Adjacent-Category Likelihoods For Ordinal Outcomes

Comparing Neighboring Categories

An ordinal response has categories with a meaningful order, but the gaps between categories need not have a numerical interpretation. A model can use that order by comparing the probability of each category with the probability of the category immediately next to it. These local comparisons define an adjacent-category likelihood.

Suppose $Y_{ij}$ is the response for observation $i$ in cluster $j$, and its possible categories are $1,\ldots,K$, ordered from lowest to highest. Write $p_{ijk}=P(Y_{ij}=k\mid u_j,x_{ij})$, the probability of category $k$ conditional on covariates $x_{ij}$ and a cluster random effect $u_j$. There are $K-1$ adjacent comparisons: category 2 versus 1, category 3 versus 2, and so on.

Definition: The adjacent-category logit for the pair $k$ and $k+1$ is $\log\{p_{ij,k+1}/p_{ijk}\}$, for $k=1,\ldots,K-1$. A common-slope adjacent-category GLMM specifies $$ \log\left(\frac{p_{ij,k+1}}{p_{ijk}}\right) =\alpha_k+x_{ij}^{T}\beta+z_{ij}^{T}u_j,\qquad k=1,\ldots,K-1. $$ Here $\alpha_k$ is a category-pair-specific intercept, $\beta$ is a shared fixed-effect coefficient vector, $z_{ij}$ is a random-effect design vector, and $u_j$ is a cluster-level random-effect vector.

The category labels must be mutually exclusive and exhaustive for the response, and their ordering must be specified before fitting the model. The probabilities are conditional probabilities: for a fixed value of $u_j$, they are positive and sum to one. A standard random-intercept version uses $z_{ij}=1$ and $u_j\sim N(0,\sigma_u^2)$, independently across clusters. Responses are assumed conditionally independent within a cluster given its random effect and covariates. The common-slope form additionally assumes that a covariate has the same coefficient in every adjacent logit; the intercepts can differ by category pair.

From Adjacent Odds to Category Probabilities

The logits describe ratios, not probabilities by themselves. To obtain a proper response distribution, every adjacent ratio must be combined with the requirement that the category probabilities sum to one.

r_{ijk}=\exp(\alpha_k+x_{ij}^{T}\beta+z_{ij}^{T}u_j) =\frac{p_{ij,k+1}}{p_{ijk}}.

Start with an unnormalized weight $w_{ij1}=1$. The ratio equation gives $w_{ij,k+1}=w_{ijk}r_{ijk}$, so the weight for category $k$ is the product of all preceding adjacent ratios. Dividing each weight by their sum gives the unique probability vector satisfying the specified ratios.

w_{ij1}=1,\qquad w_{ijk}=\exp\left\{\sum_{\ell=1}^{k-1} (\alpha_\ell+x_{ij}^{T}\beta+z_{ij}^{T}u_j)\right\}\quad(k\geq 2), \qquad p_{ijk}=\frac{w_{ijk}}{\sum_{m=1}^{K}w_{ijm}}.

This construction is a short justification of the normalization. The recurrence ensures that $w_{ij,k+1}/w_{ijk}=r_{ijk}$; dividing numerator and denominator by the same positive sum preserves every ratio. Also, each resulting probability is positive and their sum is exactly one. Conversely, any positive probabilities with the specified adjacent ratios must be proportional to these recursively determined weights, and the sum-to-one constraint fixes the proportionality constant.

Worked Example: Recovering Probabilities From Adjacent Odds

Consider a hypothetical ordinal response with four categories. Suppose the adjacent odds are $p_2/p_1=2$, $p_3/p_2=0.5$, and $p_4/p_3=3$. Find the four probabilities.

Set the first unnormalized weight to 1. The next weights follow by multiplication: $w_1=1$, $w_2=1(2)=2$, $w_3=2(0.5)=1$, and $w_4=1(3)=3$. The total weight is $1+2+1+3=7$.

Therefore $p_1=1/7$, $p_2=2/7$, $p_3=1/7$, and $p_4=3/7$, approximately 0.143, 0.286, 0.143, and 0.429. The probabilities sum to 1, and their adjacent ratios are $2/1=2$, $(1/7)/(2/7)=0.5$, and $(3/7)/(1/7)=3$, as required.

The Ordinal Response Likelihood

For one observation with observed category $y_{ij}$, its conditional likelihood contribution is the probability assigned to that category. If the response is recorded as category counts $n_{ijk}$ within a covariate pattern, the conditional likelihood is multinomial. This is the categorical likelihood built from the adjacent-category probabilities, not a set of separate binary outcomes.

L_j(u_j)=\prod_i\prod_{k=1}^{K}p_{ijk}^{\,I(Y_{ij}=k)}, \qquad L_j(u_j)=\frac{n_j!}{\prod_{k=1}^{K}n_{jk}!} \prod_{k=1}^{K}p_{jk}^{\,n_{jk}} \quad\text{for grouped counts,}

In the grouped expression, the covariates and conditional probabilities are the same for all observations in the group, $n_j=\sum_k n_{jk}$, and $I(\cdot)$ is 1 when its statement is true and 0 otherwise. With random effects, the cluster likelihood used for marginal inference averages the conditional likelihood over the random-effect distribution:

L_j=\int L_j(u_j)f(u_j)\,du_j.

Here $f$ is the specified density of $u_j$, such as the density of a mean-zero normal random effect. The adjacent logits and probabilities describe the response conditional on that random effect. In general, integrating over $u_j$ does not leave the same adjacent-logit form for the marginal probabilities.

Worked Example: Interpreting a Shared Covariate Effect

Suppose a hypothetical three-category model has $\alpha_1=\log(2)$, $\alpha_2=\log(0.5)$, and a binary treatment indicator $x$ with common coefficient $\beta=\log(2)$. For this calculation, set the random effect to zero. Find the category probabilities for $x=0$ and $x=1$, and interpret the coefficient.

For $x=0$, the adjacent odds are $\exp\{\log(2)\}=2$ and $\exp\{\log(0.5)\}=0.5$. The weights are $1,2,1$, with total 4, giving probabilities $(0.25,0.50,0.25)$.

For $x=1$, add $\log(2)$ to each adjacent logit. The odds become $2(2)=4$ and $0.5(2)=1$. The weights are $1,4,4$, with total 9, giving probabilities $(1/9,4/9,4/9)$, or approximately $(0.111,0.444,0.444)$.

The treatment coefficient is a conditional log odds ratio for either neighboring pair, holding the random effect fixed. Its odds ratio is $\exp(\beta)=2$: treatment multiplies the odds of category $k+1$ rather than $k$ by 2 for each adjacent pair. This is not a claim that every category probability doubles. The normalized probabilities show why: changes to all adjacent odds jointly determine the probability of each category.

Worked Example: A Multinomial Likelihood From Hypothetical Counts

In a hypothetical group of 28 independent responses, the observed counts in three ordered categories are $n_1=7$, $n_2=14$, and $n_3=7$. Use the fitted category proportions as the probabilities and write the grouped likelihood.

The proportions are $\hat p_1=7/28=0.25$, $\hat p_2=14/28=0.50$, and $\hat p_3=7/28=0.25$. Their adjacent odds are $0.50/0.25=2$ and $0.25/0.50=0.5$. The grouped likelihood, including the multinomial coefficient, is $$ L=\frac{28!}{7!\,14!\,7!}(0.25)^7(0.50)^{14}(0.25)^7. $$

The log likelihood excluding the multinomial coefficient is $7\log(0.25)+14\log(0.50)+7\log(0.25)=14\log(0.25)+14\log(0.50)=14\log(0.125)$, approximately $-29.11$. This calculation uses the full three-category response likelihood; the two adjacent odds are linked by the common set of category probabilities.

What the Formulation Assumes—and What to Check

The model is useful when neighboring categories are meaningful comparisons and a concise set of adjacent odds answers the scientific question. In a GLMM, its interpretation and likelihood depend on several assumptions:

The outcome categories are correctly ordered, exhaustive, and coded consistently.
The specified adjacent-logit linear predictors are appropriate. In the common-slope model, each covariate coefficient is shared across all category pairs, while the $\alpha_k$ values vary.
For a GLMM, the random-effect distribution is adequately specified, and observations are conditionally independent given the random effects and covariates.
There is adequate information for the category pairs and covariate patterns being modeled. Very sparse or empty categories can make adjacent odds unstable or effectively infinite.

Useful checks include tabulating observed category counts by important predictors and clusters, looking for empty neighboring categories, and comparing observed patterns with fitted category probabilities. Very large estimates or standard errors, convergence warnings, and probabilities near zero across covariate patterns can signal separation or sparse information. Conditional residual summaries and plots of observed versus fitted category frequencies can reveal systematic lack of fit. Diagnostics should also consider whether the shared-slope restriction is plausible; a common coefficient is a modeling assumption, not a consequence of the categories being ordered.

A positive coefficient raises every modeled adjacent odds in the $k+1$ versus $k$ direction, conditional on the random effect. It does not assign equal numerical distances to the categories. Nor should a conditional effect be reported as though it were automatically the same as a population-averaged effect, because random-effect integration changes the marginal probabilities.

How It Differs From Other Ordinal Formulations

The earlier material on proportional odds formulates the response through cumulative probabilities, such as the probability of being at or below a threshold. An adjacent-category model instead compares each pair of neighboring category probabilities. The two formulations therefore encode different patterns of association: a common slope in adjacent logits is not the same assumption as a common slope in cumulative logits.

Adjacent-category logits are appropriate when the relative probability of neighboring responses is the target contrast. They are not automatically appropriate just because the outcome has an order. If the response process is naturally about crossing thresholds, cumulative probabilities may align more closely with the question. If a predictor’s effects plausibly differ by neighboring pair, the common-slope model may be too restrictive and a more flexible specification may be needed, subject to enough data to estimate it.

A frequent mistake is to compute each neighboring odds ratio separately and treat the resulting binary comparisons as independent likelihood contributions. That reasoning double-uses the same category probabilities and can produce incoherent estimates. The correction is to define one probability vector using all adjacent ratios and normalization, then use the single categorical or multinomial likelihood for the observed response.

Key takeaway: Adjacent-category modeling starts with $K-1$ neighboring probability ratios. Recursion and normalization turn them into one valid $K$-category response distribution, which can then enter a conditional GLMM likelihood.

Check Your Understanding

Try each question, then compare your reasoning with the solutions.

For three categories, the adjacent odds are $p_2/p_1=3$ and $p_3/p_2=2$. Find the normalized category probabilities.
In a common-slope adjacent-category model, what does $\exp(\beta)=1.5$ mean for a one-unit increase in a covariate?
Why must adjacent logits be converted into a normalized probability vector before writing the categorical likelihood?
What does the random effect condition on in a GLMM, and why should a conditional coefficient not automatically be described as a marginal effect?

Solutions. (1) Set weights to $1,3,6$; their sum is 10, so the probabilities are $0.1,0.3,0.6$. (2) Holding other predictors and the random effect fixed, a one-unit increase multiplies the odds of category $k+1$ rather than $k$ by 1.5 for each adjacent pair. It does not multiply each category probability by 1.5. (3) The response has one category per observation, so its likelihood requires a coherent probability for every possible category, with all probabilities positive and summing to one. (4) The random effect represents cluster-level variation; conditional probabilities and coefficients are defined at a fixed random-effect value. Averaging over its distribution changes probabilities, so the marginal effect need not equal the conditional coefficient.