Ordered Thresholds and Admissible Category Probabilities

From Ordered Categories to Cumulative Probabilities

An ordinal response has categories with a meaningful order, such as mild, moderate, and severe. A cumulative-link model describes the probability of being at or below each cutpoint. The resulting cumulative probabilities must themselves be ordered: the probability of being at or below a higher category cannot be smaller than the probability of being at or below a lower one. This requirement is what makes the category probabilities obtained from them valid.

Let \(Y\) take one of the ordered categories \(1,\ldots,K\), where \(K\geq 2\), and let \(x\) denote a covariate value. Write \(q_j(x)=P(Y\leq j\mid x)\) for \(j=1,\ldots,K-1\). The first \(j\) categories are included in this cumulative event; category \(j\) itself is not the only category represented by \(q_j(x)\).

In a cumulative-link model with a common regression effect, a standard parameterization is \(q_j(x)=F(\theta_j-x^\mathsf{T}\beta)\). Here \(F\) is a cumulative distribution function, \(\theta_1,\ldots,\theta_{K-1}\) are thresholds, and \(x^\mathsf{T}\beta\) is the common linear predictor. Logistic \(F\) gives the cumulative logit model; standard normal \(F\) gives the cumulative probit model. The threshold ordering is \(\theta_1<\cdots<\theta_{K-1}\). This order refers to the cutpoints, not to a ranking of the response probabilities \(p_j(x)\).

What Makes Category Probabilities Admissible?

A set of fitted category probabilities is admissible if every probability is nonnegative and the probabilities sum to one. The cumulative probabilities meet this requirement precisely when they start at zero, end at one, and do not decrease as the cutpoint advances.

Proof. If the cumulative probabilities are nondecreasing, then \(p_j=q_j-q_{j-1}\geq 0\) for every \(j\). Their sum telescopes:

Conversely, if all \(p_j\geq0\), then \(q_j-q_{j-1}=p_j\geq0\), so each successive pair is ordered. Since the endpoints are \(q_0=0\) and \(q_K=1\), this gives the stated chain of inequalities. Finally, \(p_j>0\) holds exactly when \(q_j>q_{j-1}\), proving the strict version as well. \(\square\)

This result separates two checks that are easy to conflate. Differences of adjacent cumulative probabilities always telescope to one when the endpoints are set to zero and one. But if any cumulative probability decreases, at least one difference is negative, so the numbers cannot be valid category probabilities.

Why Ordered Thresholds Preserve the Cumulative Order

The cumulative-link structure gives a direct condition for admissibility. Suppose \(F\) is a nondecreasing cumulative distribution function and the same linear predictor is subtracted at every cutpoint. If \(\theta_j\leq\theta_{j+1}\), then \(\theta_j-x^\mathsf{T}\beta\leq\theta_{j+1}-x^\mathsf{T}\beta\) for every \(x\). Applying a nondecreasing \(F\) preserves that order, so \(q_j(x)\leq q_{j+1}(x)\). With a strictly increasing \(F\) and strictly ordered thresholds, these cumulative inequalities are strict at every finite predictor value.

Proof. For any adjacent thresholds, \(\theta_j\leq\theta_{j+1}\). Subtracting the same real number \(x^\mathsf{T}\beta\) from both sides gives \(\theta_j-x^\mathsf{T}\beta\leq\theta_{j+1}-x^\mathsf{T}\beta\). Because \(F\) is nondecreasing,

The boundary values \(q_0(x)=0\) and \(q_K(x)=1\) complete the ordered sequence, since a CDF takes values in \([0,1]\). The preceding theorem then gives nonnegative category probabilities summing to one. If \(F\) is strictly increasing and \(\theta_j<\theta_{j+1}\), the arguments are strictly ordered, so \(q_j(x)<q_{j+1}(x)\). The same strict inequalities hold at the boundaries for finite arguments when \(F\) takes values strictly between zero and one, as logistic and normal CDFs do. Therefore all category probabilities are positive for those links. \(\square\)

The shared predictor matters to this guarantee: it shifts every threshold argument by the same amount, leaving their order unchanged. More generally, if cumulative probabilities are specified through threshold-specific predictors \(a_j(x)\) as \(q_j(x)=F(a_j(x))\), admissibility at a given \(x\) requires \(a_j(x)\leq a_{j+1}(x)\) whenever \(F\) is strictly increasing. Merely labeling parameters as thresholds does not establish that their resulting cumulative probabilities are ordered; the relevant check is the order of the cumulative probabilities (or, for a strictly increasing \(F\), of their arguments).

Common Pitfall and Practical Checks

A common mistake is to check only that the category probabilities add to one. Differencing an unordered sequence of cumulative values still telescopes to one, as the previous example shows, but it can produce a negative entry. Another mistake is to assume that increasing thresholds alone establish the order of cumulative probabilities without checking the model formula. In the common-predictor cumulative-link model, the same predictor is subtracted from each threshold, so threshold order is preserved. With other specifications, check the actual cumulative values at the covariate values of interest.

A reliable calculation follows the same sequence each time: write down all cumulative probabilities in category order, include the boundary values zero and one, verify that the sequence is nondecreasing, and only then difference adjacent entries. This checks both nonnegativity and normalization rather than relying on the sum alone.