Thresholds as Ordered Cutpoints

Thresholds as Boundaries on the Link Scale

An ordinal response has categories with a meaningful order, such as mild, moderate, and severe. A cumulative-link model represents the chance of being at or below each category using a common link scale. Its thresholds mark the successive boundaries between those cumulative events. Their order is not just a convenient convention: it is what allows the cumulative probabilities to increase from one category boundary to the next.

Write the categories as \(1,\ldots,K\), and let \(\eta(x)\) denote the predictor value for a fixed covariate pattern \(x\). In the cumulative-link equation, the cumulative probability through category \(j\) is

P(Y\leq j\mid x)=G\bigl(\alpha_j-\eta(x)\bigr),\qquad j=1,\ldots,K-1.

Here \(G\) is a continuous, strictly increasing cumulative distribution function taking values in \((0,1)\) at finite arguments, and \(\alpha_j\) is the threshold for the boundary after category \(j\). Equivalently, applying the inverse link gives \(G^{-1}\{P(Y\leq j\mid x)\}=\alpha_j-\eta(x)\). The threshold is on the link scale; it is not itself a probability.

Definition: In a cumulative-link model with \(K\) ordered categories, the \(K-1\) thresholds \(\alpha_1,\ldots,\alpha_{K-1}\) are the cutpoints associated with the cumulative events \(Y\leq 1,\ldots,Y\leq K-1\). They are ordered cutpoints when \(\alpha_1\leq\cdots\leq\alpha_{K-1}\). Strict inequalities give each intervening category a nonzero probability at finite predictor values.

For a fixed \(x\), each threshold divides the shifted link scale at \(\alpha_j-\eta(x)\). Category \(j\), for an interior category, lies between the boundaries associated with \(j-1\) and \(j\). The first and last categories extend to the lower and upper ends of the scale. This interpretation is about cumulative probabilities; it does not require introducing a latent response.

Why the Cutpoints Must Be Ordered

Cumulative events are nested: \(Y\leq j\) is contained in \(Y\leq j+1\). Therefore their probabilities cannot decrease as \(j\) increases. Since \(G\) is strictly increasing, the cumulative-link equation translates this probability ordering directly into an ordering of thresholds.

Theorem: Suppose \(G\) is strictly increasing and the cumulative-link equation holds for a fixed covariate pattern. The cumulative probabilities are nondecreasing in category index if and only if \(\alpha_j\leq\alpha_{j+1}\) for every \(j=1,\ldots,K-2\). If the thresholds are strictly increasing, every interior category has positive probability.

Proof. For each adjacent pair of cumulative probabilities, the equation gives

P(Y\leq j\mid x)=G\bigl(\alpha_j-\eta(x)\bigr),\qquad P(Y\leq j+1\mid x)=G\bigl(\alpha_{j+1}-\eta(x)\bigr).

Because \(G\) is strictly increasing, the first quantity is at most the second exactly when \(\alpha_j-\eta(x)\leq\alpha_{j+1}-\eta(x)\). Cancelling the common predictor gives \(\alpha_j\leq\alpha_{j+1}\). This proves both directions. For an interior category \(j\), its probability is the difference between the cumulative probabilities at its two boundaries. Strictly ordered thresholds give a strictly positive difference, again because \(G\) is strictly increasing. \(\square\)

Weak ordering permits a zero-width region: if adjacent thresholds are equal, the category between them has probability zero. Strict ordering is thus the condition for positive probability in every interior category, not merely a way to avoid negative probabilities.

Spacing and Category Probabilities

The gap between two successive thresholds sets the width of the corresponding category’s region on the link scale. For a fixed predictor, a wider region has a larger probability under a strictly increasing link distribution when the regions share a center or the wider region contains the narrower one. The exact probability also depends on where the region is located: equal-width regions need not have equal probability when the link density varies across the scale.

To make the role of width precise, consider an interval with center \(c\) and width \(w>0\) on the shifted scale. Its probability is \(G(c+w/2)-G(c-w/2)\). If the center stays fixed and the width increases, the lower endpoint moves down and the upper endpoint moves up. Strict monotonicity of \(G\) means the probability increases.

Result: For any strictly increasing \(G\), fixed center \(c\), and widths \(0<w_1<w_2\), the interval probability at width \(w_2\) is strictly larger than the interval probability at width \(w_1\).

Proof. Since \(w_2>w_1\), we have \(c-w_2/2<c-w_1/2\) and \(c+w_1/2<c+w_2/2\). Strict increase of \(G\) gives \(G(c-w_2/2)<G(c-w_1/2)\) and \(G(c+w_1/2)<G(c+w_2/2)\). Subtracting the lower-endpoint value from the upper-endpoint value shows

$$ G(c+w_2/2)-G(c-w_2/2) > G(c+w_1/2)-G(c-w_1/2). $$

Thus widening the interval increases its probability. The argument does not say that intervals with the same width have equal probability; their locations can differ. \(\square\)

Worked Example: Calculating Probabilities from Ordered Logistic Cutpoints

Suppose there are four response categories, the link is logistic, \(\eta(x)=0\), and the thresholds are \(\alpha_1=-1\), \(\alpha_2=0.5\), and \(\alpha_3=2\). The logistic cumulative distribution function is \(G(t)=1/(1+e^{-t})\). The cutpoints are ordered because \(-1<0.5<2\).

The cumulative probabilities at the three boundaries are

G(-1)=0.268941421,\qquad G(0.5)=0.622459331,\qquad G(2)=0.880797078.

The category probabilities are the successive increments of these cumulative probabilities, with 0 and 1 as the lower and upper endpoints:

\begin{aligned} P(Y=1\mid x)&=0.268941421,\\ P(Y=2\mid x)&=0.622459331-0.268941421=0.353517910,\\ P(Y=3\mid x)&=0.880797078-0.622459331=0.258337747,\\ P(Y=4\mid x)&=1-0.880797078=0.119202922. \end{aligned}

They are all positive and sum to \(0.268941421+0.353517910+0.258337747+0.119202922=1.000000000\). The middle category’s threshold gap is \(0.5-(-1)=1.5\); the next gap is \(2-0.5=1.5\) as well. Although those two regions have equal width, their probabilities differ because they occupy different locations on the logistic scale.

Worked Example: Widening a Category Region

Use the logistic link with \(\eta(x)=0\) and consider a category whose boundaries are \(-1.5\) and \(1.5\). Its width is 3 and its probability is

\begin{aligned} P(\text{category}\mid x) &=G(1.5)-G(-1.5)\\ &=0.8175744761936437-0.18242552380635635\\ &=0.6351489523872874\\ &\approx 0.635148952. \end{aligned}

Now narrow the region to boundaries \(-1\) and \(1\), keeping its center at zero. The width is 2, and

\begin{aligned} P(\text{narrower category}\mid x) &=G(1)-G(-1)\\ &=0.731058579-0.268941421\\ &=0.462117158. \end{aligned}

The wider region has the larger probability: approximately \(0.635149\) rather than \(0.462117\). Both regions are centered at zero, so this comparison isolates the effect of width.

A Boundary Moves Probability Between Neighbors

A threshold is shared by the cumulative probabilities on either side of a category boundary. Moving it while holding the other thresholds and the predictor fixed changes the probabilities of the two adjacent categories in opposite directions. To state this locally, suppose \(G\) has a density \(g\) that is positive at the relevant finite arguments.

Result: For an interior threshold \(\alpha_j\), increasing \(\alpha_j\) increases the probability of category \(j\) and decreases the probability of category \(j+1\); it leaves all other category probabilities unchanged, provided the change does not alter the other thresholds.

Derivation. The probabilities for these neighboring categories are

\begin{aligned} p_j&=G(\alpha_j-\eta(x))-G(\alpha_{j-1}-\eta(x)),\\ p_{j+1}&=G(\alpha_{j+1}-\eta(x))-G(\alpha_j-\eta(x)). \end{aligned}

Differentiating with respect to \(\alpha_j\), using the assumption that \(G\) has density \(g\), gives

\frac{\partial p_j}{\partial\alpha_j}=g(\alpha_j-\eta(x))>0,\qquad \frac{\partial p_{j+1}}{\partial\alpha_j}=-g(\alpha_j-\eta(x))<0.

Every other category probability involves neither \(\alpha_j\) nor a cumulative boundary that changes with it, so its derivative is zero. The two displayed derivatives are equal in magnitude and opposite in sign, consistent with probability being reassigned across the shared boundary rather than added to or removed from the total. \(\square\)

Worked Example: Detecting an Invalid Cutpoint Order

Suppose there are three categories, \(G\) is logistic, \(\eta(x)=0.3\), and the proposed thresholds are \(\alpha_1=0.8\) and \(\alpha_2=0.2\). The first two cumulative probabilities would be

P(Y\leq1\mid x)=G(0.8-0.3)=G(0.5)=0.622459331, \qquad P(Y\leq2\mid x)=G(0.2-0.3)=G(-0.1)=0.475020813.

The second cumulative probability is smaller than the first, which cannot happen for nested events. Subtracting them would give the middle category probability \(0.475020813-0.622459331=-0.147438518\), an impossible negative value.

If the thresholds are instead ordered as \(\alpha_1=0.2\) and \(\alpha_2=0.8\), then the cumulative probabilities are \(G(-0.1)=0.475020813\) and \(G(0.5)=0.622459331\). The middle probability is now \(0.622459331-0.475020813=0.147438518\), which is positive. The order condition prevents the invalid reversal.

Common Mistake: Treating Thresholds as Probabilities

A threshold such as \(\alpha_j=1.2\) is not a probability of 1.2, nor does a threshold gap of 1.2 mean a category has probability 1.2. Thresholds are coordinates on the link scale; the link distribution maps those coordinates, after accounting for the predictor, into cumulative probabilities. The category probability then depends on the difference between two successive cumulative probabilities.

A related mistake is to assume that equal threshold spacing guarantees equal category probabilities. Equal spacing gives equal widths on the link scale, but the link distribution’s density can vary with location. As the logistic example showed, two intervals of the same width can therefore carry different probability. A threshold’s position matters as well as its distance from its neighbor.

Key takeaway: Ordered thresholds define successive boundaries on the link scale. Their ordering ensures nondecreasing cumulative probabilities; their gaps set category-region widths; and both gap and location affect the resulting category probabilities.

Check Your Understanding

Use the cumulative-link equation and the role of cutpoints to answer these questions.

For four categories, how many thresholds are there, and which cumulative events do they mark?
Why does strict increase of \(G\) make the ordering of cumulative probabilities equivalent to the ordering of thresholds?
If two adjacent thresholds are equal, what is the probability of the category between them?
Why can two category regions with equal threshold gaps have different probabilities?
When an interior threshold increases and all others remain fixed, which neighboring category probabilities change, and in what directions?