Calculating Fitted Category Probabilities

From Fitted Values to Category Probabilities

A fitted cumulative-link model naturally gives probabilities for events of the form “the response is at or below category \(j\).” In many applications, however, the quantity of interest is the probability of each individual response category: for example, the fitted probability of mild, moderate, or severe symptoms. The calculation has two stages: evaluate the fitted cumulative probabilities at the covariate value, then take differences between adjacent cumulative probabilities.

The order of the response categories matters throughout. We use categories \(1,\ldots,K\), in their specified order, and the cumulative probability at cutpoint \(j\) means the probability of category \(j\) or any category below it. Throughout this tutorial, use the cumulative-link convention \(\operatorname{link}\{P(Y\leq j\mid \boldsymbol{x})\}=\alpha_j-\boldsymbol{x}^{\mathsf T}\boldsymbol{\beta}\). The calculations below use the fitted version of this equation.

For the cumulative logit model, \(G(z)=1/(1+\exp(-z))\), the logistic cumulative distribution function. For the cumulative probit model, \(G(z)=\Phi(z)\), where \(\Phi\) is the standard normal cumulative distribution function. In either case, first compute \(\hat{\eta}\), then evaluate the cumulative probabilities at the threshold-minus-predictor values, and only then take adjacent differences. The first category uses \(\hat{q}_1-\hat{q}_0=\hat{q}_1\); the last uses \(\hat{q}_K-\hat{q}_{K-1}=1-\hat{q}_{K-1}\).

Worked Calculations

Why Differencing Produces a Probability Distribution

The calculation is valid when the cumulative probabilities are ordered. In the usual fitted cumulative logit or probit model, this ordering follows from ordered thresholds: the same predictor is subtracted at every cutpoint, and the link CDF is nondecreasing. The following result states precisely why adjacent differences then form valid category probabilities.

Proof. For each \(j\) from 1 to \(K-2\), ordered thresholds give \(\hat{\alpha}_j-\hat{\eta}\leq\hat{\alpha}_{j+1}-\hat{\eta}\). Since \(G\) is nondecreasing, \(\hat{q}_j\leq\hat{q}_{j+1}\). Also \(0\leq\hat{q}_1\), because CDF values are nonnegative, and \(\hat{q}_{K-1}\leq1\), because CDF values are at most one. Thus the full sequence satisfies \(0=\hat{q}_0\leq\hat{q}_1\leq\cdots\leq\hat{q}_{K-1}\leq\hat{q}_K=1\). Every adjacent difference \(\hat{p}_j=\hat{q}_j-\hat{q}_{j-1}\) is therefore nonnegative. The sum telescopes:

This proves both claims. \(\square\)

There is also a converse bookkeeping identity: once category probabilities have been computed, summing them up through category \(j\) recovers the cumulative probability at that cutpoint. This is useful both conceptually and as a check on a table of results.

Proof. Substitute the definition of each category probability into the sum. The intermediate cumulative terms cancel in adjacent pairs:

The last equality uses \(\hat{q}_0=0\). Thus differencing and cumulative summation reverse one another for these fitted values. \(\square\)

Checks, Rounding, and a Common Pitfall

A practical calculation should include two checks. First, each category probability should be at least zero. Second, the probabilities should sum to one. For a correctly evaluated model with ordered thresholds, these properties follow from the theorem; they are not optional adjustments made after the fact. If an apparent category probability is negative, check the threshold order, the signs in \(\hat{\alpha}_j-\hat{\eta}\), and the order in which the cumulative probabilities were differenced. A negative value is not a probability to report.

For hand calculations, displayed cumulative values are usually rounded. A small discrepancy in the final sum, such as 0.999 or 1.001, can result from rounding intermediate numbers. Keep additional digits while differencing, and round only the final category probabilities. For the final category, calculate \(1-\hat{q}_{K-1}\); this makes its relationship to the upper endpoint explicit. Do not divide all the results by their sum merely to force the rounded total to one, because that can conceal a mistaken cumulative calculation.

• Pitfall: treating cumulative probabilities as category probabilities. A value such as \(\hat{q}_2=P(Y\leq2\mid\boldsymbol{x})\) includes both categories 1 and 2; it is not the probability of category 2 alone. The correction is \(\hat{p}_2=\hat{q}_2-\hat{q}_1\).
• Pitfall: changing the sign convention midway. If the fitted cumulative equation uses \(\hat{\alpha}_j-\hat{\eta}\), use that exact argument in \(G\). Substituting \(\hat{\eta}-\hat{\alpha}_j\) without also changing the model convention generally gives different probabilities.
• Pitfall: omitting an endpoint. The first category is \(\hat{q}_1-0\), and the last is \(1-\hat{q}_{K-1}\). There are \(K-1\) fitted cutpoint cumulatives but \(K\) category probabilities.

The point of this calculation is to express a fitted ordinal model on the scale most directly useful for prediction: a full probability distribution over the response categories at the specified covariate values. Reporting the cumulative values alone leaves that distribution implicit. Showing the differences and the two checks makes the conversion transparent and helps catch arithmetic or convention errors.