Partial Proportional-Odds Models

Why Use Partial Proportional Odds?

A proportional-odds model gives each predictor the same slope at every cumulative cutpoint. That shared-slope restriction is useful: it summarizes the predictor’s association with an ordinal response using one effect. But a predictor’s association may differ across thresholds, while other predictors may still reasonably share a common effect. A partial proportional-odds model allows this mixture: some slopes are shared across cutpoints, and others vary.

The word “partial” describes the structure of the slopes, not a different response scale. The model still describes cumulative probabilities, such as the probability of being in category \(k\) or below. Its key requirements are to state which slopes are shared, which vary, and where the resulting cumulative probabilities are valid.

Model Definition

Let \(Y_i\) be an ordinal response for observation \(i\), with ordered categories \(1,\ldots,K\), where \(K\geq 2\). For \(k=1,\ldots,K-1\), define the cumulative probability \(C_{ik}=P(Y_i\leq k\mid x_i,z_i)\). Let \(x_i\) be a \(p\)-vector of predictors with shared slopes and \(z_i\) an \(r\)-vector of predictors with cutpoint-specific slopes. With the cumulative-logit sign convention used here, the model is

The \(\alpha_k\) are thresholds, \(\beta\) is the vector of shared slopes, and \(\gamma_k\) is the vector of slopes specific to cutpoint \(k\). The inverse-logit function is \(\Lambda(t)=1/(1+\exp(-t))\), so \(C_{ik}=\Lambda(\alpha_k-x_i^\mathsf{T}\beta-z_i^\mathsf{T}\gamma_k)\). When \(r=0\), the model reduces to the proportional-odds model. When \(p=0\), all modeled slopes vary across cutpoints.

The thresholds are ordered, \(\alpha_1<\cdots<\alpha_{K-1}\). That is necessary for the usual proportional-odds model, but with varying slopes it is not sufficient to ensure valid probabilities at every predictor value. For a covariate configuration of interest, the cumulative logits must be ordered:

Because \(\Lambda\) is strictly increasing, these inequalities are equivalent to \(C_k\leq C_{k+1}\). Strict inequalities give positive probabilities for the intermediate categories; equality makes the corresponding category probability zero. A model intended for a particular covariate domain must satisfy the ordering throughout that domain, or its predictions can fail to define a probability distribution.

From Cumulative Probabilities to Category Probabilities

Write \(q_{ij}=P(Y_i=j\mid x_i,z_i)\) for the probability of category \(j\). The first category has probability \(C_{i1}\), an intermediate category is the difference between two successive cumulative probabilities, and the last category is the complement of the final cumulative probability.

The result follows directly by summing the category probabilities: each intermediate cumulative probability cancels once as a subtraction and once as an addition.

Proof. The first probability is nonnegative because \(C_{i1}\geq0\). For \(2\leq j\leq K-1\), \(q_{ij}=C_{ij}-C_{i,j-1}\geq0\) by the assumed ordering. The final probability is nonnegative because \(C_{i,K-1}\leq1\). For the sum, when \(K>2\),

For \(K=2\), the sum is \(C_{i1}+(1-C_{i1})=1\) as well. Thus the category probabilities are nonnegative and sum to one. \(\square\)

Likelihood for Independent Observations

Suppose \(Y_1,\ldots,Y_n\) are conditionally independent given their covariates, and observation \(i\) is in category \(y_i\). The probability assigned to that observation is \(q_{i,y_i}\). Multiplying these probabilities gives the likelihood; taking logarithms gives a sum.

Derivation. Conditional independence means the joint probability of the observed response vector, given the covariates and model parameters, is the product of its individual conditional probabilities:

This joint probability, as a function of the model parameters, is the likelihood. Taking its logarithm yields the sum of the logarithms. If an observed category has probability zero, the likelihood is zero and the log-likelihood is \(-\infty\). The indicator form is safe when all category probabilities are positive; otherwise, an unobserved category can contribute the undefined product \(0\log(0)\). The observed-category sum avoids that ambiguity.

Interpreting Shared and Varying Slopes

At cutpoint \(k\), the cumulative odds of being in category \(k\) or below rather than above \(k\) are

Holding the other predictors fixed, increasing shared predictor \(x_{im}\) by one unit multiplies these cumulative odds by \(\exp(-\beta_m)\) at every cutpoint. In this sign convention, a positive \(\beta_m\) lowers the odds of being at or below each cutpoint. Equivalently, it raises the odds of being above each cutpoint by \(\exp(\beta_m)\).

For a predictor \(z_{im}\) whose coefficient varies by cutpoint, a one-unit increase multiplies the cumulative odds at cutpoint \(k\) by \(\exp(-\gamma_{km})\). Those multipliers can differ across cutpoints. A varying slope therefore represents a change in the predictor’s association with the cumulative divisions of the response; it is not a single odds ratio for every possible comparison of categories.

Why Threshold Ordering Is Not Enough

With shared slopes, the term involving the predictors is the same at every cutpoint, so ordered thresholds keep the cumulative logits ordered. Cutpoint-specific slopes change that: the difference between neighboring cumulative logits depends on the covariates. This is why model specification includes an admissibility check on the predictor domain, rather than relying on the threshold ordering alone.

Common Mistake and Practical Meaning

• Mistake: Assume that \(\alpha_1<\cdots<\alpha_{K-1}\) guarantees valid category probabilities in a partial proportional-odds model.
• Correction: Check that the full cumulative logits, including their cutpoint-specific predictor terms, are ordered for the covariate values where predictions are required. If not, differencing cumulative probabilities can produce a negative category probability.
• Mistake: Interpret a varying coefficient as one common odds ratio across all cutpoints.
• Correction: At cutpoint \(k\), the cumulative-odds multiplier is \(\exp(-\gamma_{km})\); it can vary with \(k\).

The partial proportional-odds structure is useful when a common cumulative effect is appropriate for some predictors but not for all. Its interpretation remains anchored to cumulative odds, and its probability implications remain anchored to ordered cumulative probabilities. Those two checks—what each slope means and whether the cumulative probabilities are ordered—belong together.