Deriving Category Probabilities From Ordinal Thresholds

From Cumulative Probabilities to Category Probabilities

A cumulative-logit model predicts the probability that an ordinal pharmacodynamic outcome is at or below each possible threshold. Clinical decisions, however, often concern the probability of a particular outcome category: for example, the chance of no response, mild response, or strong response. Recovering those probabilities requires differencing cumulative probabilities. That subtraction is simple, but it is valid only when the cumulative probabilities are ordered correctly.

Let \(Y\) take one of \(J\) ordered categories, labeled \(1,2,\ldots,J\), from lowest to highest. For a given predictor value \(x\), write \(F_j(x)=P(Y\leq j\mid x)\), for \(j=1,\ldots,J-1\). The cumulative-logit model uses a logistic function \(G(z)=1/(1+\exp(-z))\) and has the form \(F_j(x)=G(\kappa_j-\eta(x))\). Here \(\kappa_j\) is the threshold between categories \(j\) and \(j+1\), and \(\eta(x)\) is the predictor contribution. There is no separate intercept in \(\eta(x)\) when the thresholds supply the intercept-like terms; including both without an identifying constraint would make the model’s location ambiguous.

Definition: The probability of category 1 is \(p_1(x)=F_1(x)\). For an interior category \(k\), where \(2\leq k\leq J-1\), \(p_k(x)=F_k(x)-F_{k-1}(x)\). The probability of the highest category is \(p_J(x)=1-F_{J-1}(x)\). These are conditional probabilities given the predictor value \(x\).

The category probabilities are not obtained by treating each cumulative probability as if it described a separate category. For example, \(F_2(x)\) includes outcomes in categories 1 and 2; subtracting \(F_1(x)\) leaves only category 2. The final probability is the complement of the cumulative probability through category \(J-1\).

Why Threshold Ordering Matters

A cumulative distribution cannot decrease as the category boundary moves upward: the event \(Y\leq j-1\) is contained in the event \(Y\leq j\). Thus, a valid set of cumulative probabilities must satisfy \(F_1(x)\leq F_2(x)\leq\cdots\leq F_{J-1}(x)\). In the cumulative-logit form above, \(G\) is strictly increasing, and the same \(\eta(x)\) is used at every threshold. Therefore the required threshold constraint is \(\kappa_1<\kappa_2<\cdots<\kappa_{J-1}\). Strict ordering gives positive probabilities for all interior categories at finite predictor values; weak ordering would allow some categories to have probability zero.

This ordering is a structural condition on the model, not a condition to check separately at every \(x\): because the predictor contribution is common to the thresholds, ordered thresholds imply ordered cumulative probabilities for every predictor value. A common way to enforce strict ordering during estimation is to set \(\kappa_1=\gamma_1\) and \(\kappa_j=\kappa_{j-1}+\exp(\delta_j)\) for \(j=2,\ldots,J-1\). Since \(\exp(\delta_j)>0\), each successive threshold is larger. The exact parameterization used by software may differ, but fitted thresholds must still obey the ordering constraint.

Worked Example: Three Ordered Response Categories

Suppose a hypothetical three-category tolerability outcome has thresholds \(\kappa_1=-0.8\) and \(\kappa_2=0.6\), and the patient’s predictor value gives \(\eta(x)=0\). Calculate the probability of each category.

The thresholds are ordered because \(-0.8<0.6\). The cumulative probabilities are \(F_1(x)=G(-0.8)\approx0.3100255\) and \(F_2(x)=G(0.6)\approx0.6456563\). Differencing before rounding gives:

\begin{aligned} p_1(x)&=F_1(x)\approx0.3100255\approx0.3100,\\ p_2(x)&=F_2(x)-F_1(x)\\ &\approx0.6456563-0.3100255\\ &\approx0.3356308\approx0.3356,\\ p_3(x)&=1-F_2(x)\approx1-0.6456563\\ &\approx0.3543437\approx0.3543. \end{aligned}

Thus, the model assigns about 31.00% probability to category 1, 33.56% to category 2, and 35.43% to category 3. The displayed four-decimal probabilities sum to 0.9999, not 1.0000, because each has been rounded separately. Using the unrounded values, their sum is 1.

Justification: Nonnegativity and Total Probability

The formulas produce a probability distribution when the cumulative probabilities are ordered and lie between zero and one. The logistic function guarantees \(0<F_j(x)<1\) for finite thresholds and predictor values. Threshold ordering guarantees \(F_j(x)\geq F_{j-1}(x)\), so each interior difference is nonnegative. The first probability is positive, and the last is positive for finite inputs. The probabilities also sum to one because the cumulative terms cancel:

\begin{aligned} \sum_{k=1}^{J}p_k(x) &=F_1(x)+\sum_{k=2}^{J-1}\left[F_k(x)-F_{k-1}(x)\right] +\left[1-F_{J-1}(x)\right]\\ &=1. \end{aligned}

This cancellation is sometimes called telescoping: every cumulative probability appears once with a positive sign and once with a negative sign, except that the remaining constant is 1. The result is an algebraic check of the formula. It does not, by itself, establish that the probabilities are valid: if thresholds are unordered, negative category probabilities can still sum to one.

Worked Example: Four Categories at a Nonzero Predictor Value

For a hypothetical four-category pharmacodynamic response, let \(\eta(x)=0.4\) and let the ordered thresholds be \(-1.2,-0.2,\) and \(0.9\). Calculate each category probability and check the distribution.

Subtracting \(\eta(x)\) from each threshold gives the cumulative-logit values \(-1.6,-0.6,\) and \(0.5\). Therefore \(F_1(x)=G(-1.6)\approx0.1679816\), \(F_2(x)=G(-0.6)\approx0.3543437\), and \(F_3(x)=G(0.5)\approx0.6224593\). The thresholds are increasing, so the cumulative probabilities are increasing as required. Calculate each difference using the unrounded cumulative values:

\begin{aligned} p_1(x)&\approx0.1679816\approx0.1680,\\ p_2(x)&\approx0.3543437-0.1679816 =0.1863621\approx0.1864,\\ p_3(x)&\approx0.6224593-0.3543437 =0.2681156\approx0.2681,\\ p_4(x)&\approx1-0.6224593 =0.3775407\approx0.3775. \end{aligned}

All four probabilities are nonnegative. Before rounding, they sum to \(0.1679816+0.1863621+0.2681156+0.3775407=1.0000000\), up to the displayed precision. The rounded values sum to 1.0000 as well. The middle calculations illustrate why the cumulative probabilities must be subtracted in category order: each difference isolates the probability of the category between the two boundaries.

Failure Modes and Numerical Checks

An implementation or calculation should be checked for more than a total of one. For each predictor value used for prediction, inspect the cumulative probabilities in order and then inspect the category probabilities. Negative category probabilities indicate that at least one cumulative probability decreased across thresholds. In a proportional cumulative-logit model with a common \(\eta(x)\), this points to unordered thresholds, a coding or indexing error, or a calculation that does not match the fitted model. A sum different from one can indicate a missing category, an incorrect endpoint formula, or an arithmetic error.

Finite-precision rounding creates a separate issue. Calculate each difference from the full-precision cumulative probabilities, then round the resulting category probability for display. Subtracting already-rounded cumulative values can change the last reported digit, especially when a category has a small probability. A displayed sum such as 0.9999 is not, by itself, evidence of a faulty model if the unrounded values sum to one. Conversely, rounding must not be used to disguise a genuinely negative probability.

Worked Example: Detecting an Unordered-Threshold Problem

Suppose a hypothetical three-category response is assigned thresholds \(\kappa_1=0.5\) and \(\kappa_2=-0.1\), with \(\eta(x)=0\). Check whether these values produce valid category probabilities.

The thresholds violate the required ordering because \(0.5>-0.1\). The cumulative probabilities are \(F_1(x)=G(0.5)\approx0.6224593\) and \(F_2(x)=G(-0.1)\approx0.4750208\). The interior category probability is therefore:

p_2(x)=F_2(x)-F_1(x) \approx0.4750208-0.6224593 =-0.1474385\approx-0.1474.

The other probabilities are \(p_1(x)\approx0.6225\) and \(p_3(x)=1-F_2(x)\approx0.5250\). Although the three values sum to one before rounding, \(p_2(x)\) is negative, so they are not a valid probability distribution. A sum-to-one check alone would miss the problem. The correction is to constrain the thresholds to increase, not to replace the negative value with zero or renormalize the probabilities; either of those changes would no longer represent the specified cumulative model.

How This Construction Relates to Other Models

The threshold-based construction represents an ordinal outcome through cumulative events and recovers category probabilities by differencing. It respects the ordering of categories and, with common predictor effects across thresholds, uses one shared predictor contribution to shift the cumulative distribution. Its probabilities are therefore linked: changing a threshold or predictor can affect several category probabilities at once.

A direct multinomial model instead specifies category probabilities through a set of category-specific linear predictors and normalizes them to sum to one. That approach does not require ordinal threshold ordering, but it does not use the category order in the same way. Adjacent-category and continuation-ratio models are other ordinal formulations that model different comparisons between categories. They are alternatives when the scientific question or data structure calls for a different conditioning scheme; their probability formulas should not be substituted into the cumulative-threshold equations. Whichever model is used, predicted category probabilities should be checked for nonnegativity and a total of one.

For PK/PD interpretation, these probabilities describe the model-predicted distribution of an ordered response at specified predictor values. They can help distinguish, for instance, a shift from mild to moderate response from a shift into the most severe category. They are predictions conditional on the stated model and predictors, not guarantees about an individual patient. Sparse observations in some categories, extreme predictors that push cumulative probabilities close to zero or one, and coding mistakes in category order can all make predictions difficult to assess. Check the category labels and their intended order before interpreting the output.

Common Mistakes

Using a cumulative probability as a category probability. \(F_j(x)\) includes every category up to \(j\); for an interior category, subtract \(F_{j-1}(x)\).
Reversing the subtraction. The probability of category \(j\) is \(F_j(x)-F_{j-1}(x)\), not the reverse. A negative result is a warning, not a small probability to ignore.
Checking only that the probabilities sum to one. Unordered cumulative probabilities can yield a negative category probability while the algebraic sum remains one. Check each probability as well as the sum.
Rounding before differencing. Retain full precision through subtraction and round only the final category probabilities for reporting.

Key takeaway: For an ordered response, obtain category probabilities by differencing adjacent cumulative probabilities, with the first and last categories formed from the cumulative probability and its complement. Ordered thresholds ensure nonnegative differences; telescoping ensures the probabilities sum to one.

Check Your Understanding

Answer each question, then compare with the solutions below.

For four ordered categories with cumulative probabilities \(F_1=0.20\), \(F_2=0.55\), and \(F_3=0.80\), calculate all category probabilities.
Why does \(\kappa_1<\kappa_2<\cdots<\kappa_{J-1}\) ensure nonnegative interior probabilities in the specified cumulative-logit model?
Three displayed category probabilities are 0.2500, 0.3333, and 0.4167. Their sum is one. Is this sufficient to establish that the underlying cumulative model is valid? Explain.
A cumulative probability for a higher threshold is smaller than the cumulative probability for the preceding threshold. What is the implication for the corresponding interior category probability, and what should be checked?

Solutions. 1. \(p_1=0.20\), \(p_2=0.55-0.20=0.35\), \(p_3=0.80-0.55=0.25\), and \(p_4=1-0.80=0.20\). These are nonnegative and sum to 1. 2. The logistic function is increasing, so ordered thresholds with the same \(\eta(x)\) imply \(F_j(x)\geq F_{j-1}(x)\); hence \(p_j(x)=F_j(x)-F_{j-1}(x)\geq0\). 3. No. A sum of one does not rule out a negative category probability that is offset by other values; inspect the unrounded category probabilities and their cumulative ordering. 4. The difference \(F_j(x)-F_{j-1}(x)\) is negative, so the proposed values are invalid. Check threshold order, category indexing, and whether the cumulative probabilities were calculated from the same fitted model and predictor value.