter to interpretation.

Worked Example: Interpreting a Predictor Coefficient

Consider a hypothetical three-category model with upper threshold $\alpha_2=0.5$, predictor coefficient $\beta=0.6$ per dose unit, and subject random intercept $b_i=-0.2$. Let $\eta_{ij}=\beta x_{ij}+b_i$. At $x_{ij}=0$, $\eta_{ij}=-0.2$, so $P(Y_{ij}\leq2\mid b_i)=F\{0.5-(-0.2)\}=F(0.7)=0.6682$. Thus $P(Y_{ij}>2\mid b_i)=0.3318$, and the odds above category 2 are $0.3318/0.6682=0.4966$.

At $x_{ij}=1$, $\eta_{ij}=0.6-0.2=0.4$. The upper-category odds are $\exp(0.4-0.5)=\exp(-0.1)=0.9048$. The odds ratio comparing $x=1$ with $x=0$, for the same individual random effect, is $0.9048/0.4966\approx1.822$, which equals $\exp(0.6)$ up to rounding. Thus one additional dose unit multiplies the conditional odds of being above either cumulative boundary by about 1.822 under this model. It does not mean that the probability of each individual category increases by that factor.

Worked Example: The Effect of a Subject Random Intercept

Suppose a hypothetical three-category model has $\alpha_1=-0.5$, $\alpha_2=1.0$, and fixed predictor contribution $\beta x_{ij}=0.6$. For an individual with random intercept $b_i=-0.2$, $\eta_{ij}=0.6-0.2=0.4$. The cumulative probabilities are $C_1=F(-0.5-0.4)=F(-0.9)=0.2890505$ and $C_2=F(1.0-0.4)=F(0.6)=0.6456563$.

The conditional category probabilities are $0.2890505$ for category 1, $0.6456563-0.2890505=0.3566058$ for category 2, and $1-0.6456563=0.3543437$ for category 3. They sum to one. Here the negative random intercept reduces $\eta_{ij}$, moving the conditional cumulative probabilities upward compared with the same covariates at $b_i=0$, and therefore tending to shift responses toward lower categories. This is an individual-conditional prediction, not a population-averaged prediction.

Justification, Diagnostics, and Alternatives

The category-probability formula follows directly from nested events: the event $Y_{ij}=k$ is the event $Y_{ij}\leq k$ minus the event $Y_{ij}\leq k-1$. Hence $P(Y_{ij}=k\mid b_i)=C_k-C_{k-1}$. For the lowest category, the lower cumulative event is empty and its probability is zero; for the highest category, the cumulative event through $K$ has probability one. This derivation also explains why unordered or independently estimated cumulative probabilities can produce invalid negative category probabilities: they may fail to be nondecreasing.

Before interpreting fitted probabilities, check that the category definitions have a defensible order, estimated thresholds are ordered, and predicted category probabilities are between zero and one and sum to one. Warning signs include sparse or empty categories, unstable threshold estimates, numerical convergence problems, or predictions that poorly represent observed category frequencies across important covariate or subject groups. Such patterns can reflect limited information, an unsuitable predictor or random-effects structure, or a mismatch between the assumed cumulative-logit form and the data. They warrant investigation; they do not by themselves identify one particular cause.

Cumulative link models make use of ordering and can summarize effects across category boundaries. A nominal-category model instead treats outcomes as unordered and estimates category-specific comparisons, which may be preferable when the order is not meaningful, but it does not exploit ordinal structure in the same way. A binary model formed by collapsing categories answers a narrower question and discards distinctions among the categories merged together. Neither alternative should be selected merely because it is easier to fit: the outcome scale and scientific question should guide the choice.

A central limitation of the cumulative-logit parameterization is its common predictor slope across thresholds. It can be a useful compact model, but an effect that differs substantially between lower and upper category boundaries may not be represented well. Also, conditional effects for a specified random effect should not be reported as though they were population-averaged effects: averaging probabilities over a nonzero random-effects distribution generally changes them. The earlier material on conditional and population-averaged effects distinguishes these interpretations.

Common Mistakes

Using a cumulative probability as a category probability. $P(Y\leq2)$ includes more than one category. Obtain an interior category probability by subtracting the preceding cumulative probability.
Reversing the effect direction. With $\alpha_k-\eta_{ij}$, a positive predictor coefficient lowers the cumulative logit and shifts probability toward higher categories, conditional on other terms.
Interpreting thresholds as category means. A threshold locates a cumulative boundary on the logit scale; it is not a mean or a probability for one category.
Rounding too early. Subtract unrounded cumulative probabilities before reporting category probabilities. Otherwise the final displayed digit may be wrong.
Ignoring the conditioning in a mixed model. A probability conditional on $b_i$ describes an individual with that random effect; it is not automatically a population-average probability.

Key takeaway: A cumulative-logit mixed-effects model specifies ordered cumulative probabilities through thresholds and a subject-specific linear predictor. Differencing adjacent cumulative probabilities gives the category probabilities; their interpretation depends on the sign convention, covariate units, and whether predictions are conditional on or averaged over random effects.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

For a three-category response, suppose $C_1=0.25$ and $C_2=0.70$. What are the three category probabilities?
In the model $\operatorname{logit}\{P(Y\leq k\mid b_i)\}=\alpha_k-\eta$, what is the direction of the shift in response categories when $\eta$ increases?
If a predictor coefficient is $0.4$ per unit, what is the multiplicative change in conditional odds of being above a cumulative boundary for a one-unit increase?
Why must thresholds be ordered in a cumulative-link model?

Solutions. (1) The category probabilities are $P(Y=1)=C_1=0.25$, $P(Y=2)=C_2-C_1=0.70-0.25=0.45$, and $P(Y=3)=1-C_2=0.30$; they sum to 1. (2) Increasing $\eta$ decreases each cumulative probability $P(Y\leq k\mid b_i)$, so it shifts probability toward higher categories. (3) The conditional upper-category odds are multiplied by $\exp(0.4)=1.492$ per predictor unit. (4) Ordered thresholds make cumulative probabilities nondecreasing, so adjacent differences yield nonnegative category probabilities.