What Nonproportional Slopes Imply Across Cutpoints

From Cumulative Probabilities to Category Probabilities

In an ordinal response model, the fitted probabilities at successive cutpoints must describe a coherent ordering of the response categories. With a common slope at every cutpoint, ordered thresholds ensure this ordering. When slopes differ by cutpoint, however, the cumulative probabilities can change order as covariates change. A reversed pair does not merely make an interpretation awkward: it produces a negative fitted probability for a category.

Let $Y$ take the ordered values $1,\ldots,K$, and let $x$ be a covariate vector. For cutpoint $j$, where $j=1,\ldots,K-1$, define $q_j(x)=P(Y\leq j\mid x)$. Consider the cumulative-logit specification

\operatorname{logit}\{q_j(x)\} =\log\!\left(\frac{q_j(x)}{1-q_j(x)}\right) =\alpha_j-x^\mathsf{T}\beta_j.

Here $\alpha_j$ is the threshold for cutpoint $j$, and $\beta_j$ is its slope vector. The slopes may differ across cutpoints. Write $\operatorname{expit}(t)=1/(1+\exp(-t))$, so that $q_j(x)=\operatorname{expit}(\alpha_j-x^\mathsf{T}\beta_j)$. The logistic function is strictly increasing and takes values strictly between zero and one.

Definition: The fitted category probabilities are $p_1(x)=q_1(x)$, $p_j(x)=q_j(x)-q_{j-1}(x)$ for $j=2,\ldots,K-1$, and $p_K(x)=1-q_{K-1}(x)$. These formulas follow because the event $Y=j$, for an interior category, is the difference between $Y\leq j$ and $Y\leq j-1$.

These differences sum to one by cancellation. Their nonnegativity is not automatic: an interior probability $p_j(x)$ is nonnegative exactly when $q_{j-1}(x)\leq q_j(x)$. Thus each covariate value requires the cumulative probabilities to be ordered from the first cutpoint to the last.

The Ordering Condition for Nonparallel Slopes

The cumulative probabilities are ordered if and only if their linear predictors are ordered, because the expit function is strictly increasing. This gives a direct check in terms of thresholds and slopes.

Theorem: At a fixed covariate value $x$, the fitted category probabilities are all nonnegative and sum to one if and only if $$ \alpha_j-x^\mathsf{T}\beta_j \leq \alpha_{j+1}-x^\mathsf{T}\beta_{j+1}, \qquad j=1,\ldots,K-2. $$ The resulting probabilities are all strictly positive if and only if each of these inequalities is strict.

Proof. By definition, $p_1(x)=q_1(x)>0$ and $p_K(x)=1-q_{K-1}(x)>0$, since each logistic cumulative probability is strictly between zero and one. For an interior category, $p_j(x)\geq0$ if and only if $q_{j-1}(x)\leq q_j(x)$. Since expit is strictly increasing, this is equivalent to $\alpha_{j-1}-x^\mathsf{T}\beta_{j-1}\leq\alpha_j-x^\mathsf{T}\beta_j$. Requiring this for every interior category gives exactly the stated adjacent inequalities. If they hold, all category probabilities are nonnegative; moreover, their sum is $q_1+(q_2-q_1)+\cdots+(q_{K-1}-q_{K-2})+(1-q_{K-1})=1$. The inequalities are strict exactly when all interior differences are positive. $\square$

Worked Example: Valid Nonparallel Slopes at a Covariate Value

Suppose $Y$ has three categories, $x$ is scalar, and $\alpha_1=-1,\ \alpha_2=1,\ \beta_1=0.5,\ \beta_2=1$. The slopes differ. At $x=1$, calculate the cumulative and category probabilities.

The two cumulative-logit predictors are $\eta_1=-1-(1)(0.5)=-1.5$ and $\eta_2=1-(1)(1)=0$. Therefore $q_1=\operatorname{expit}(-1.5)\approx0.1824$ and $q_2=\operatorname{expit}(0)=0.5000$.

For three categories, $p_1=q_1$, $p_2=q_2-q_1$, and $p_3=1-q_2$. Hence $p_1\approx0.1824$, $p_2\approx0.5000-0.1824=0.3176$, and $p_3=1-0.5000=0.5000$. Each is nonnegative and their sum is $0.1824+0.3176+0.5000=1.0000$. The predictor ordering, $-1.5<0$, confirms validity at this $x$.

Why Validity Can Depend on the Covariate Domain

For adjacent cutpoints, the ordering condition can be rearranged as \[ (\alpha_{j+1}-\alpha_j)-x^\mathsf{T}(\beta_{j+1}-\beta_j)\geq0. \] This is a constraint on $x$, not just on the thresholds. Unequal slopes can satisfy it on some covariate values and fail on others. In particular, validity at the observed values does not by itself imply validity at every possible value of a continuous covariate.

Worked Example: Validity on a Restricted Range

Consider three categories with $\alpha_1=-2,\ \alpha_2=0,\ \beta_1=0.2,\ \beta_2=0.8$, and suppose predictions are required only for $0\leq x\leq2$. The adjacent predictor difference is \[ \eta_2(x)-\eta_1(x) =(0-0.8x)-(-2-0.2x)=2-0.6x. \] On the specified range this is at least $2-0.6(2)=0.8>0$, so the cumulative probabilities are ordered throughout that range.

At $x=2$, the predictors are $\eta_1=-2-0.2(2)=-2.4$ and $\eta_2=0-0.8(2)=-1.6$. Thus $q_1\approx0.0832$ and $q_2\approx0.1680$, giving $p_1\approx0.0832$, $p_2\approx0.1680-0.0832=0.0848$, and $p_3\approx1-0.1680=0.8320$. They are positive and sum to one, up to rounding. The unequal slopes are compatible with valid probabilities on the stated range; checking that range is essential to the conclusion.

When Slopes Differ, Crossing May Produce Invalid Probabilities

A crossing occurs when two cumulative probabilities become equal and then reverse order. At the equality point, the intervening category has probability zero. Beyond it, the difference is negative. The following result describes what happens if predictions are required over an unrestricted covariate space.

Theorem: Suppose the covariate vector $x$ may take every value in $\mathbb{R}^p$. The cumulative probabilities are ordered for every $x$ if and only if all cutpoint slope vectors are equal and the thresholds are ordered: $\beta_1=\cdots=\beta_{K-1}$ and $\alpha_1\leq\cdots\leq\alpha_{K-1}$.

Proof. For each adjacent pair, ordering at every $x$ requires \[ (\alpha_{j+1}-\alpha_j)-x^\mathsf{T}(\beta_{j+1}-\beta_j)\geq0 \quad\text{for every }x\in\mathbb{R}^p. \] If $\beta_{j+1}-\beta_j\neq0$, choose $x=t(\beta_{j+1}-\beta_j)$, where $t>0$. The left side becomes \[ \alpha_{j+1}-\alpha_j-t\|\beta_{j+1}-\beta_j\|^2, \] which is negative for sufficiently large $t$. This contradicts ordering for every $x$. Therefore adjacent slopes must be equal, and repeating the argument gives a common slope vector. With equal slopes, the predictor difference is the constant $\alpha_{j+1}-\alpha_j$, so ordering at every $x$ holds exactly when the thresholds are ordered. $\square$

Worked Example: A Crossing That Creates a Negative Probability

Take three categories with $\alpha_1=-1,\ \alpha_2=1,\ \beta_1=1,\ \beta_2=-1$. At $x=-2$, the predictors are $\eta_1=-1-(-2)(1)=1$ and $\eta_2=1-(-2)(-1)=-1$. Consequently, $q_1=\operatorname{expit}(1)\approx0.7311$ while $q_2=\operatorname{expit}(-1)\approx0.2689$.

The middle-category probability is therefore $p_2=q_2-q_1\approx0.2689-0.7311=-0.4622$, which is impossible for a probability. The thresholds are ordered, but that does not rescue the model at this covariate value: the slopes reverse the predictor order. In fact, $\eta_2(x)-\eta_1(x)=2+2x$, which is negative for $x<-1$, zero at $x=-1$, and positive for $x>-1$. At the crossing $x=-1$, the middle-category probability is zero.

Practical Checks and a Common Mistake

For a proposed set of cutpoint-specific slopes, validity should be checked over the covariate domain where fitted probabilities will be used. With one covariate and a specified interval, each adjacent predictor difference is a linear function, so its minimum on a closed interval occurs at an endpoint. With several covariates, the same issue is a set of linear inequalities over the chosen domain. For a finite collection of covariate patterns, evaluate every adjacent predictor difference at each pattern.

A common mistake is to assume that ordered thresholds alone guarantee valid category probabilities. That reasoning omits the covariate-dependent slope terms. For instance, $\alpha_1<\alpha_2$ does not ensure $\alpha_1-x^\mathsf{T}\beta_1<\alpha_2-x^\mathsf{T}\beta_2$ at all $x$ when the slopes differ. The correction is to check the adjacent predictor inequalities, or equivalently the cumulative-probability ordering, wherever predictions are required. A negative difference is not a small numerical inconvenience; it means the specified cumulative probabilities do not define a categorical distribution at that covariate value.

Key takeaway: Cutpoint-specific slopes can yield valid category probabilities on a restricted covariate domain, but the adjacent cumulative logits must remain ordered throughout that domain. If validity is required for every $x\in\mathbb{R}^p$, the cutpoint slopes must be equal and the thresholds ordered.

Check Your Understanding

For each question, focus on the ordering of adjacent cumulative probabilities.

For three response categories, write the three category probabilities in terms of $q_1(x)$ and $q_2(x)$.
State the condition on the two cumulative-logit predictors that ensures the middle-category probability is nonnegative.
If $\eta_2(x)-\eta_1(x)=3-2x$, for which values of $x$ are the three fitted category probabilities valid?
Why can ordered thresholds fail to guarantee nonnegative category probabilities when slopes vary by cutpoint?
If a model must give valid probabilities for every real value of a scalar covariate, what must be true of the cutpoint slopes?