Worked Cumulative Link Model Problems

From an Ordinal Model to Answers

A cumulative-link model is useful only when its fitted quantities can be translated back into the ordered response categories. A complete solution typically moves through three steps: state the response order and model, interpret its thresholds and slope in that order, and turn fitted cumulative probabilities into probabilities for individual categories.

Use the convention that \(Y\) takes the ordered values \(1,\ldots,K\), and let \(x\) be a covariate vector. For \(j=1,\ldots,K-1\), write

P(Y\leq j\mid x)=F(\theta_j-x^\mathsf{T}\beta).

Here \(F\) is a cumulative distribution function, \(\theta_1<\cdots<\theta_{K-1}\) are thresholds, and \(\beta\) is the slope vector. For the cumulative-logit model, \(F(t)=1/(1+\exp(-t))\); for the cumulative-probit model, \(F(t)=\Phi(t)\), the standard normal cumulative distribution function. The minus sign fixes the direction of interpretation: increasing \(x^\mathsf{T}\beta\) shifts probability toward higher response categories. Thresholds take the place of a separate intercept in these cumulative equations.

Definition: The cumulative probability at cutpoint \(j\) is \(C_j(x)=P(Y\leq j\mid x)\), the probability of being in category \(j\) or any lower category. The probability of category \(j\) is not generally \(C_j(x)\): for an interior category it is the difference between two adjacent cumulative probabilities.

Worked Cumulative-Logit Problems

The examples below follow the order a reader should use in an applied calculation: establish what the categories and covariates mean, evaluate the cumulative probabilities, and only then obtain category probabilities.

Worked Example: Specify the Model and Interpret a Threshold

A study records symptom severity in four ordered categories: 1 = mild, 2 = moderate, 3 = severe, and 4 = very severe. Let \(x=0\) denote control and \(x=1\) denote treatment. Consider a cumulative-logit model with thresholds \((-1,0.4,1.6)\) and treatment slope \(\beta=0.6\).

The model has three cumulative equations, one for each cutpoint:

\operatorname{logit}P(Y\leq j\mid x)=\theta_j-0.6x,\qquad j=1,2,3.

At the second cutpoint, \(Y\leq2\) means mild or moderate, while \(Y>2\) means severe or very severe. For a control participant, the cumulative logit is \(0.4-0.6(0)=0.4\). Thus

P(Y\leq2\mid x=0)=\frac{\exp(0.4)}{1+\exp(0.4)} =\frac{1.4918}{2.4918}\approx0.5987.

The threshold \(0.4\) is the cumulative log-odds at this cutpoint for the reference value \(x=0\); it is not itself a probability. For treatment, the cumulative logit is \(0.4-0.6=-0.2\), giving \(P(Y\leq2\mid x=1)\approx0.4502\). The corresponding odds of being in category 2 or below, rather than above category 2, are \(\exp(-0.2)\approx0.8187\) under treatment and \(\exp(0.4)\approx1.4918\) under control. Their ratio is \(\exp(-0.6)\approx0.5488\). In this model, treatment is associated with lower cumulative odds of being at or below this cutpoint, and hence a shift toward higher severity categories.

Worked Example: Calculate All Fitted Category Probabilities

Suppose a four-category response is modeled with the cumulative-logit thresholds \(\theta_1=-1\), \(\theta_2=0.5\), and \(\theta_3=2\), and slope \(\beta=0.8\) for a scalar predictor \(x\). Find the fitted category probabilities at \(x=1\).

First calculate the three cumulative logits, \(\theta_j-\beta x\). They are \(-1-0.8=-1.8\), \(0.5-0.8=-0.3\), and \(2-0.8=1.2\). Applying the logistic CDF gives

\begin{aligned} C_1(1)&=\frac{1}{1+\exp(1.8)}\approx0.1419,\\ C_2(1)&=\frac{1}{1+\exp(0.3)}\approx0.4256,\\ C_3(1)&=\frac{1}{1+\exp(-1.2)}\approx0.7685. \end{aligned}

The first category probability is \(P(Y=1\mid x=1)=C_1(1)\). The two interior category probabilities are adjacent differences, and the last category is the probability above the final cutpoint:

\begin{aligned} P(Y=1\mid x=1)&=C_1(1)\approx0.1419,\\ P(Y=2\mid x=1)&=C_2(1)-C_1(1)\approx0.4256-0.1419=0.2837,\\ P(Y=3\mid x=1)&=C_3(1)-C_2(1)\approx0.7685-0.4256=0.3429,\\ P(Y=4\mid x=1)&=1-C_3(1)\approx1-0.7685=0.2315. \end{aligned}

Rounding explains any tiny discrepancy in the sum: \(0.1419+0.2837+0.3429+0.2315=1.0000\). The largest fitted probability is for category 3, but that does not mean the model directly fits a separate category-3 equation; it is obtained by subtracting two cumulative probabilities.

Why the Differences Give a Valid Distribution

The conversion from cumulative probabilities to category probabilities is more than a computational shortcut. It follows directly from how the events are nested: \(\{Y\leq1\}\subset\{Y\leq2\}\subset\cdots\subset\{Y\leq K-1\}\).

Theorem: Suppose \(\theta_1<\cdots<\theta_{K-1}\), \(F\) is a nondecreasing CDF, and \(\eta=x^\mathsf{T}\beta\) is finite. Define \(C_j=F(\theta_j-\eta)\). Set \(p_1=C_1\), \(p_j=C_j-C_{j-1}\) for \(2\leq j\leq K-1\), and \(p_K=1-C_{K-1}\). Then each \(p_j\geq0\) and \(\sum_{j=1}^K p_j=1\). If \(F\) is strictly increasing, all these probabilities are positive.

Proof. Since the thresholds are ordered, \(\theta_j-\eta<\theta_{j+1}-\eta\). Monotonicity of \(F\) therefore gives \(C_j\leq C_{j+1}\). Also, every CDF value lies between zero and one. Hence \(p_1=C_1\geq0\), each difference \(p_j=C_j-C_{j-1}\geq0\), and \(p_K=1-C_{K-1}\geq0\). Adding the probabilities cancels the intermediate cumulative terms:

p_1+\cdots+p_K =C_1+(C_2-C_1)+\cdots+(C_{K-1}-C_{K-2})+(1-C_{K-1})=1.

If \(F\) is strictly increasing, the strict threshold ordering makes each adjacent cumulative difference positive; for finite arguments, the logistic and normal CDFs also lie strictly between zero and one. Thus the first and last probabilities are positive as well. \(\square\)

This check is useful in hand calculations and software output: a negative category probability often signals that thresholds were entered in the wrong order, cumulative probabilities were subtracted in the wrong direction, or the wrong category order was used.

Worked Probit Problem

Worked Example: Probit Probabilities at a Given Covariate Value

Let \(Y\) have four ordered categories, with probit thresholds \((-1,0.4,1.3)\) and slope \(\beta=0.5\) for a scalar covariate \(x\). Calculate category probabilities at \(x=1.2\).

For a cumulative-probit model, \(C_j(x)=\Phi(\theta_j-\beta x)\). Here \(\beta x=0.5(1.2)=0.6\), so the three normal arguments are \(-1-0.6=-1.6\), \(0.4-0.6=-0.2\), and \(1.3-0.6=0.7\). Standard normal CDF values give

C_1(1.2)=\Phi(-1.6)\approx0.0548,\quad C_2(1.2)=\Phi(-0.2)\approx0.4207,\quad C_3(1.2)=\Phi(0.7)\approx0.7580.

Taking differences and complements,

\begin{aligned} P(Y=1\mid x=1.2)&=0.0548,\\ P(Y=2\mid x=1.2)&=0.4207-0.0548=0.3659,\\ P(Y=3\mid x=1.2)&=0.7580-0.4207=0.3373,\\ P(Y=4\mid x=1.2)&=1-0.7580=0.2420. \end{aligned}

The rounded probabilities sum to \(0.0548+0.3659+0.3373+0.2420=1.0000\). The threshold values are on the standard normal latent scale in this probit specification; they are not probabilities. For example, at \(x=0\), the probability of category 1 or below is \(\Phi(-1)\approx0.1587\), not \(-1\).

A Slope Calculation That Applies Across Cutpoints

For a cumulative-logit model, a slope has a direct odds interpretation. This follows by applying the odds transformation to the model equation, rather than by interpreting the slope as a change in category probability.

Result: In the cumulative-logit model, for a change from covariate vector \(x\) to \(x+d\), the odds ratio for \(Y\leq j\) versus \(Y>j\) is \(\exp(-d^\mathsf{T}\beta)\) at every cutpoint \(j\).

Derivation. The model states that the cumulative logit is \(\theta_j-x^\mathsf{T}\beta\). For any event with probability \(C_j(x)\), its odds are \(C_j(x)/(1-C_j(x))\). By the definition of the logistic link,

\frac{C_j(x)}{1-C_j(x)}=\exp(\theta_j-x^\mathsf{T}\beta).

At \(x+d\), the odds are \(\exp(\theta_j-(x+d)^\mathsf{T}\beta)\). Dividing these by the odds at \(x\) cancels the threshold and the terms involving \(x\), leaving

\frac{C_j(x+d)/(1-C_j(x+d))}{C_j(x)/(1-C_j(x))} =\exp(-d^\mathsf{T}\beta).

For a one-unit increase in predictor \(x_r\), this ratio is \(\exp(-\beta_r)\). It describes cumulative odds at a cutpoint; it is not an odds ratio for one individual category. The cumulative-probit slope is interpreted on the probit (standard normal) scale instead: changing \(x\) to \(x+d\) changes the normal argument at every cutpoint by \(-d^\mathsf{T}\beta\).

Common Pitfall: Confusing Cumulative and Category Probabilities

A frequent error is to report \(P(Y\leq2\mid x)\) as the probability of category 2. The event \(Y\leq2\) includes categories 1 and 2, so category 2 alone requires subtraction: \(P(Y=2\mid x)=C_2(x)-C_1(x)\). Similarly, the last category is the upper tail \(1-C_{K-1}(x)\), not \(C_{K-1}(x)\).

Another error is to interpret \(\theta_j\) as a fitted probability. It is a threshold on the linear-predictor scale, and its cumulative probability depends on both the link and the covariate value. At the reference value \(x=0\), the probability at cutpoint \(j\) is \(F(\theta_j)\); away from that reference, it is \(F(\theta_j-x^\mathsf{T}\beta)\).

A reliable workflow is to write down the category order, evaluate all \(K-1\) cumulative probabilities in order, take adjacent differences, and check nonnegativity and a total of one. Keep the signs in the specified model visible throughout: reversing the response order or changing the model's sign convention changes the interpretation.

Check Your Understanding

Answer each question using the cumulative-link convention in this tutorial.

For a five-category response, how many thresholds and cumulative probabilities are needed to recover all category probabilities?
If \(C_1=0.20\) and \(C_2=0.65\), what is the probability of category 2?
In a cumulative-logit model with slope \(\beta=0.4\), what is the odds ratio for \(Y\leq j\) versus \(Y>j\) after a one-unit increase in the predictor?
Why is a threshold not itself the probability of being at or below its cutpoint?
What two checks can you perform on a computed set of category probabilities?