Category Probabilities by Differencing Cumulative Probabilities

From Cumulative Probabilities to Category Probabilities

In an ordinal response, the categories have a meaningful order, but a cumulative probability does not give the probability of one category. It gives the probability of being at or below a cutpoint. To obtain the probability of an individual category, take the difference between the cumulative probability at its upper boundary and the cumulative probability immediately before it.

This subtraction is especially important in a cumulative link model: the model specifies cumulative probabilities, while interpretation often requires the probability of each response category. The first and last categories need boundary conventions, because there is no category below the first or above the last.

The boundary definitions make the formula apply uniformly to all categories. In particular, the first category has probability \(p_1(x)=q_1(x)\), since \(q_0(x)=0\). The last has probability \(p_K(x)=1-q_{K-1}(x)\), since \(q_K(x)=1\). For a middle category \(j\), its probability is \(q_j(x)-q_{j-1}(x)\).

Why Adjacent Differences Give Category Probabilities

The derivation follows directly from the events involved. The event \(\{Y\leq j-1\}\) is contained in \(\{Y\leq j\}\). The part of \(\{Y\leq j\}\) that is not in \(\{Y\leq j-1\}\) is exactly \(\{Y=j\}\). Thus, subtracting the probability of the smaller event from that of the larger one leaves the probability of category \(j\).

Proof. For \(1\leq j\leq K\), the events \(\{Y=j\}\) and \(\{Y\leq j-1\}\) are disjoint, and their union is \(\{Y\leq j\}\). Conditional on \(x\), finite additivity of probability therefore gives

Rearranging gives the stated difference. For \(j=1\), the event \(\{Y\leq 0\}\) is empty, so its probability is \(q_0(x)=0\). For \(j=K\), \(\{Y\leq K\}\) contains every possible value of \(Y\), so its probability is \(q_K(x)=1\). These cases give the first- and last-category formulas as well. \(\square\)

The reverse calculation is also useful: adding the category probabilities up to category \(j\) recovers the cumulative probability at \(j\). This gives a direct check that differencing and cumulative summation are inverse operations.

Proof. Substitute \(p_r(x)=q_r(x)-q_{r-1}(x)\) into the sum:

Every intermediate cumulative probability appears once with a positive sign and once with a negative sign, so those terms cancel. Taking \(j=K\) and using \(q_K(x)=1\) proves that the category probabilities sum to one. \(\square\)

Worked Three-Category Calculations

With three ordered categories, the boundary and middle-category calculations can be written out explicitly:

The first cumulative probability is already the probability of category 1. The second cumulative probability combines categories 1 and 2, so subtracting the first leaves category 2. The probability left after category 2 is the probability of category 3.

Differencing a Cumulative-Logit Model

In a cumulative-logit model with shared slopes, a common parameterization is \(\operatorname{logit}\{q_j(x)\}=\alpha_j-x^\mathsf{T}\beta\) for \(j=1,\ldots,K-1\). Here \(\alpha_j\) is the cutpoint for cumulative category \(j\), and the same slope vector \(\beta\) appears at every cutpoint. The corresponding cumulative probability is \(q_j(x)=\operatorname{expit}(\alpha_j-x^\mathsf{T}\beta)\), where \(\operatorname{expit}(u)=1/(1+\exp(-u))\).

The category probabilities follow by differencing these cumulative probabilities—not by applying the inverse-logit separately and treating each result as a category probability. For three categories, the model gives \(p_1=q_1\), \(p_2=q_2-q_1\), and \(p_3=1-q_2\). The shared slope specifies how the cumulative logits depend on \(x\); the subtraction converts the resulting cumulative probabilities into category probabilities.

Checks and a Common Mistake

Differencing is simple arithmetic, but it is easy to confuse what the cumulative model supplies with what the category probabilities represent. A cumulative probability includes all categories up to a cutpoint; a category probability isolates one outcome. The subtraction must therefore use adjacent cutpoints, while the first and last categories use the boundary values zero and one.

• First category: use \(q_1\), not \(q_1-q_0\) with an unspecified lower boundary; by definition \(q_0=0\).
• Middle category: subtract the cumulative probability at the previous cutpoint, \(q_{j-1}\), from \(q_j\).
• Last category: subtract \(q_{K-1}\) from one, not from zero.
• Check: category probabilities should add to one, and their successive sums should reproduce the cumulative probabilities.

A common mistake is to read \(q_2=0.70\) in a three-category example as the probability of category 2. That is incorrect: \(q_2=P(Y\leq2\mid x)\) includes both categories 1 and 2. The correction is \(p_2=q_2-q_1\). Similarly, in a cumulative-logit model, the inverse-logit of a cutpoint gives a cumulative probability; it gives the category probability directly only for the first category.

For differences to represent valid category probabilities, the supplied cumulative probabilities must form a nondecreasing sequence from \(q_0=0\) to \(q_K=1\). Then each adjacent difference is nonnegative, and the telescoping calculation shows the differences sum to one. The central practical rule is therefore: identify the cumulative probabilities at the two boundaries of the category, subtract the lower cumulative probability from the upper one, and use zero or one at the endpoints.