Continuation-Ratio Likelihoods For Sequential Categories

Ordered Outcomes as Successive Transitions

An ordered response can sometimes be understood as a sequence of stages: an observation either stops at its current category or continues to a later one. Examples might include the highest stage of a process reached, or an ordered rating where each level represents moving beyond the preceding level. A continuation-ratio likelihood describes the probability of stopping at each stage conditional on having reached that stage.

This conditional definition is the key distinction from a model that directly describes cumulative probabilities or compares neighboring categories. It also makes the response-process interpretation important: the categories must have a meaningful order, and the sequential conditioning must match the question being modeled.

Definition: Let \(Y_i\) take one of \(K\) ordered categories \(1,\ldots,K\). For \(k=1,\ldots,K-1\), define the continuation-ratio probability \(q_{ik}=P(Y_i=k\mid Y_i\ge k,\ b_i)\). It is the probability that observation \(i\) stops in category \(k\), conditional on reaching category \(k\), given random effects \(b_i\). Category \(K\) is the terminal category, so its conditional stopping probability is 1.

The event \(Y_i\ge k\) means that the observation has not stopped in any earlier category. Thus \(q_{i1}=P(Y_i=1\mid b_i)\), while \(q_{i2}\) is conditional on not stopping in category 1. A low value of \(q_{ik}\) means a greater chance of continuing beyond category \(k\), among observations that reach it.

From Transition Probabilities to Category Probabilities

To end in category \(k<K\), an observation must continue through every earlier category and then stop at \(k\). The probability of category \(K\) is the probability of continuing at every earlier stage. Therefore,

P(Y_i=k\mid b_i) = q_{ik}\prod_{h=1}^{k-1}(1-q_{ih}),\quad k=1,\ldots,K-1, \qquad P(Y_i=K\mid b_i) = \prod_{h=1}^{K-1}(1-q_{ih}).

An empty product, as in the formula for \(k=1\), equals 1. These probabilities are nonnegative and sum to 1. To see why, the probability of reaching stage \(k\) is \(\prod_{h<k}(1-q_{ih})\); the probability of reaching stage \(k\) and stopping there is that quantity times \(q_{ik}\). The sum over all stopping stages plus the probability of continuing through the last modeled stage telescopes to 1.

Model: A common continuation-ratio GLMM uses \(\operatorname{logit}(q_{ik})=\alpha_k+x_i^\mathsf{T}\beta_k+z_{ik}^\mathsf{T}b_i\), where \(\alpha_k\) is a transition-specific intercept, \(x_i\) is a covariate vector, \(\beta_k\) contains fixed effects, \(z_{ik}\) specifies the random-effect contribution, and \(b_i\sim N(0,D)\). A shared fixed-effect vector \(\beta_k=\beta\) across stages is an additional restriction, not a requirement of the continuation-ratio formulation.

The model is conditional on the random effects. Its usual assumptions include a correctly specified ordered-category and transition structure, a suitable link and linear predictor for each transition, and a specified random-effects distribution. For independent subjects, their random effects are independent draws from the stated distribution and are independent of the covariates. For repeated observations within a subject or cluster, the model assumes conditional independence given that cluster’s random effects unless further dependence is explicitly modeled.

A random effect is not part of the observed response. It represents latent cluster-level variation in transition probabilities. Consequently, the model first specifies category probabilities conditional on \(b_i\); population-level probabilities require averaging over the random-effects distribution.

The Likelihood Factorization

The factorization follows from the chain rule for conditional probabilities. For one observation, the probability of category \(k<K\) is the probability of continuing at stages \(1,\ldots,k-1\), multiplied by the probability of stopping at \(k\). For category \(K\), it is the probability of continuing at every modeled stage. This gives the category-probability formulas above without treating the stages as independent unconditional outcomes.

For a cluster \(i\) with one ordinal outcome, let \(p_{ik}(b_i)=P(Y_i=k\mid b_i)\). Its contribution to the marginal likelihood is obtained by integrating the conditional contribution over the random-effects distribution:

L_i = \int \left\{ \prod_{k=1}^{K}p_{ik}(b)^{\mathbb{1}(Y_i=k)} \right\} \phi(b;0,D)\,db, \qquad L=\prod_i L_i.

Here \(\mathbb{1}(Y_i=k)\) equals 1 when \(Y_i=k\) and 0 otherwise, and \(\phi(b;0,D)\) is the density of the specified random-effects distribution. The integral is generally evaluated numerically. For several conditionally independent observations in a cluster, the conditional contributions are multiplied within the integral, so the same random effect is integrated over the entire cluster contribution.

Worked Example: Factoring a Three-Category Likelihood

Suppose 20 hypothetical independent observations are classified as low, middle, or high, numbered \(1,2,3\). Eight stop in category 1, six continue and stop in category 2, and six continue through to category 3. Assume a model with no covariates or random effects, with \(q_1=0.40\) and \(q_2=0.50\). The observations are independent and these transition probabilities apply to each one.

The category probabilities are \(P(Y=1)=q_1=0.40\), \(P(Y=2)=(1-q_1)q_2=(0.60)(0.50)=0.30\), and \(P(Y=3)=(1-q_1)(1-q_2)=(0.60)(0.50)=0.30\). They sum to \(0.40+0.30+0.30=1.00\).

The likelihood kernel for the observed category counts is equivalently factored by transition. The first transition has 8 stops and 12 continuations; among those 12, the second has 6 stops and 6 continuations:

L(q_1,q_2) \propto q_1^8(1-q_1)^{12}q_2^6(1-q_2)^6.

Substituting the stated probabilities gives \(0.40^8(0.60)^{12}(0.50)^6(0.50)^6 =0.40^8(0.60)^{12}(0.50)^{12} \approx 3.48\times10^{-10}\). The proportionality omits the multinomial coefficient for the category counts, which does not depend on \(q_1\) or \(q_2\). The estimated transition probabilities from these counts are \(\hat q_1=8/20=0.40\) and \(\hat q_2=6/12=0.50\); the denominator for the second transition is 12 because only those 12 observations reached it.

Interpreting Transition Effects

Under a logit link, \(\beta_k\) describes a covariate’s effect on the log odds of stopping at category \(k\) rather than continuing beyond it, conditional on reaching \(k\) and on the random effects. In particular, a one-unit increase in a covariate with coefficient \(\beta_k\) multiplies these conditional stopping odds by \(\exp(\beta_k)\), holding the other predictors fixed. This is not generally an odds ratio for any two final categories.

Worked Example: A Shared Transition Effect

Consider a hypothetical three-category outcome with no random effects. Suppose the baseline transition probabilities are \(q_1=0.40\) and \(q_2=0.50\). A binary covariate \(x\) has the same log-odds coefficient at both transitions, \(\beta=\log(0.5)\approx-0.6931\). Thus a one-unit increase in \(x\) halves the conditional odds of stopping at either transition. This example assumes the same coefficient applies at both stages.

For \(x=0\), the category probabilities are \(0.40\), \((1-0.40)(0.50)=0.30\), and \((1-0.40)(1-0.50)=0.30\).

At transition 1, the baseline stopping odds are \(0.40/0.60=2/3\). For \(x=1\), the odds become \((2/3)(0.5)=1/3\), giving \(q_1= (1/3)/(1+1/3)=0.25\). At transition 2, baseline odds are \(0.50/0.50=1\); halving them gives odds \(0.5\), and hence \(q_2=0.5/(1+0.5)=1/3\).

Therefore, for \(x=1\), the category probabilities are \(0.25\), \((0.75)(1/3)=0.25\), and \((0.75)(2/3)=0.50\). They sum to 1. The coefficient describes conditional transition odds, not a direct multiplicative change in each category probability: the probability of category 2, for example, depends on both reaching that category and stopping there.

Random Effects and Population-Averaged Probabilities

A conditional transition probability \(q_{ik}(b)\) applies to a cluster with random effect \(b\). Averaging it over the unconditional random-effects distribution gives \(E_b[q_{ik}(b)]\), but that is not generally the probability of stopping at stage \(k\) among the population that reaches stage \(k\). Reaching stage \(k\) depends on earlier transitions, which can also depend on \(b\). The random-effects distribution among those at risk is therefore reweighted toward values of \(b\) associated with reaching that stage.

Worked Example: Reweighting Among Those Who Reach a Stage

Suppose, hypothetically, that a random effect \(b\) is \(+0.5\) or \(-0.5\), each with probability \(0.5\). There are three categories, and both transition probabilities are \(q_k(b)=\operatorname{logit}^{-1}(b)\), where \(\operatorname{logit}^{-1}(u)=1/(1+\exp(-u))\). Thus \(q(+0.5)\approx0.6225\) and \(q(-0.5)\approx0.3775\).

The unconditional average of the stage-2 conditional probabilities is \((0.6225+0.3775)/2=0.5000\). But the probability of reaching stage 2 is \(1-q(b)\): it is \(0.3775\) for \(b=+0.5\) and \(0.6225\) for \(b=-0.5\). Among observations that reach stage 2, the stopping probability is therefore

P(Y=2\mid Y\ge2) = \frac{E_b[(1-q_1(b))q_2(b)]}{E_b[1-q_1(b)]} = \frac{0.5(0.3775)(0.6225)+0.5(0.6225)(0.3775)} {0.5(0.3775)+0.5(0.6225)} \approx 0.4700.

The calculation weights each random-effect value by its probability of reaching stage 2. The unweighted average, \(0.5000\), is \(E_b[q_2(b)]\); it is not the population stopping probability conditional on reaching stage 2. This distinction matters whenever transition probabilities vary with random effects.

Diagnostics, Comparisons, and Limitations

Useful checks focus on the transitions as well as on the final categories. Examine the number of observations at risk at each stage; later transitions may be estimated from much smaller subsets. Compare observed and model-predicted category frequencies, and, where meaningful, compare transition-specific stopping rates among those at risk. Check whether fitted probabilities are extreme or whether a transition has no observed stops or no continuations, which can create separation or weakly identified coefficients. For mixed models, also check whether numerical integration is adequate and whether the fitted random-effects structure is estimable.

A continuation-ratio formulation is not automatically a literal model of a physical process. Its conditional probabilities are mathematically definable for ordered categories, but a stage-by-stage interpretation is strongest when the response process supports it. A shared coefficient across transitions can be too restrictive if covariate effects differ by stage. Conversely, transition-specific effects can require many parameters, especially when few observations reach later stages.

The main alternatives describe different conditional comparisons. Cumulative-link likelihoods model cumulative probabilities across cutpoints; proportional odds adds a common-slope restriction on cumulative log odds. Adjacent-category likelihoods compare neighboring category probabilities. Continuation-ratio likelihoods instead model stopping versus continuing among those who have reached a category. These formulations can yield different fitted probabilities and answer different conditional questions; one is not a mere relabeling of another.

A frequent mistake is to interpret \(\exp(\beta_k)\) as the odds ratio for being in category \(k\) rather than any other category. The correction is to state the conditioning: it is the ratio of conditional stopping odds at transition \(k\), comparing covariate values while holding other predictors and random effects fixed. Another mistake is to average conditional transition probabilities over the original random-effects distribution and call the result the probability among those who reached that transition. As the example shows, the latter probability must account for the random-effects distribution among those at risk.

Key takeaway: A continuation-ratio model builds final category probabilities from successive conditional stop-or-continue probabilities. Interpret its coefficients at those transitions, and distinguish cluster-conditional probabilities from probabilities averaged over the appropriate at-risk population.

Check Your Understanding

Try each question, then compare with the solutions below.

For three categories, let \(q_1=0.20\) and \(q_2=0.60\). Find the probabilities of all three categories.
At stage 2, 18 of 30 observations that reached that stage stop there. What is the estimated continuation-ratio probability \(q_2\), and what is its denominator?
A transition-specific logit coefficient is \(\log(1.5)\) for a one-unit covariate increase. Interpret its exponentiated value.
Why might \(E_b[q_2(b)]\) differ from \(P(Y=2\mid Y\ge2)\)?

Solutions. (1) \(P(Y=1)=0.20\), \(P(Y=2)=(0.80)(0.60)=0.48\), and \(P(Y=3)=(0.80)(0.40)=0.32\); these sum to 1. (2) \(\hat q_2=18/30=0.60\). Its denominator is 30 because those are the observations that reached stage 2. (3) Holding other predictors and random effects fixed, a one-unit increase multiplies the conditional odds of stopping at that transition rather than continuing by \(1.5\), among observations reaching it. (4) The random-effects distribution among those reaching stage 2 can differ from the original distribution because reaching stage 2 depends on earlier transition probabilities, which may depend on \(b\). The at-risk probability must weight \(q_2(b)\) by the probability of reaching stage 2.