Thresholds and Baseline Category Probabilities

Thresholds and a Reference Setting

In a cumulative link model, each threshold marks a boundary between adjacent ordered response categories. Its location is meaningful only together with the predictor setting at which it is interpreted. When the linear predictor is zero, a threshold directly determines a cumulative probability through the link distribution. At a different reference setting, the linear predictor must be accounted for before interpreting that threshold.

This connection makes thresholds interpretable in terms of the response scale: they can be translated into the chance of being at or below a category boundary, and therefore into the probabilities of individual categories. The calculations below focus on that translation at zero or a specified reference setting.

Definition: Let $Y$ take ordered values $1,\ldots,K$, and let $G$ be a continuous, strictly increasing cumulative distribution function (CDF) with limits 0 and 1 at negative and positive infinity. Write $h=G^{-1}$ for its inverse. A cumulative link model has the form $$ h\{P(Y\leq j\mid x)\}=\theta_j-x^\mathsf{T}\beta,\qquad j=1,\ldots,K-1, $$ where $\theta_1<\cdots<\theta_{K-1}$ are thresholds and $x^\mathsf{T}\beta$ is the linear predictor. Equivalently, $$ P(Y\leq j\mid x)=G(\theta_j-x^\mathsf{T}\beta). $$

The logistic and standard normal CDFs satisfy the stated conditions on $G$. Their inverses are, respectively, the logit function $h(q)=\log\{q/(1-q)\}$ and the standard normal quantile function. The inverse is well-defined for every $q$ strictly between 0 and 1 because continuity and the two tail limits ensure that every such probability is attained, while strict increase ensures it is attained only once.

At a reference predictor value $x_0$, write $\eta_0=x_0^\mathsf{T}\beta$. If the cumulative probability at boundary $j$ is $q_j=P(Y\leq j\mid x_0)$, then $$ q_j=G(\theta_j-\eta_0). $$ Thus, the quantity mapped back from $q_j$ by $G^{-1}$ is $\theta_j-\eta_0$, not necessarily $\theta_j$ itself. At $x_0=0$, the linear predictor is zero and the threshold equals $G^{-1}(q_j)$.

From Thresholds to Baseline Category Probabilities

For three ordered categories, there are two thresholds. The first cumulative probability is the probability of category 1; the second is the combined probability of categories 1 and 2. Subtracting the first cumulative probability from the second isolates the probability of category 2. The remaining probability belongs to category 3.

Theorem: In a three-category cumulative link model, let $\eta_0=x_0^\mathsf{T}\beta$, and define $q_1=P(Y\leq1\mid x_0)$ and $q_2=P(Y\leq2\mid x_0)$. Then $$ \begin{aligned} P(Y=1\mid x_0)&=q_1,\\ P(Y=2\mid x_0)&=q_2-q_1,\\ P(Y=3\mid x_0)&=1-q_2. \end{aligned} $$ The cumulative probabilities are $q_j=G(\theta_j-\eta_0)$. In particular, at $x_0=0$, they are $G(\theta_j)$.

Proof. Since the response can take only the values 1, 2, or 3, the event $Y\leq1$ is exactly the event $Y=1$. Therefore $P(Y=1\mid x_0)=q_1$. The event $Y\leq2$ is the disjoint union of $Y=1$ and $Y=2$, so $q_2=P(Y=1\mid x_0)+P(Y=2\mid x_0)$. Subtracting $q_1$ gives the second equality. The three category probabilities sum to 1, so the remaining probability is $1-q_2$. The cumulative link equation gives $q_j=G(\theta_j-\eta_0)$, and setting $x_0=0$ gives $\eta_0=0$. $\square$

Worked Example: Logistic Thresholds at Zero Predictors

Suppose a three-category model uses the logistic CDF and has thresholds $\theta_1=\log(1/4)$ and $\theta_2=\log(7/3)$. Find the cumulative and category probabilities when $x=0$.

For the logistic CDF, $G(t)=1/(1+\exp(-t))$. At zero predictors, the cumulative probabilities are $q_j=G(\theta_j)$. First, $$ q_1=\frac{1}{1+\exp\{-\log(1/4)\}} =\frac{1}{1+4} =0.20. $$

For the second threshold, $$ q_2=\frac{1}{1+\exp\{-\log(7/3)\}} =\frac{1}{1+3/7} =\frac{7}{10} =0.70. $$ The category probabilities are consequently $$ P(Y=1\mid 0)=0.20,\qquad P(Y=2\mid 0)=0.70-0.20=0.50,\qquad P(Y=3\mid 0)=1-0.70=0.30. $$ They are all positive and sum to $0.20+0.50+0.30=1$. Also, $\theta_1<\theta_2$, as required for ordered thresholds.

Recovering Thresholds from Cumulative Probabilities

The same relationship can be used in the opposite direction. To recover thresholds from cumulative probabilities, first apply the inverse link to each cumulative probability. If the reference linear predictor is nonzero, add it back to obtain the threshold. The conditions on $G$ matter: an inverse value exists uniquely for every probability strictly between 0 and 1 when $G$ is continuous, strictly increasing, and has the stated tail limits.

Theorem: Suppose $G$ is continuous and strictly increasing on the real line, with limits 0 and 1 at negative and positive infinity. Fix a reference setting $x_0$, with $\eta_0=x_0^\mathsf{T}\beta$. For any $0<q_1<\cdots<q_{K-1}<1$, the unique thresholds satisfying $P(Y\leq j\mid x_0)=q_j$ for every $j$ are $$ \theta_j=\eta_0+G^{-1}(q_j),\qquad j=1,\ldots,K-1. $$ They are strictly ordered.

Proof. The tail limits provide real numbers $a$ and $b$ such that $G(a)<q_j<G(b)$ for any fixed $q_j\in(0,1)$. Continuity and the intermediate value theorem give a value $z_j\in(a,b)$ with $G(z_j)=q_j$. Strict increase makes that value unique, so $z_j=G^{-1}(q_j)$. The cumulative link equation at $x_0$ requires $G(\theta_j-\eta_0)=q_j$. By uniqueness, $\theta_j-\eta_0=G^{-1}(q_j)$, giving the stated formula. Since $q_j<q_{j+1}$ and $G^{-1}$ is strictly increasing, $G^{-1}(q_j)<G^{-1}(q_{j+1})$; adding the same $\eta_0$ preserves this order. This proves existence, uniqueness, and ordering. $\square$

Worked Example: Recovering Probit Thresholds from Baseline Probabilities

At $x=0$, suppose a three-category cumulative probit model has category probabilities $0.25$, $0.55$, and $0.20$. Find the thresholds.

The first cumulative probability is $q_1=0.25$. The second includes categories 1 and 2, so $q_2=0.25+0.55=0.80$. At $x=0$, each threshold is the standard normal quantile of its cumulative probability: $$ \theta_1=\Phi^{-1}(0.25)\approx-0.6745,\qquad \theta_2=\Phi^{-1}(0.80)\approx0.8416. $$ The thresholds are ordered because $-0.6745<0.8416$. Checking in the forward direction, $\Phi(\theta_1)\approx0.25$ and $\Phi(\theta_2)\approx0.80$, so differencing gives category probabilities $0.25$, $0.80-0.25=0.55$, and $1-0.80=0.20$, as specified.

When the Reference Predictor Is Not Zero

A threshold is not itself the cumulative probability when the reference linear predictor is nonzero. The cumulative probability depends on the difference $\theta_j-\eta_0$. Consequently, recovering a threshold from a cumulative probability requires adding $\eta_0$ after applying the inverse CDF. This adjustment is shared across thresholds at the same reference setting.

Worked Example: Logistic Probabilities at a Nonzero Reference Setting

Consider a three-category cumulative logit model with one predictor $x$, slope $\beta=0.4$, and thresholds to be determined. At the reference value $x_0=1.5$, suppose the desired category probabilities are $0.30$, $0.45$, and $0.25$. Find the thresholds.

The reference linear predictor is $\eta_0=x_0\beta=1.5(0.4)=0.6$. The cumulative probabilities at the reference are $q_1=0.30$ and $q_2=0.30+0.45=0.75$. For a logistic CDF, $G^{-1}(q)=\log\{q/(1-q)\}$. Therefore $$ \theta_1=0.6+\log\left(\frac{0.30}{0.70}\right) \approx0.6-0.8473=-0.2473, $$ and $$ \theta_2=0.6+\log\left(\frac{0.75}{0.25}\right) =0.6+1.0986=1.6986. $$ To check, subtract the reference predictor from each threshold: $\theta_1-0.6\approx-0.8473$ and $\theta_2-0.6\approx1.0986$. Applying the logistic CDF gives $q_1=0.30$ and $q_2=0.75$. The resulting category probabilities are $0.30$, $0.75-0.30=0.45$, and $1-0.75=0.25$. The thresholds remain ordered.

Validity Checks and Common Pitfalls

A set of cumulative probabilities must be strictly increasing and lie between 0 and 1 to correspond to finite, strictly ordered thresholds under the stated assumptions. For three categories, positive category probabilities automatically give $0<q_1<q_2<1$: the cumulative probabilities are $q_1=p_1$ and $q_2=p_1+p_2$, and positivity of all three categories ensures the required inequalities.

Using a threshold as though it were a probability. A threshold can be negative or larger than 1. Apply $G$ to $\theta_j-\eta_0$ to obtain a cumulative probability.
Ignoring the reference predictor. At $x_0\ne0$, the relevant quantity is $\theta_j-x_0^\mathsf{T}\beta$. Recovering the threshold requires adding $x_0^\mathsf{T}\beta$ to the inverse-CDF value.
Subtracting cumulative probabilities in the wrong order. For category 2, use $q_2-q_1$, not $q_1-q_2$. The difference must be nonnegative because cumulative probabilities are ordered.
Assuming strict increase alone guarantees an inverse for every probability. A strictly increasing CDF can have jumps and may omit some values between 0 and 1. Continuity, strict increase, and the two tail limits together ensure a unique inverse value for every probability in that interval.

For example, a mixture assigning probability $1/2$ to a standard normal variable and probability $1/2$ to a point mass at zero has a jump in its CDF at zero. Its CDF is strictly increasing, but it does not attain every probability between 0 and 1. Thus strict increase alone is not enough to justify applying an inverse to an arbitrary cumulative probability. Logistic and standard normal links do satisfy the stronger conditions used here.

Key takeaway: At a reference setting with linear predictor $\eta_0$, cumulative probability $q_j$ corresponds to threshold $\theta_j=\eta_0+G^{-1}(q_j)$. At zero linear predictor, this reduces to $G^{-1}(q_j)$. Category probabilities are obtained from successive cumulative probabilities and their complements.

Check Your Understanding

Use the cumulative-link equation to answer each question.

In a three-category model at zero predictors, the cumulative probabilities are $0.15$ and $0.65$. What are the three category probabilities?
For a logistic model at zero predictors, write the thresholds in terms of cumulative probabilities $q_1$ and $q_2$.
At a reference setting with linear predictor $0.8$, a cumulative probability is $0.40$. Express the corresponding threshold using the inverse link.
Why must the second cumulative probability exceed the first if all three category probabilities are positive?
What additional property, beyond strict increase, ensures that every probability between 0 and 1 has a unique inverse-CDF value?