Comparing Partial Proportional Odds With Multinomial Logit

Two Ways to Relax the Ordinal Structure

For an ordinal response, the ordering of categories carries information: “mild,” “moderate,” and “severe” are not merely three unrelated labels. A proportional-odds model uses that ordering by modelling cumulative probabilities and sharing each predictor’s slope across cutpoints. A partial proportional-odds model relaxes that sharing for selected predictors. A baseline-category multinomial-logit model instead assigns a separate linear predictor to each nonbaseline category.

The practical comparison is not simply “constrained versus unconstrained.” Partial proportional odds retains a cumulative structure for predictors whose slopes remain shared, but predictor-specific slope relaxation can make its cumulative probabilities cross. Multinomial logit always gives a coherent category-probability vector, but generally uses more parameters. Those distinctions affect likelihood, precision, and the validity of fitted probabilities.

In the multinomial model, the category probabilities are obtained by normalizing the exponentiated linear predictors, including the baseline predictor zero. In the partial proportional-odds model, the category probabilities must instead be formed by differencing successive cumulative probabilities. The distinction is consequential: positive multinomial probabilities are built into the softmax normalization, whereas cumulative-logit equations do not automatically guarantee that their fitted cumulative probabilities are ordered at every \(x\).

Coherence of Cumulative Probabilities

Write \(F_j(x)=P(Y\leq j\mid x)\). The implied category probabilities are \(p_1(x)=F_1(x)\), \(p_j(x)=F_j(x)-F_{j-1}(x)\) for \(2\leq j\leq J-1\), and \(p_J(x)=1-F_{J-1}(x)\). The following criterion makes the issue precise.

Proof. If the cumulative probabilities are ordered, then \(p_1=F_1\geq0\), each difference \(p_j=F_j-F_{j-1}\geq0\), and \(p_J=1-F_{J-1}\geq0\). Their sum telescopes: $$ F_1+\sum_{j=2}^{J-1}(F_j-F_{j-1})+(1-F_{J-1})=1. $$ Conversely, if all category probabilities are nonnegative, then for each \(j<J-1\), $$ F_{j+1}-F_j=p_{j+1}\geq0. $$ Thus the cumulative probabilities are ordered. \(\square\)

The example also shows why ordering the cutpoints at a reference value is not enough. In the equations above, the cumulative logits are ordered at \(x=0\), but their different slopes reverse that order for \(x=4\). If slope differences are allowed, coherence is a condition on the predictions over the relevant covariate region, not just on the intercepts.

Likelihood and Parameter Counts

For observations \(i=1,\ldots,n\), let \(p_{ij}=P(Y_i=j\mid x_i)\). Apart from the multinomial coefficient, the log-likelihood is $$ \ell=\sum_{i=1}^n\sum_{j=1}^J I(Y_i=j)\log(p_{ij}). $$ At a covariate pattern observed in category counts \(n_1,\ldots,n_J\), its contribution is \(\sum_j n_j\log(p_j)\). This gives a direct way to compare the consequences of two proposed probability vectors at that pattern. It does not, by itself, establish which full model has the larger maximized likelihood.

A useful first comparison is the number of free regression parameters. Suppose there are \(p\) predictors and \(r\) of them have cutpoint-specific slopes; the remaining \(p-r\) slopes are shared. With \(J-1\) cutpoints, partial proportional odds has \(J-1\) cutpoint parameters, \(p-r\) shared slopes, and \((J-1)r\) relaxed slopes. Its total is therefore $$ (J-1)+(p-r)+(J-1)r=(J-1)+p+(J-2)r. $$ The baseline-category multinomial logit has \(J-1\) intercepts and \((J-1)p\) slopes, for a total of \((J-1)(p+1)\) parameters. The difference in counts is \((J-2)(p-r)\), with the multinomial model having that many more parameters when \(J>2\) and at least one slope remains shared in the partial model.

What a Larger Likelihood Does—and Does Not—Show

For two models fitted to the same observations, if one model’s set of possible probability distributions contains the other’s, its maximized log-likelihood cannot be smaller. The reason is direct: maximizing over a larger set cannot yield a value below the maximum over a subset. But partial proportional odds and baseline-category multinomial logit are generally different restrictions on the conditional probabilities, not simply a small model and a large model. A cumulative-logit specification with cutpoint-specific slopes does not generally translate into linear baseline-category logits; conversely, multinomial logits do not generally imply cumulative logits linear in the predictors. Thus the parameter-count difference does not make these two model families nested, and their maximized likelihoods have no general ordering from that count alone.

This matters when interpreting likelihood comparisons. A higher fitted likelihood for one model is evidence about fit to the observed data, but it does not automatically justify a nested-model likelihood-ratio test. The comparison must account for whether the models are nested and for whether the fitted partial model produces coherent probabilities. A model that assigns a negative category probability at a relevant predictor value is not rescued by a favorable likelihood value at some other pattern.

Choosing Between Flexibility and Structure

The partial proportional-odds model is attractive when the ordinal cumulative structure is substantively useful and only selected predictors show evidence of nonparallel slopes. It preserves shared effects for the other predictors and can use fewer parameters than a fully category-specific regression. Its main additional responsibility is to check that cumulative probabilities remain ordered over the intended prediction domain.

The baseline-category multinomial logit provides valid category probabilities by construction and permits category-specific linear effects without a cumulative-order constraint. That flexibility generally costs parameters and does not use the ordering as directly. Its probability coherence is an advantage, not a guarantee of better predictive performance or a reason to ignore the ordinal structure.

Common Mistakes

• Assuming cutpoint-specific slopes preserve valid probabilities. They need not. Compute successive cumulative probabilities and check their ordering.
• Calling multinomial logit a nested version of partial proportional odds. Their regression restrictions are generally different. A larger parameter count alone does not establish nesting.
• Interpreting a local likelihood contribution as a full model comparison. Contributions at one covariate pattern do not equal the maximized log-likelihood over all observations and parameters.
• Concluding that fewer parameters always mean smaller standard errors. Sharing slopes can improve precision when the equality restriction is appropriate; misspecification can instead distort estimates and uncertainty.