Multiple Predictors in a Cumulative Link Model

Combining Covariates in One Cumulative Link Model

An ordinal response often depends on several characteristics at once. For example, a patient’s ordered symptom rating may be associated with both age and a continuous measure of disease severity. A cumulative link model combines the covariates into one linear predictor; that predictor then shifts all cumulative probabilities together. The key calculation is to form the predictor for the particular covariate values of interest before evaluating any probabilities.

Suppose the response \(Y\) has \(K\) ordered categories, numbered \(1,\ldots,K\), and an observation has \(q\) covariates \(x_1,\ldots,x_q\). Write their coefficients as \(\beta_1,\ldots,\beta_q\). The cumulative link model uses ordered thresholds and a specified increasing distribution function \(G\), such as the logistic distribution function for a cumulative logit model or the standard normal distribution function for a cumulative probit model.

The predictor is a sum of contributions, one for each covariate. A coefficient can be positive while its covariate value is negative, so the contribution \(\beta_r x_r\) can have either sign. The predictor is not itself a probability; it shifts the arguments supplied to \(G\). For a given observation, use the same \(\eta(x)\) at every threshold. The thresholds differ across cumulative splits, but the slopes are shared.

For clarity, define \(Q_j(x)=P(Y\leq j\mid x)\), and set \(Q_0(x)=0\) and \(Q_K(x)=1\). Then category probabilities are \(P(Y=j\mid x)=Q_j(x)-Q_{j-1}(x)\). This gives the endpoint formulas \(P(Y=1\mid x)=Q_1(x)\) and \(P(Y=K\mid x)=1-Q_{K-1}(x)\), as well as differences for the intermediate categories. Since \(G\) is increasing and the thresholds are ordered, the cumulative probabilities are ordered, so these differences are nonnegative.

Calculate the Predictor Before the Probabilities

A reliable calculation has two stages. First substitute the specified covariate values into the linear predictor and add the contributions. Then, for each threshold, subtract that predictor from the threshold, evaluate \(G\), and difference the cumulative probabilities to obtain the category probabilities. Keeping these stages separate helps prevent sign errors.

The calculated probabilities belong to the entire covariate combination, not to one predictor in isolation. To study one predictor’s change, compare two predictor combinations that differ in that covariate and hold the others fixed. The coefficient describes the change in the linear predictor per unit increase in its covariate; the resulting change in a particular category probability also depends on the thresholds, the other covariate values, and the link distribution.

What Changes When One Covariate Changes?

Let \(x_r\) change by an amount \(\Delta\), while all other covariates remain fixed. The new predictor is the old predictor plus \(\beta_r\Delta\). This simple contrast has a direct interpretation on the cumulative-link scale. In the cumulative logit model, it also gives a constant odds ratio at every threshold.

Proof. Denote the original predictor by \(\eta\). After the change, the predictor is \(\eta_{\mathrm{new}}=\eta+\beta_r\Delta\), because only the \(r\)th contribution changes. In a cumulative logit model, \(\operatorname{logit}\{P(Y\leq j\mid x)\}=\theta_j-\eta\). The new cumulative log odds are \(\theta_j-\eta-\beta_r\Delta\), which differ from the original by \(-\beta_r\Delta\). Exponentiating gives the stated multiplier for the cumulative odds; taking the reciprocal gives the multiplier for the odds of being above the threshold. This holds for every \(j\) because the same slope \(\beta_r\) appears at every threshold. \(\square\)

How Category Probabilities Respond Locally

The cumulative probabilities have a simple directional response to a covariate: when \(G\) has a density \(g\), their derivatives follow from the chain rule. The category probabilities are differences of cumulative probabilities, so their derivatives involve two density values. This explains why the direction of change for an interior category is not determined by the sign of a coefficient alone.

Derivation. Since \(Q_j(x)=G(\theta_j-\eta(x))\) and \(\partial\eta(x)/\partial x_r=\beta_r\), the chain rule gives \(\partial Q_j(x)/\partial x_r=g(\theta_j-\eta)(-\beta_r)\), proving the cumulative derivative. For an interior category, \(P(Y=j\mid x)=Q_j(x)-Q_{j-1}(x)\). Subtract the derivative of \(Q_{j-1}\) from that of \(Q_j\): \(-\beta_r g(\theta_j-\eta)-[-\beta_r g(\theta_{j-1}-\eta)]\) \(=\beta_r[g(\theta_{j-1}-\eta)-g(\theta_j-\eta)]\). For the first category \(P(Y=1\mid x)=Q_1(x)\), so its derivative is the first cumulative derivative. For the last category \(P(Y=K\mid x)=1-Q_{K-1}(x)\), so its derivative is the negative of the last cumulative derivative. \(\square\)

Common Pitfalls and a Practical Check

A frequent mistake is to apply each coefficient directly to a probability, as if a coefficient of \(0.7\) meant that a category probability increases by \(0.7\). It does not. The coefficients add to form \(\eta\), which shifts the cumulative-link arguments; probabilities are obtained only after applying \(G\) and differencing the cumulative values. In a logit model, exponentiating a coefficient gives an odds multiplier for a specified cumulative contrast, not an additive probability change.

Another common error is to change one covariate but inadvertently change another as well. A comparison intended to describe \(x_r\) should explicitly state the fixed values of the remaining predictors. Also check the sign convention in the model equation: with \(G(\theta_j-\eta)\), a positive coefficient lowers \(P(Y\leq j\mid x)\) as its covariate increases, whereas a model written with the opposite sign convention would require a corresponding change in interpretation.

As a final arithmetic check, verify that cumulative probabilities are nondecreasing in \(j\), each category probability is nonnegative, and all category probabilities sum to one. These checks can reveal a sign mistake, a subtraction error, or use of inconsistent predictor values across thresholds.