Why Cumulative Probabilities Must Be Ordered

Why the Order of Cumulative Probabilities Matters

For an ordinal response, categories have a meaningful order, such as mild, moderate, and severe. The cumulative probability at cutpoint \(j\) is the probability that the response is in category \(j\) or below. These probabilities cannot be chosen independently: the event of being at or below category \(j\) is contained in the event of being at or below category \(j+1\). Its probability therefore cannot be larger.

This basic constraint is also essential for cumulative link models. Such a model assigns cumulative probabilities using ordered thresholds and an increasing inverse link. The ordering of the thresholds then carries through to the ordering of the cumulative probabilities, so differencing them gives category probabilities that are not negative.

The Probability Ordering Condition

Each cumulative event contains the preceding one: \(\{Y\leq j\}\) is a subset of \(\{Y\leq j+1\}\). This gives \(C_j(x)\leq C_{j+1}(x)\). In addition, probabilities lie between zero and one. These conditions are not merely convenient properties of cumulative probabilities. Together, they are exactly what is required for their successive differences to be valid category probabilities.

Proof. First suppose the stated cumulative inequalities hold. For every category \(j\), \(C_j(x)\geq C_{j-1}(x)\), so \(p_j(x)=C_j(x)-C_{j-1}(x)\geq 0\). The category probabilities sum to

Conversely, suppose the category probabilities \(p_1(x),\ldots,p_K(x)\) are nonnegative and sum to one, and define \(C_j(x)=\sum_{r=1}^{j}p_r(x)\) for \(j=1,\ldots,K-1\). Then \(C_1(x)\geq 0\), and \(C_{j+1}(x)-C_j(x)=p_{j+1}(x)\geq 0\). Also, \(C_{K-1}(x)=1-p_K(x)\leq 1\). Thus the cumulative probabilities satisfy the required bounds and ordering. \(\square\)

This result gives a quick diagnostic. If a proposed cumulative sequence decreases at any step, the difference at that step is a negative category probability. If the cumulative values are ordered and bounded, the differences are nonnegative and sum to one; no additional probability check is needed.

How Ordered Thresholds Ensure the Required Order

In a cumulative link model, let \(g\) be an increasing link function and let \(F=g^{-1}\) be its inverse, also increasing. With the common cumulative-link convention, the model is

or, equivalently,

Here \(\theta_1,\ldots,\theta_{K-1}\) are thresholds, \(\beta\) is a shared slope vector, and the same \(x^\mathsf{T}\beta\) appears at every cutpoint. The logit link, for example, has the logistic inverse \(F(t)=1/(1+\exp(-t))\), which is strictly increasing and takes values between zero and one.

Proof. Fix \(x\), and write \(a=x^\mathsf{T}\beta\). Subtracting the same \(a\) from each ordered threshold preserves their order: \(\theta_j-a\leq\theta_{j+1}-a\). Since \(F\) is nondecreasing, applying \(F\) preserves this inequality:

The range of \(F\) gives \(0\leq C_j(x)\leq 1\). The preceding theorem then shows that the differences, including the first and last category probabilities, are nonnegative and sum to one. \(\square\)

The shared predictor is important to this argument: it means that changing the cutpoint changes only the threshold, not the covariate contribution. If a model instead gives each cutpoint a different predictor, ordered thresholds alone do not establish the ordering at every \(x\). The guarantee stated here is for the cumulative link form with a common predictor across cutpoints.

Strictness, Zero Probabilities, and a Common Pitfall

Nondecreasing cumulative probabilities guarantee probabilities that are nonnegative, not necessarily strictly positive. If two consecutive cumulative probabilities are equal, then the category between them has probability zero. For example, if \(C_1=C_2=0.40\), then \(p_2=C_2-C_1=0\). With strictly ordered thresholds and a strictly increasing inverse link, the intermediate cumulative probabilities are strictly ordered, so the corresponding interior category probabilities are positive. This statement concerns the interior categories; the first and last category probabilities depend on the cumulative values relative to zero and one.

A common mistake is to check only that each cumulative probability lies in \([0,1]\). The invalid sequence \(0.30,0.25,0.80\) shows why this fails: bounded values can still decrease, creating a negative difference. The correction is to check the complete chain \(0\leq C_1\leq\cdots\leq C_{K-1}\leq1\). In a cumulative link model, ordered thresholds and a common predictor provide a structural reason for this chain, rather than requiring a separate check at each covariate value.