Testing Structured Threshold Constraints

Why Test a Threshold Structure?

An ordinal model can use a separate threshold at each boundary between response categories. That flexibility is useful, but it may spend parameters describing a pattern that could be summarized more simply. A symmetric or equally spaced threshold structure makes a specific claim about those boundaries. A likelihood-ratio test can assess whether the simpler structure loses enough fit to count as a meaningful restriction.

The test is not a test of the proportional-odds assumption: both models in the comparison can retain the same common slopes. Here the question is narrower: do the thresholds need to be unrestricted, or is a specified structure adequate? Because the restrictions also change the cumulative probabilities, it is important to inspect fitted category probabilities alongside the test statistic.

Models and Threshold Restrictions

Let \(Y\) have ordered categories \(1,\ldots,K\), with \(K\geq 3\), and write \(J=K-1\) for the number of thresholds. In a cumulative-logit model with no separate intercept, let \(P(Y\leq j\mid x)=G(\alpha_j-x^\mathsf{T}\beta)\), where \(G(t)=1/(1+\exp(-t))\), \(j=1,\ldots,J\), and \(\alpha_1<\cdots<\alpha_J\). The slope vector \(\beta\) is common across thresholds in both models being compared. Excluding a separate intercept avoids confounding its location with the thresholds.

The unrestricted model estimates all \(J\) thresholds freely, subject to their required order. A symmetric threshold structure about zero imposes \(\alpha_j=-\alpha_{K-j}\) for \(j=1,\ldots,J\). An equally spaced threshold structure imposes \(\alpha_j=a+(j-1)d\), where \(a\) is a location parameter and \(d>0\) is the common spacing. The spacing must be positive to preserve threshold order.

Definition: A likelihood-ratio test compares a larger model with a nested restricted model using \(D=2\{\ell_U-\ell_R\}\), where \(\ell_U\) and \(\ell_R\) are their maximized log-likelihoods. The restriction is assessed by comparing \(D\) with a chi-square distribution whose degrees of freedom equal the difference in the models’ parameter dimensions, under the regularity conditions for Wilks’ theorem.

The restrictions concern the thresholds, while the common slopes remain free in both models. Thus the slope parameters are nuisance parameters for this comparison; they do not add to the difference in dimension. If a model instead includes a freely estimated intercept, the location convention must be made identifiable before testing. In particular, symmetry about a freely estimated center is a different restriction from symmetry about zero and generally has a different parameter count.

Counting the Degrees of Freedom

The unrestricted threshold vector has \(J=K-1\) free coordinates. For symmetric thresholds about zero, each pair \((\alpha_j,\alpha_{K-j})\) contributes one independent restriction. When \(K\) is even, there is also a middle threshold that must equal zero. The total number of independent restrictions is therefore \(\lfloor K/2\rfloor\).

Result: Comparing unrestricted thresholds with symmetry about zero uses \(\lfloor K/2\rfloor\) degrees of freedom. Equally spaced thresholds have two free parameters, \(a\) and \(d\), so for \(K\geq4\) their comparison with unrestricted thresholds uses \(K-3\) degrees of freedom. With \(K=3\), any two ordered thresholds are equally spaced, so that restriction adds no testable constraint.

To verify the symmetric count, if \(K\) is odd, the \(K-1\) thresholds form \((K-1)/2\) pairs and there is no middle threshold. There are \((K-1)/2=\lfloor K/2\rfloor\) restrictions. If \(K\) is even, there are \((K-2)/2\) pairs plus the middle-threshold restriction, for a total of \(K/2=\lfloor K/2\rfloor\). For equal spacing, the unrestricted dimension is \(K-1\), while the restricted dimension is two; their difference is \(K-3\).

Wilks’ theorem gives the asymptotic reference distribution \(D\xrightarrow{d}\chi^2_r\), where \(r\) is the dimension difference, when the null model is correctly specified, the true restricted parameter is an interior point of a smooth identifiable parameter space, the information is nonsingular, and the usual likelihood regularity conditions hold. The observations must provide enough information for the fitted parameters to be estimable. Sparse categories, separation, or estimates near a boundary can make the chi-square approximation unreliable.

Worked Examples

Worked Example: Testing Symmetry With Three Categories

Suppose an intercept-only cumulative-logit model has three categories and observed counts \((10,50,40)\), so \(n=100\). The unrestricted fitted probabilities are the observed proportions \((0.10,0.50,0.40)\). Compare them with symmetry about zero.

With three categories, symmetry requires \(\alpha_1=-\alpha_2\). If \(q=P(Y=1)=P(Y=3)\), the constrained probabilities are \((q,1-2q,q)\). The constrained log-likelihood, apart from the multinomial coefficient, is \(\ell(q)=50\log q+50\log(1-2q)\), since the first and third counts sum to 50.

Differentiating gives \(\ell'(q)=50/q-100/(1-2q)\). Setting this to zero yields \(50(1-2q)=100q\), hence \(\hat q=0.25\); this is the maximum because the log-likelihood is strictly concave for \(0<q<1/2\). The constrained probabilities are therefore \((0.25,0.50,0.25)\).

The likelihood-ratio statistic is \(D=2\{10\log(0.10/0.25)+50\log(0.50/0.50)+40\log(0.40/0.25)\}\). Using \(\log(0.4)=-0.9162907\) and \(\log(1.6)=0.4700036\), this is \(2\{10(-0.9162907)+40(0.4700036)\}=19.2745\). There is one restriction, since \(\lfloor 3/2\rfloor=1\); the asymptotic reference distribution is \(\chi^2_1\). The observed discrepancy is substantial evidence against symmetric thresholds.

The threshold estimates show the same mismatch. Unrestricted, they are \(\operatorname{logit}(0.10)=-2.1972\) and \(\operatorname{logit}(0.60)=0.4055\), which are not negatives of each other. Under symmetry they become \(\operatorname{logit}(0.25)=-1.0986\) and \(\operatorname{logit}(0.75)=1.0986\).

Worked Example: Degrees of Freedom for Equal Spacing

An ordinal response has five categories, so the unrestricted model estimates four ordered thresholds. An equally spaced structure writes them as \(\alpha_1=a\), \(\alpha_2=a+d\), \(\alpha_3=a+2d\), and \(\alpha_4=a+3d\), with \(d>0\). It has two threshold parameters rather than four, so the likelihood-ratio test has \(4-2=2\) degrees of freedom.

Suppose the maximized log-likelihoods from fitting the two nested models to the same observations are \(\ell_U=-312.40\) and \(\ell_R=-315.05\). Then \(D=2\{(-312.40)-(-315.05)\}=5.30\). For two degrees of freedom, the chi-square upper-tail probability is \(\exp(-5.30/2)\approx0.0707\). At a 5% level, this comparison does not reject equal spacing. It does not establish that the structure is exactly true; it says the likelihood loss is not unusually large under the stated approximation.

Worked Example: Checking Category Probabilities After Restriction

For four categories at a fixed covariate value, suppose unrestricted threshold estimates are \((-0.2,0.8,1.5)\). The logistic cumulative probabilities are \(G(-0.2)=0.4501660027\), \(G(0.8)=0.6899744811\), and \(G(1.5)=0.8175744762\). The category probabilities are the first cumulative probability, successive differences, and one minus the last cumulative probability.

Using the unrounded cumulative values gives \(p_1=0.4501660027\), \(p_2=0.6899744811-0.4501660027=0.2398084784\), \(p_3=0.8175744762-0.6899744811=0.1275999951\), and \(p_4=1-0.8175744762=0.1824255238\). Rounded to five decimal places, these are \((0.45017,0.23981,0.12760,0.18243)\). Their displayed sum is 1.00001 because each entry was rounded; the unrounded probabilities sum to one.

For four categories, thresholds \((-0.8,0,0.8)\) are both symmetric about zero and equally spaced. They produce cumulative probabilities \(G(-0.8)=0.3100255\), \(G(0)=0.5\), and \(G(0.8)=0.6899745\), hence category probabilities approximately \((0.31003,0.18997,0.18997,0.31003)\). This illustrates why a threshold restriction should be checked in terms of its fitted probabilities: even if the restricted model is more compact, it may redistribute probability substantially across categories. These values illustrate the probability calculation; a formal test still requires comparing maximized likelihoods.

What the Test Does—and Does Not—Say

The likelihood-ratio statistic is nonnegative because the restricted parameter space is contained in the unrestricted one: maximizing over the larger space cannot give a smaller likelihood. Its size measures the loss of fit caused by the specified restriction, relative to the sampling variation represented by the chi-square approximation. It does not say which category probabilities are most affected, so report or inspect fitted probabilities at substantively relevant covariate values as well.

A symmetric-threshold test and an equally-spaced-threshold test are each comparisons with the same unrestricted model. Neither restricted model is generally a subset of the other, so their likelihoods should not be compared to each other with an ordinary nested likelihood-ratio test. If comparing either with the unrestricted model, fit both to the same observations using the same response categories and predictor specification.

A frequent mistake is to count the number of thresholds as the chi-square degrees of freedom. That is incorrect: the reference degrees of freedom are the number of independent restrictions, or equivalently the difference in parameter dimensions. Another mistake is to round cumulative probabilities before taking their differences; as the last example shows, that can change a category probability at the reported precision. Keep full precision through the subtraction and round only the final probabilities.

Key takeaway: State the exact threshold restriction, count its independent constraints, and compare maximized log-likelihoods using the corresponding degrees of freedom. Then inspect the fitted category probabilities: a nonsignificant test does not guarantee that the restricted predictions are harmless for the application.

Check Your Understanding

Use the definitions and calculations above to answer these questions.

For six response categories, how many degrees of freedom are used to test symmetry about zero against unrestricted thresholds? Explain the count.
For five categories, how many degrees of freedom are used to test equally spaced thresholds against unrestricted thresholds?
If the restricted and unrestricted maximized log-likelihoods are \(-80.2\) and \(-77.7\), respectively, calculate the likelihood-ratio statistic.
Why does a symmetric-threshold test not by itself establish that the restricted model’s category probabilities are acceptable?
When calculating a category probability from two cumulative probabilities, why should the subtraction be performed before rounding?