The Cumulative Link Equation

From Cumulative Probabilities to a Regression Equation

For an ordered response, the probability of being at or below a given category is often more informative than the probability of one category considered alone. Cumulative link models describe these probabilities using a common predictor and a separate threshold for each cumulative split. The equation’s minus sign is a convention with a useful consequence: a positive coefficient moves cumulative probability toward higher response categories as its predictor increases.

Let $Y$ take values in the ordered categories $1,\ldots,K$, where $K\geq 2$. For $j=1,\ldots,K-1$, the $j$th cumulative probability at predictor value $x$ is $P(Y\leq j\mid x)$. The model relates each such probability to the same predictor through a link function.

Definition: A cumulative link model specifies the conditional cumulative probabilities of an ordered response by $$ g[P(Y\leq j\mid x)]=\theta_j-x^\mathsf{T}\beta,\qquad j=1,\ldots,K-1. $$ Here $g$ is a strictly increasing bijection from $(0,1)$ onto $\mathbb{R}$; $\theta_j$ is the threshold for cumulative split $j$; $x$ is a vector of predictor values; and $\beta$ is the vector of regression coefficients. The expression $x^\mathsf{T}\beta$ is the linear predictor. The minus sign is part of this stated convention.

A threshold belongs to a split in the ordered response: $j$ separates categories $1,\ldots,j$ from categories $j+1,\ldots,K$. The predictor term is shared across these splits in the displayed equation, while the threshold can differ from one split to another. A coherent model must give ordered cumulative probabilities between zero and one; its admissible parameter values must respect that requirement.

The bijection requirement matters. Strict increase alone gives an inverse only on the link’s image, which may not be all of $\mathbb{R}$. Since the linear predictor $\theta_j-x^\mathsf{T}\beta$ is real-valued and the model must translate it back to a probability, we require $g$ to map $(0,1)$ onto $\mathbb{R}$. Its inverse $g^{-1}$ is then defined for every real linear predictor and takes values in $(0,1)$.

Links, Thresholds, and Predictors

The link is a function applied to a probability. It puts the probability on a real-valued scale where the model equation is linear in the predictors. The inverse link returns from that scale to the probability scale. Common choices include the logit link $g(p)=\log\{p/(1-p)\}$, the probit link $g(p)=\Phi^{-1}(p)$, and the complementary log-log link $g(p)=\log\{-\log(1-p)\}$. Each is a strictly increasing bijection from $(0,1)$ to $\mathbb{R}$; their inverse links are, respectively, the logistic function, the standard normal distribution function $\Phi$, and $1-\exp\{-\exp(\eta)\}$.

The thresholds locate the cumulative splits on the link scale. They are not probabilities: a threshold can be any real number, whereas a cumulative probability must lie between zero and one. The predictor vector $x$ records the covariate values for the observation, and $\beta$ contains the corresponding coefficients. The equation’s term $x^\mathsf{T}\beta$ is a scalar, even when there are several predictors.

Result: Under the cumulative link equation, the cumulative probability is $$ P(Y\leq j\mid x)=g^{-1}(\theta_j-x^\mathsf{T}\beta). $$ Proof. By definition, $g[P(Y\leq j\mid x)]=\theta_j-x^\mathsf{T}\beta$. Because $g$ is a bijection from $(0,1)$ onto $\mathbb{R}$, its inverse is defined at the real number on the right. Apply $g^{-1}$ to both sides to obtain the stated formula. Its value is in $(0,1)$, as required for a probability. $\square$

Worked Example: A Logit Model with One Predictor

Suppose $Y$ has four ordered categories and the link is logit. Let the thresholds be $\theta_1=-1$, $\theta_2=0.5$, and $\theta_3=2$, and let the coefficient be $\beta=0.8$. Calculate the three cumulative probabilities at $x=1$.

For the logit link, $g^{-1}(\eta)=1/(1+\exp(-\eta))$. At $x=1$, the linear predictor for split $j$ is $\eta_j=\theta_j-(1)(0.8)$. Therefore,

\eta_1=-1.8,\qquad \eta_2=-0.3,\qquad \eta_3=1.2.

Applying the inverse logit separately gives

\begin{aligned} P(Y\leq 1\mid x=1)&=\frac{1}{1+\exp(1.8)}\approx 0.142,\\ P(Y\leq 2\mid x=1)&=\frac{1}{1+\exp(0.3)}\approx 0.426,\\ P(Y\leq 3\mid x=1)&=\frac{1}{1+\exp(-1.2)}\approx 0.769. \end{aligned}

These values lie between zero and one and increase across the cumulative splits, as cumulative probabilities for an ordered response must. The calculation applies the model to the cumulative events; it does not treat the thresholds themselves as probabilities.

What the Minus Sign Means

Because the link is strictly increasing, its inverse is also strictly increasing. Thus, for a fixed threshold, increasing the linear predictor $x^\mathsf{T}\beta$ decreases $\theta_j-x^\mathsf{T}\beta$, and therefore decreases the cumulative probability $P(Y\leq j\mid x)$. This direction is the central sign-convention point. A positive coefficient does not mean that the probability of being at or below a category increases.

Result: Suppose the covariate vector changes from $x$ to $x+d$, with the same thresholds and coefficients. If $d^\mathsf{T}\beta>0$, then for every $j$, $$ P(Y\leq j\mid x+d)<P(Y\leq j\mid x). $$ If $d^\mathsf{T}\beta<0$, the inequality is reversed; if $d^\mathsf{T}\beta=0$, these cumulative probabilities are unchanged. Proof. The linear predictor at $x+d$ is $$ (x+d)^\mathsf{T}\beta=x^\mathsf{T}\beta+d^\mathsf{T}\beta. $$ Consequently, the argument of $g^{-1}$ at $x+d$ is $$ \theta_j-(x+d)^\mathsf{T}\beta =\theta_j-x^\mathsf{T}\beta-d^\mathsf{T}\beta. $$ When $d^\mathsf{T}\beta>0$, this argument is strictly smaller than $\theta_j-x^\mathsf{T}\beta$. Since $g^{-1}$ is strictly increasing, applying it preserves the strict inequality. The cases $d^\mathsf{T}\beta<0$ and $d^\mathsf{T}\beta=0$ follow by the same comparison. $\square$

For a single predictor, if $\beta>0$, increasing $x$ lowers every cumulative probability. In terms of the ordered response, less probability lies at or below each split and more probability lies above it. This describes the direction of the cumulative effects without claiming that every individual category probability must increase: probabilities for middle categories depend on more than one cumulative split.

Worked Example: Reading the Direction of an Effect

Use the four-category logit model from the previous example. Compare $x=0$ with $x=1$. The coefficient is positive, so the model predicts lower cumulative probabilities at $x=1$. Verify this directly for the first split.

At $x=0$, the first linear predictor is $\theta_1-0.8(0)=-1$, giving

P(Y\leq1\mid x=0)=\frac{1}{1+\exp(1)}\approx 0.269.

At $x=1$, the predictor is $-1-0.8=-1.8$, so

P(Y\leq1\mid x=1)=\frac{1}{1+\exp(1.8)}\approx 0.142.

The cumulative probability decreases by approximately $0.269-0.142=0.127$. The sign agrees with the result: the predictor change is $d=1$, so $d\beta=0.8>0$, which implies a decrease in $P(Y\leq j\mid x)$ for every split.

Choosing and Applying a Link

The equation has the same structure for different links, but the inverse link determines how a given real-valued predictor becomes a probability. A numerical predictor value alone therefore does not determine a cumulative probability; the chosen link and threshold are also needed. The following calculation isolates that distinction.

Worked Example: The Same Linear Predictor Under Two Links

Suppose a cumulative split has linear predictor $\eta=\log(4)$. Under the logit link, the inverse link gives

g^{-1}(\eta)=\frac{\exp(\eta)}{1+\exp(\eta)} =\frac{4}{1+4}=0.8.

Under the probit link, the inverse link is $\Phi$, so the same numerical value gives

g^{-1}(\eta)=\Phi(\log 4)\approx\Phi(1.386)\approx0.917.

Thus, the same value on the linear-predictor scale gives different probabilities under these two link choices. This comparison illustrates the inverse-link calculation; it does not say that fitted thresholds or coefficients from models with different links should be treated as directly comparable.

With several predictors, the equation still uses a single scalar linear predictor at each cumulative split. For example, if $x=(x_1,x_2)^\mathsf{T}$ and $\beta=(\beta_1,\beta_2)^\mathsf{T}$, then $x^\mathsf{T}\beta=x_1\beta_1+x_2\beta_2$. The sign convention applies to this combined contribution, not to either predictor in isolation.

Worked Example: Combining Two Predictor Contributions

Consider a cumulative logit model with two predictors, $x=(2,1)^\mathsf{T}$, coefficient vector $\beta=(0.4,-0.3)^\mathsf{T}$, and threshold $\theta_j=1.2$ for the split of interest. Find the cumulative probability at this split.

First calculate the linear predictor:

x^\mathsf{T}\beta=(2)(0.4)+(1)(-0.3)=0.8-0.3=0.5.

The threshold-minus-predictor value is therefore $1.2-0.5=0.7$. Transforming back with the inverse logit gives

P(Y\leq j\mid x) =\frac{1}{1+\exp(-0.7)} \approx 0.668.

Both covariates contribute to the model through their weighted sum. Although the first coefficient is positive and the second negative, the combined predictor is positive at these particular covariate values. The cumulative probability follows from that combined value and the specified threshold.

Equivalent Sign Conventions and a Common Mistake

Some presentations write the equation with a plus sign, such as $g[P(Y\leq j\mid x)]=\alpha_j+x^\mathsf{T}\gamma$. For the same link and predictor values, this is only a reparameterization if $\alpha_j=\theta_j$ and $\gamma=-\beta$. Substituting gives $\alpha_j+x^\mathsf{T}\gamma=\theta_j-x^\mathsf{T}\beta$. The numerical coefficient signs consequently reverse when moving between these conventions. A coefficient’s direction should always be interpreted from the equation actually being used.

A common mistake is to read a positive $\beta$ in the equation $g[P(Y\leq j\mid x)]=\theta_j-x^\mathsf{T}\beta$ as evidence that the cumulative probability rises with $x$. That reasoning overlooks the minus sign. The correction is to compare the linear-predictor arguments first: increasing $x$ when $\beta>0$ makes $\theta_j-x\beta$ smaller. Because the inverse link is increasing, the resulting cumulative probability is smaller, not larger.

A second check is to distinguish the link from its inverse. The equation applies $g$ to a probability; calculating a probability from the equation requires $g^{-1}$. The inverse-link calculation is valid for every real predictor here because the link has been specified as a bijection onto $\mathbb{R}$, not merely as a strictly increasing function.

Key takeaway: In the cumulative link equation, each cumulative probability is obtained by applying $g^{-1}$ to a threshold minus a shared linear predictor. Under this minus-sign convention, a predictor change with positive dot product with $\beta$ lowers every cumulative probability.

Check Your Understanding

Use the equation and sign convention stated in this tutorial.

In $g[P(Y\leq j\mid x)]=\theta_j-x^\mathsf{T}\beta$, what does the threshold $\theta_j$ correspond to in an ordered response?
Why is it not enough to require that $g$ be strictly increasing if its inverse must accept every real linear predictor?
If $d^\mathsf{T}\beta<0$, does $P(Y\leq j\mid x+d)$ rise, fall, or remain unchanged relative to $P(Y\leq j\mid x)$? Explain.
For a probit link and linear predictor $\eta$, which function gives the corresponding cumulative probability?
How must the coefficient vector change to express the same linear predictor using $\alpha_j+x^\mathsf{T}\gamma$ instead of $\theta_j-x^\mathsf{T}\beta$?

What You'll Learn