Why Thresholds Replace the Intercept

Why the Intercept Creates a Redundancy

In a cumulative link model, each threshold separates adjacent regions of an underlying cumulative response scale. A common intercept also shifts the location of that scale. If both are freely estimated, they can move together without changing any fitted category probability. The data therefore cannot distinguish their separate values.

This is a structural feature of the model, not a matter of having a small sample or an inconvenient estimation algorithm. The standard parameterization avoids the redundancy by letting the thresholds carry the common location and omitting a separate intercept.

The sign convention matters: the intercept and the predictor term enter with minus signs on the right-hand side. The same argument works under other sign conventions, but its algebra must follow the stated model. Write \(\eta(x)=\gamma+x^\mathsf{T}\beta\). Since \(F\) is strictly increasing, the cumulative probability is

The individual category probabilities follow from these cumulative probabilities: the probability of category \(1\) is \(F(\theta_1-\eta)\); for \(2\leq r\leq K-1\), it is \(F(\theta_r-\eta)-F(\theta_{r-1}-\eta)\); and for category \(K\), it is \(1-F(\theta_{K-1}-\eta)\). Thus, if the cumulative probabilities stay the same, all category probabilities stay the same as well.

The Location Shift That Changes Nothing

For any real constant \(c\), add \(c\) to every threshold and to the common intercept. Their difference is unchanged:

Consequently, all cumulative probabilities—and hence all category probabilities—remain exactly the same. The model describes the differences between threshold locations and the common intercept, not their individual locations.

Proof. For each threshold index \(j\) and covariate value \(x\), the transformed cumulative linear predictor is

Applying the same function \(F\) to equal quantities gives the same \(P(Y\leq j\mid x)\). Each category probability is either one of these cumulative probabilities, a difference of two consecutive ones, or one minus the final cumulative probability. Therefore every category probability is unchanged. Since any \(c\neq 0\) gives a different parameter vector with the same distribution, the parameters are not jointly identifiable. \(\square\)

This invariance also appears directly in the likelihood. For observed responses \(y_1,\ldots,y_n\) at covariate vectors \(x_1,\ldots,x_n\), the likelihood is the product of the corresponding category probabilities. Because each probability is unchanged by the joint shift, the likelihood is unchanged. It has an entire one-dimensional family of equivalent parameter values, rather than a unique maximum in those coordinates.

Absorbing the Intercept Into the Thresholds

Define adjusted thresholds \(\delta_j=\theta_j-\gamma\). Substituting this definition gives a model with no separate common intercept:

This is not a restriction on the original model’s fitted probabilities. For any original parameter values, \(\delta_j=\theta_j-\gamma\) gives exactly the same cumulative probabilities. Conversely, for any adjusted thresholds \(\delta_1,\ldots,\delta_{K-1}\), the original model can represent them by choosing \(\gamma=0\) and \(\theta_j=\delta_j\). The common location has simply been assigned to the thresholds instead of estimated twice.

Derivation. Starting from the intercept model,

This proves that every intercept-model parameter vector yields probabilities in the no-intercept model. In the other direction, given any \((\delta_1,\ldots,\delta_{K-1},\beta)\), set \(\gamma=0\) and \(\theta_j=\delta_j\). The original equation then gives \(\delta_j-x^\mathsf{T}\beta\), the same cumulative-link value. The threshold ordering is preserved: subtracting the same \(\gamma\) from every threshold leaves their pairwise differences unchanged. Thus the two parameterizations yield the same set of distributions. \(\square\)

The adjusted thresholds also have a direct interpretation. At the reference covariate profile \(x=0\), they satisfy \(F^{-1}\{P(Y\leq j\mid x=0)\}=\delta_j\). They are the cumulative locations for that profile; changing the covariates shifts the cumulative link values through the slope terms.

What Happens to the Likelihood

The redundancy is present for individual-response and grouped-count likelihoods alike. For individual responses, write \(p_r(x)\) for the model probability of category \(r\) at covariate vector \(x\). The likelihood is \(L=\prod_{i=1}^n p_{y_i}(x_i)\). For grouped counts, the likelihood is a product of category probabilities raised to their observed counts, multiplied by a factor that does not depend on the model parameters. In either case, identical fitted category probabilities give an identical likelihood.

Thus, fitting the redundant version cannot identify a unique value of \(\gamma\) and the thresholds separately. An optimizer may report a singular information matrix, an arbitrary point along the flat direction, or a convergence warning. Those outcomes do not establish that the fitted probabilities themselves are undefined; they indicate that the parameterization contains a direction the data cannot resolve.

Reference Coding and the Common Location

With categorical predictors, reference coding gives a useful interpretation of the no-intercept convention. If \(x=0\) represents the reference category of a factor, the thresholds describe the cumulative probabilities for that reference profile. The factor’s coefficients describe how other levels shift the cumulative link relative to it. Adding a separate common intercept would not supply a new, independently estimable location: its effect can still be absorbed into every threshold.

This issue is distinct from whether the slope coefficients are identifiable. Even after removing the common intercept, the observed design must contain enough information to estimate the slopes. For example, a predictor that never varies cannot have its slope learned from the data. Removing the redundant intercept resolves this particular location aliasing; it does not guarantee that every remaining parameter is estimable.

Common Mistake

Wrong reasoning: “The intercept belongs to the regression part, while thresholds belong to the ordinal response, so they describe different things and can both be estimated.”

Correction: In the cumulative link equation, each threshold and the common intercept enter only through their difference, \(\theta_j-\gamma\). Shifting both by the same amount leaves every cumulative probability unchanged. They are therefore not separately identified, regardless of which part of the model one informally associates them with. Use thresholds without a separate common intercept, or impose an explicit identifying constraint if a different parameterization is required.