Continuation-Ratio Logits as Sequential Decisions

Sequential Decisions Among Ordered Categories

Some ordered responses can be viewed as the outcome of a sequence of decisions. A patient may enter one disease stage or progress to a later stage; a student may complete one educational level or proceed to another. At each stage, the relevant comparison is between being in the current category and being in a later category, among observations that have reached the point where either outcome is possible.

A forward continuation-ratio logit formalizes that comparison. “Forward” means that the categories are considered in their stated order, from the first category to the last. The denominator is not just the next category: it includes all later categories, because those outcomes represent proceeding beyond the current one.

The conditional probabilities are defined only when \(S_j(x)>0\): there must be some probability of reaching the decision at step \(j\). When both conditional outcomes have positive probability, the logit is finite. The final category \(K\) has no forward continuation logit, because there is no later category to proceed to.

What the Odds Compare

The conditioning event \(Y\geq j\) defines the eligible risk set at step \(j\). The comparison is therefore local to that set: among observations whose outcome is category \(j\) or later, how do the odds of being in category \(j\) compare with the odds of being in any later category?

Conditioning does not alter the odds ratio, because both conditional probabilities have the same denominator \(S_j(x)\). It does, however, give the comparison its sequential interpretation: individuals in categories earlier than \(j\) are not eligible for the decision at step \(j\).

Proof. Set \(T_j(x)=\sum_{k=j+1}^K p_k(x)\). Since \(S_j(x)=p_j(x)+T_j(x)>0\), the conditional probability of category \(j\) is \(p_j(x)/S_j(x)\), and the conditional probability of proceeding is \(T_j(x)/S_j(x)\). If both are positive, their ratio is

Its logarithm is finite if and only if the numerator and denominator are both positive. This proves both claims. \(\square\)

From a Finite Logit to Conditional Probabilities

A finite continuation-ratio logit can be translated into the two conditional probabilities at that step. This is an algebraic interpretation of a single decision; it does not by itself specify a model for the logits or a regression structure.

Proof. Let \(q=P(Y=j\mid Y\geq j,x)\) and \(r=P(Y>j\mid Y\geq j,x)\). These are positive and satisfy \(q+r=1\). By definition, \(\exp(\theta_j(x))=q/r\). Write \(a=\exp(\theta_j(x))\), so \(q=ar\). Using \(q+r=1\), we obtain \(ar+r=1\), or \(r=1/(1+a)\). It follows that \(q=a/(1+a)\). Substituting \(a=\exp(\theta_j(x))\) proves the result. \(\square\)

Continuation Ratios Versus Adjacent-Category Odds

An adjacent-category logit compares the probability of category \(j\) with the probability of the immediately following category \(j+1\). A forward continuation-ratio logit instead compares category \(j\) with the combined probability of every later category. At the final decision, \(j=K-1\), these coincide because category \(K\) is the only later category. Earlier in the sequence, they generally differ.

For example, with four categories, the step-2 continuation comparison is \(p_2/(p_3+p_4)\), not \(p_2/p_3\). Replacing the sum of later probabilities by only the next-category probability changes the question being answered: it no longer compares stopping at category 2 with proceeding beyond category 2.

Boundary Cases and a Common Mistake

The finite-logit definition uses positive probabilities for both outcomes, but valid distributions can lie on the boundary. The appropriate conclusion depends first on whether the eligible risk set has positive probability.

• If \(S_j(x)>0\), \(p_j(x)=0\), and \(T_j(x)>0\), the conditional probability of category \(j\) is zero and the logit is \(-\infty\).
• If \(S_j(x)>0\), \(p_j(x)>0\), and \(T_j(x)=0\), the conditional probability of proceeding is zero and the logit is \(+\infty\).
• If \(p_j(x)=T_j(x)=0\), then \(S_j(x)=0\). No observations are eligible, so the conditional probabilities and logit are undefined.

Common mistake: “A zero probability for category \(j\) always makes the logit \(-\infty\).” This overlooks the conditioning event. For probabilities \((1,0,0)\), category 2 has probability zero, but at step 2 the total eligible probability is also zero. There is no conditional probability at that step to take a logit of. The correction is to check \(S_j(x)>0\) first, and only then inspect the probabilities of entering and proceeding.

The sequential interpretation is most useful when category order corresponds to a meaningful progression and the risk set at each stage makes sense for the scientific question. It should not be imposed merely because categories can be arranged in some order: the conditional odds then may not describe an actual process of stopping or proceeding.