Likelihood-Ratio Tests of Parallel Slopes

Why Compare Parallel-Slope and Less-Restricted Models?

The proportional-odds assumption says that a predictor has the same slope at every cumulative cutpoint of an ordinal response. A likelihood-ratio test asks whether allowing those slopes to differ improves the fit enough to justify the additional parameters. The comparison is useful because it puts the assumption into a nested-model framework: the proportional-odds model is a special case of a less-restricted cumulative-link model.

The test compares the models’ maximized likelihoods for the same observations and response coding. Its degrees of freedom come from the number of independent slope parameters added in the less-restricted model, not from the number of observations or response categories alone. A chi-square reference distribution is justified only under regularity conditions; nesting by itself is not enough.

The Nested Cumulative-Link Models

Let \(Y_i\) take ordered categories \(1,\ldots,J\), and let \(x_i\) be a vector of \(p\) predictors. Use an increasing cumulative link \(G\), such as the logistic or standard normal distribution function. For each cutpoint \(j=1,\ldots,J-1\), write the less-restricted model as

Here \(\alpha_j\) is a cutpoint and \(\beta_j\) is a \(p\)-dimensional slope vector that may depend on the cutpoint. The proportional-odds model restricts these vectors to be equal. The minus sign is a common cumulative-link convention; changing the sign convention does not change the likelihood-ratio test.

For clarity, set \(F_{ij}=P(Y_i\le j\mid x_i)\), with \(F_{i0}=0\) and \(F_{iJ}=1\). The probability assigned to category \(j\) is \(F_{ij}-F_{i,j-1}\). The model must therefore produce nonnegative category probabilities (strictly positive probabilities are part of the usual regularity conditions below). With cutpoint-specific slopes, ordered cutpoints alone do not guarantee valid probabilities for every possible \(x_i\); admissibility must be ensured for the design points or covariate support under consideration.

If observations are conditionally independent given their predictors, the log-likelihood for either model is

Let \(\hat\ell_0\) denote the maximized log-likelihood under proportional odds and \(\hat\ell_1\) the maximized log-likelihood under the less-restricted model, using the same data and likelihood. The likelihood-ratio statistic is

Counting the Added Slope Parameters

Both models estimate the same \(J-1\) cutpoints. The proportional-odds model estimates \(p\) common slopes; the less-restricted model estimates \((J-1)p\) cutpoint-specific slopes. When these parameters are identifiable and the restrictions are independent, the difference in dimension is

This count can also be seen directly from the null restrictions. Taking \(\beta_1\) as the reference slope, proportional odds imposes \(\beta_j-\beta_1=0\) for \(j=2,\ldots,J-1\). There are \(J-2\) vector equalities, each with \(p\) components. These are independent when the slope parameters are identifiable.

For example, when \(J=2\), there is only one cumulative cutpoint, so there is no cross-cutpoint slope equality to test: \((J-2)p=0\). When \(J=4\), each predictor contributes two slope parameters beyond its common slope, giving \(2p\) degrees of freedom.

Why the Statistic Is Nonnegative

The comparison has an immediate property that is also a practical check on model fitting.

Proof. Every proportional-odds parameter value is available in the less-restricted model by setting \(\beta_1=\cdots=\beta_{J-1}\). Thus the less-restricted model maximizes the same likelihood over a set containing all parameter values available to the proportional-odds model. Its maximum cannot be smaller, so \(\hat\ell_1\ge\hat\ell_0\). Multiplication of their difference by 2 gives \(\Lambda\ge0\). \(\square\)

A materially negative value usually signals that the models were not actually nested as fitted, used different observations or likelihoods, or that an optimization routine failed to find the maximum. It is not a legitimate outcome to interpret as evidence in favor of the restricted model.

Chi-Square Calibration and Its Conditions

The standard calibration is an application of Wilks’ theorem. In a regular nested parametric model, if the null model is true, the null parameter is an interior point of the larger model, and the \(q\) restrictions are independent, then

For this comparison, the usual conditions include the following:

• The observations are independent under the fitted likelihood (or the likelihood correctly accounts for their dependence), and both models use the same observations and response categories.
• The cumulative-link models are correctly specified in a neighborhood of the null parameter, with differentiable likelihood contributions and finite information.
• The parameters are identifiable; the relevant information matrix is nonsingular, and the design provides enough independent information to estimate the added slopes.
• The null parameter lies in the interior of the admissible parameter space of the less-restricted model. In particular, the fitted category probabilities must not be at a boundary, and the varying-slope model must be valid locally around the null for the covariate values or support in use.
• The maximum-likelihood estimators exist and have the usual consistency and asymptotic normality properties, and the number \(q\) of independent restrictions is fixed as \(n\) grows.

In a fixed-design analysis, a useful formulation is that the design and information must identify the parameters and the null must have a locally open neighborhood within the alternative’s admissible parameter space. With a covariate support extending without bound, unconstrained cutpoint-specific slopes may fail to preserve valid probabilities throughout that support. The ordinary Wilks conclusion should not be assumed in such a case without checking the model’s parameter space.

Common Pitfalls and Interpretation

Counting the wrong parameters. The test does not add one degree of freedom per response category. There are \(J-1\) cumulative cutpoints, and the common-slope restriction removes \((J-2)p\) independent slope parameters. Cutpoints appear in both models, so they cancel in the dimension difference.

Comparing incomparable fits. The likelihoods must come from nested models fitted to the same observations with the same response definition and likelihood. Different missing-data subsets, weights, or likelihood conventions can invalidate the direct comparison. The alternative must also contain the null as a parameter submodel.

Treating the chi-square approximation as automatic. A large likelihood improvement does not by itself guarantee the reference distribution is valid. Sparse data, unidentified slopes, boundary estimates, or failed optimization can undermine the regularity conditions. In those situations the usual chi-square p-value may be unreliable; the test’s assumptions and fit diagnostics need attention.

Finally, rejection indicates that the joint common-slope restriction is inconsistent with the data to the extent measured by this test. It does not identify which predictor or cutpoint is responsible, nor does failure to reject prove proportional odds. Those are separate interpretive questions.