Wald Tests for Cutpoint-Specific Slope Differences

What a Wald Test Compares

In a cumulative-logit model, each cutpoint defines a cumulative probability. With \(K\) ordered response categories, there are \(J=K-1\) cutpoints. Write the model with cutpoint-specific slopes as

where \(x\) is a vector of \(p\) predictors, \(\alpha_j\) is the threshold at cutpoint \(j\), and \(\beta_j\) is the vector of slopes there. The parallel-regression restriction says that the slopes do not vary across cutpoints: \(\beta_1=\cdots=\beta_J\). A Wald test assesses this restriction, or selected parts of it, using the unrestricted estimates and their estimated covariance matrix.

Using cutpoint 1 as the reference gives \(J-1\) contrasts per predictor. These are sufficient to test equality across all cutpoints: if every \(\beta_{jr}-\beta_{1r}\) is zero, then every slope equals \(\beta_{1r}\). The reference is a convenient way to encode the null, not a claim that cutpoint 1 is intrinsically special.

Building the Wald Statistic

Let \(\theta\) collect all model parameters, including thresholds and slopes, and let \(\hat\theta\) be its estimate from the fitted model. A linear null hypothesis can be written \(A\theta=c\), where \(A\) has \(q\) linearly independent rows. Let \(\widehat{\operatorname{Cov}}(\hat\theta)\) be the estimated covariance matrix from that fit. The estimated contrast vector is \(A\hat\theta-c\), and its estimated covariance matrix is obtained by applying the same linear transformation to the full fitted covariance matrix.

Under the null and standard regularity conditions, \(W\) is compared with a chi-square distribution with \(q\) degrees of freedom. The conditions include independent observations, an identifiable and correctly specified model, a true parameter in the interior of the parameter space, a nonsingular information matrix, and a fixed number of parameters as sample size increases. The usual large-sample normal approximation for the estimator and a consistent covariance estimate are also required. In finite samples, the chi-square reference distribution is an approximation.

For a single contrast \(d=a^\mathsf{T}\theta\), the Wald statistic is \(z=(\hat d-d_0)/\operatorname{SE}(\hat d)\), where \(\operatorname{SE}(\hat d)\) is the square root of its estimated variance. Squaring this statistic gives a one-degree-of-freedom Wald statistic. A two-sided confidence interval is \(\hat d\pm z_{1-\gamma/2}\operatorname{SE}(\hat d)\), with confidence level \(1-\gamma\).

Joint Tests and Confidence Regions

Testing one contrast answers whether a particular slope difference is distinguishable from zero. To assess a predictor across all cutpoints, test its \(J-1\) reference-cutpoint contrasts jointly. To test parallel regression for all \(p\) predictors, stack the contrasts for every predictor; the null then has \(p(J-1)\) independent restrictions. The degrees of freedom are the number of independent restrictions, not the number of slope estimates appearing in the model.

The corresponding joint confidence region for a \(q\)-dimensional contrast vector \(d\) is an ellipsoid. If \(\hat d\) and \(\widehat V\) are its estimate and estimated covariance matrix, the large-sample \(1-\gamma\) region is the set of values \(d_0\) satisfying

Here \(\chi^2_{q,\,1-\gamma}\) denotes the \(1-\gamma\) quantile of a chi-square distribution with \(q\) degrees of freedom. The joint null value is tested by checking whether it lies inside this region. A joint test can reject even when none of the separate unadjusted tests is individually compelling, because it evaluates the combined displacement while accounting for covariance among contrasts.

Why the Covariance Matrix Matters

The covariance between slope estimates is part of the uncertainty in their difference. More generally, every joint test depends on covariance among the contrasts. A covariance matrix for the transformed contrasts must be computed from the fitted covariance matrix using \(A\widehat{\operatorname{Cov}}(\hat\theta)A^\mathsf{T}\), not assembled by treating coefficient estimates as independent. The fitted model’s thresholds and other parameters may also be correlated with its slopes; the appropriate covariance is the one reported for the full fitted model.

Justification of the Wald Calibration

Proof. Under the null, \(A\hat\theta-c=A(\hat\theta-\theta_0)\). By the assumed asymptotic normality and linear transformation,

Consistency of the covariance estimate gives \(nA\widehat{\operatorname{Cov}}(\hat\theta)A^\mathsf{T}\xrightarrow{p}A\Sigma A^\mathsf{T}\). The limit is positive definite, so its inverse exists; continuity of matrix inversion and Slutsky’s theorem allow the estimated covariance in the Wald statistic to be replaced asymptotically by this limit. Let \(Z\sim N(0,A\Sigma A^\mathsf{T})\). The limiting quadratic form is \(Z^\mathsf{T}(A\Sigma A^\mathsf{T})^{-1}Z\). Whitening \(Z\) by the inverse square root of its covariance produces a \(q\)-dimensional standard normal vector, whose squared Euclidean length has a \(\chi^2_q\) distribution. Therefore \(W\xrightarrow{d}\chi^2_q\). \(\square\)

The covariance transformation used above follows directly from the definition of covariance. If \(U=A\hat\theta\), then \(E[U]=AE[\hat\theta]\), and

Thus the off-diagonal entries of the transformed covariance are not optional: they encode how the estimated contrasts move together. This also explains why a joint Wald test is not generally the sum of separate squared \(z\)-statistics.

Common Pitfalls

• Testing each cutpoint slope against zero instead of testing their differences. These are different hypotheses. Parallel regression requires equality across cutpoints; the slopes may all be nonzero and still be parallel.
• Ignoring covariance. The variance of \(\hat\beta_{jr}-\hat\beta_{1r}\) is the sum of the two variances minus twice their covariance. Omitting that term can change both the standard error and the conclusion.
• Counting restrictions incorrectly. For \(p\) predictors and \(J\) cutpoints, the full parallel-slope null has \(p(J-1)\) independent restrictions, assuming the contrasts are identifiable.
• Reading a nonsignificant result as proof of parallel slopes. A Wald test that does not reject says the data do not provide strong evidence against the specified equality at the chosen level; it does not establish exact equality.

Wald tests are convenient because they use a fitted model’s estimates and covariance matrix. Their approximation can be poor when estimates are imprecise, the likelihood is irregular, or the fitted covariance matrix is nearly singular. Those conditions call for careful interpretation of the result rather than a mechanical reading of the reference \(p\)-value.