When Sequential Logits Misrepresent the Process

Algebraic Fit Is Not a Sequential Story

A continuation-ratio model can describe categorical probabilities through a sequence of conditional probabilities. That can be a natural representation when people really move through successive stages, or when each conditional comparison corresponds to a genuine risk set. But the same algebra can be applied to categories that are merely different labels. In that case, the model may fit or factor the probabilities without representing how outcomes were generated.

The distinction matters because a sequential interpretation invites substantive claims: who remains at risk at each step, what decision or event occurs there, and what a covariate effect means at that step. Those claims do not follow just from writing down conditional probabilities. The ordering and risk sets must be justified by the study design or scientific process.

This definition makes the ordering consequential: \(Y\geq j\) is defined by the labels. In an actual sequential process, the labels should correspond to successive stages and the risk set should identify those still eligible for the event at that stage. For nominal categories, the same conditioning event can be computed from a chosen ordering, but it may have no process-based interpretation.

A Factorization Does Not Establish a Mechanism

Write \(p_j(x)=P(Y=j\mid x)\), and let \(S_j(x)=\sum_{\ell=j}^{K}p_\ell(x)\) be the probability of being in the forward risk set at step \(j\). If \(S_j(x)>0\), then \(q_j(x)=p_j(x)/S_j(x)\). This gives a mathematical representation of the category probabilities. It does not say that an individual literally faced a decision at each step.

Proof. Since \(S_1=1\), and \(S_{j+1}=S_j-p_j=S_j(1-q_j)\), it follows successively that \(S_j=\prod_{r=1}^{j-1}(1-q_r)\). Therefore, for \(j<K\), \(p_j=q_jS_j=q_j\prod_{r=1}^{j-1}(1-q_r)\). For the last category, \(p_K=S_K=\prod_{r=1}^{K-1}(1-q_r)\). For the converse, define the probabilities by the displayed formulas. Their sum is one because \[ \sum_{j=1}^{K-1}q_j\prod_{r=1}^{j-1}(1-q_r) +\prod_{r=1}^{K-1}(1-q_r)=1. \] To verify the identity, set \(A_j=\prod_{r=1}^{j}(1-q_r)\), with \(A_0=1\). Each summand in the sum is \(A_{j-1}-A_j\), so the total is \(\sum_{j=1}^{K-1}(A_{j-1}-A_j)+A_{K-1}=A_0=1\). All probabilities are positive because every \(q_j\) and \(1-q_j\) is positive. The risk-set probabilities obtained from these probabilities are \(S_j=\prod_{r=1}^{j-1}(1-q_r)\), so the conditional probability at step \(j\) is indeed \(q_j\). \(\square\)

The theorem explains the central diagnostic caution: continuation probabilities can always be recovered from positive category probabilities for a chosen order. Thus, the existence of a continuation-ratio factorization is not evidence that the response arose through sequential decisions. A probabilistic representation is not, by itself, a causal or temporal account.

When the Sequential Interpretation Is Defensible

A continuation-ratio interpretation is more plausible when the data-generating process identifies a sequence and the risk set at each step. For example, a clinical pathway might classify a patient by the first of several ordered escalation stages reached during follow-up. If patients first face a decision about an initial escalation, and only those not yet escalated remain at risk for a later stage, the successive conditional probabilities can match the pathway.

Even then, the analyst must check the details. The outcome categories must be mutually exclusive in the way the model assumes, the order must match the actual sequence, and the conditional denominator must represent the people eligible at that stage. A list of severity labels alone does not prove that patients passed through the associated risk sets. The process may include transitions, repeated events, or eligibility rules that a single categorical response does not capture.

Conditional odds must also use the correct risk-set denominator. For example, if a stage has event mass \(0.20\) and non-event mass \(0.30\), its conditional event probability is \(0.20/(0.20+0.30)=0.40\), and its conditional odds are \(0.20/0.30=2/3\). If another conditional comparison has probability \(0.40\), its odds are \(0.40/0.60=2/3\); the ratio of these odds is \((2/3)/(2/3)=1\), not 2. The event and non-event masses, rather than the total risk-set mass alone, determine the odds.

How Arbitrary Ordering Changes Regression Meaning

In a continuation-ratio regression, each logit describes the odds of category \(j\) rather than a later category, conditional on being in the risk set at \(j\). If the labels have no progression, changing their order changes both the conditioning sets and the comparisons. Consequently, regression coefficients can answer different conditional questions under different orders even when the underlying category probabilities are unchanged.

A Practical Diagnostic

Before interpreting a continuation-ratio model as sequential, ask what makes someone enter and leave each risk set. The question should have an answer in the study design or substantive mechanism, not merely in the way the categories were numbered. If the only reason category \(j\) is considered before category \(j+1\) is that it was listed first, the fitted conditional logits are order-dependent descriptions rather than natural stage-specific effects.

Common Mistakes

• Taking a factorization as proof of a sequence. Every positive category distribution has continuation probabilities for a chosen order. Their existence does not show that individuals passed through sequential decisions.
• Assuming an ordered label means an ordered process. Coding nominal options as 1, 2, and 3 creates a computational order, not necessarily a meaningful progression.
• Using the wrong risk set. At step \(j\), the forward continuation probability conditions on \(Y\geq j\). Its odds compare category \(j\) with the remaining later categories within that set.
• Interpreting a coefficient without naming the conditional comparison. State both the event category and the risk set; otherwise, the coefficient’s meaning may be unclear or mistaken.
• Ignoring changes under reordering. If plausible reorderings change the scientific interpretation, that is a warning that the sequential story lacks a basis in the response process.