Ordinal Biomarkers With Cumulative-Link Models

Why Model an Ordinal Biomarker?

A biomarker may be recorded in ordered categories such as absent, mild, moderate, and severe. The ordering contains useful information, but it does not imply that the numerical gaps between adjacent categories are equal. Assigning scores such as 0, 1, 2, and 3 and treating them as continuous measurements imposes a spacing assumption that may not be justified. A cumulative-link model instead models probabilities of being at or below each category boundary.

For repeated biomarker measurements, responses from the same subject are generally related. A cumulative-logit mixed model represents that dependence using subject-level effects while retaining the ordered nature of the response. The model can use time or a flexible time basis in its predictor; here the focus is the ordinal response model, its thresholds, and the interpretation of its effects.

Model Definition and Assumptions

Definition: Let $Y_{ij}$ be the ordinal biomarker for subject $i$ at visit $j$, with categories $1,\ldots,K$, ordered from lowest to highest. Let $x_{ij}$ be a vector of observed predictors and $b_i$ a vector of subject-level random effects. A cumulative-logit model specifies, for $k=1,\ldots,K-1$, $$ \operatorname{logit}\{P(Y_{ij}\leq k\mid b_i,x_{ij})\} =\alpha_k-x_{ij}^{T}\beta-z_{ij}^{T}b_i. $$ Here $\alpha_1<\alpha_2<\cdots<\alpha_{K-1}$ are thresholds, $\beta$ contains fixed-effect coefficients, and $z_{ij}$ selects or constructs the predictors for the random effects. The inverse-logit function is $\operatorname{logit}^{-1}(u)=1/(1+\exp(-u))$.

The minus signs in this parameterization determine the interpretation: a positive predictor effect shifts probability toward higher categories. The thresholds locate the category boundaries on the cumulative logit scale. There is no separate fixed intercept in the predictor as written; including both an unrestricted intercept and all thresholds would create a location-identification problem, because adding a constant to the intercept and thresholds can leave the probabilities unchanged.

The ordered thresholds ensure coherent cumulative probabilities. For a fixed subject and visit, define $\eta_{ij}=x_{ij}^{T}\beta+z_{ij}^{T}b_i$. Then $F_{ij,k}=P(Y_{ij}\leq k\mid b_i,x_{ij})=\operatorname{logit}^{-1}(\alpha_k-\eta_{ij})$. Since the inverse-logit is increasing and $\alpha_k<\alpha_{k+1}$, $F_{ij,k}<F_{ij,k+1}$. Category probabilities follow by subtraction:

\begin{aligned} P(Y_{ij}=1\mid b_i,x_{ij})&=F_{ij,1},\\ P(Y_{ij}=k\mid b_i,x_{ij})&=F_{ij,k}-F_{ij,k-1},\quad k=2,\ldots,K-1,\\ P(Y_{ij}=K\mid b_i,x_{ij})&=1-F_{ij,K-1}. \end{aligned}

Consequently every category probability is nonnegative, and they sum to one by cancellation. The model’s main assumptions are that the categories have a defensible order; the cumulative-logit form is suitable; the thresholds are ordered; and, conditional on observed predictors and the subject effects, repeated responses are independent. A common random-effects specification is $b_i\sim N_q(0,D)$, independently across subjects, where $D$ is a covariance matrix. A random intercept is a simple choice; random slopes can allow subjects to differ in their time trends.

The usual cumulative-logit specification also assumes proportional odds: a predictor has the same coefficient at every cumulative cutpoint. In other words, it shifts the cumulative logits by the same amount for the contrasts $Y\leq1$ versus $Y>1$, $Y\leq2$ versus $Y>2$, and so on. This is a substantive modeling restriction, not a consequence of the outcomes being ordinal.

Worked Probabilities From Thresholds

Worked Example: Converting Thresholds to Category Probabilities

Suppose a hypothetical biomarker has four ordered categories and thresholds $\alpha_1=-1$, $\alpha_2=0.5$, and $\alpha_3=1.5$. For a particular visit, let the fixed and subject-level effects combine to give $\eta_{ij}=0.7$. The thresholds are ordered, so the model’s cumulative probabilities are ordered as required.

Calculate each cumulative probability using $F_{ij,k}=\operatorname{logit}^{-1}(\alpha_k-\eta_{ij})$:

\begin{aligned} F_{ij,1}&=\operatorname{logit}^{-1}(-1-0.7) =\operatorname{logit}^{-1}(-1.7)=0.1545,\\ F_{ij,2}&=\operatorname{logit}^{-1}(0.5-0.7) =\operatorname{logit}^{-1}(-0.2)=0.4502,\\ F_{ij,3}&=\operatorname{logit}^{-1}(1.5-0.7) =\operatorname{logit}^{-1}(0.8)=0.6900. \end{aligned}

The category probabilities are $p_1=F_{ij,1}$, $p_2=F_{ij,2}-F_{ij,1}$, $p_3=F_{ij,3}-F_{ij,2}$, and $p_4=1-F_{ij,3}$. Substitution gives $p_1=0.1545$, $p_2=0.4502-0.1545=0.2957$, $p_3=0.6900-0.4502=0.2398$, and $p_4=1-0.6900=0.3100$. The four probabilities sum to $1.0000$, up to rounding.

These are conditional probabilities for the specified predictor value and subject effect. They do not assert that every subject with the same observed predictors has these probabilities: subjects with different random effects have different conditional distributions.

Interpreting Fixed and Subject-Level Effects

For a one-unit increase in a predictor $x_r$, holding the other predictors and random effects fixed, the cumulative log odds of being at or below any cutpoint change by $-\beta_r$. Equivalently, the odds of being above that cutpoint are multiplied by $\exp(\beta_r)$. This common odds ratio across cutpoints is the proportional-odds interpretation. It is conditional on the subject effect in a mixed model.

A subject-level random intercept shifts all cumulative logits for that subject together. It does not give a separate coefficient for each category. A positive random intercept in the stated parameterization increases $\eta_{ij}$, lowering each $P(Y_{ij}\leq k\mid b_i,x_{ij})$ and shifting mass toward higher categories. With repeated observations, the shared effect induces dependence: two visits from the same subject tend to have more similar ordinal outcomes than visits from different subjects with the same observed predictors.

Worked Example: Effect of a Subject-Level Intercept

Use the four-category thresholds from the previous example, but suppose the fixed predictor contribution is $0.7$ and a subject’s random intercept is $b_i=0.6$, with random-effect design value $z_{ij}=1$. Thus $\eta_{ij}=0.7+0.6=1.3$. The thresholds remain ordered, so the conditional category probabilities are valid.

The cumulative probabilities are $F_{ij,1}=\operatorname{logit}^{-1}(-1-1.3)=0.0911$, $F_{ij,2}=\operatorname{logit}^{-1}(0.5-1.3)=0.3100$, and $F_{ij,3}=\operatorname{logit}^{-1}(1.5-1.3)=0.5498$.

Therefore the probabilities for categories 1 through 4 are $0.0911$, $0.3100-0.0911=0.2189$, $0.5498-0.3100=0.2398$, and $1-0.5498=0.4502$. They sum to $1.0000$. Compared with the calculation at $\eta_{ij}=0.7$, this subject has less probability in the lowest category and more in the highest category, consistent with a positive random intercept shifting the response upward.

The value $b_i=0.6$ is a hypothetical conditional subject effect, not a population-wide increase in the biomarker. In a model, its distribution describes variation among subjects, while the fixed effects describe systematic predictor associations.

Proportional Odds in a Time Example

Worked Example: Interpreting a Time Coefficient

Suppose the predictor is time in months, with coefficient $\beta_{\text{time}}=0.08$ per month, and the subject effect is held at zero for illustration. Let the thresholds again be $-1$, $0.5$, and $1.5$. The thresholds satisfy the ordering condition, and the same time coefficient is used at all three cutpoints as required by proportional odds.

For a five-month increase, the linear predictor changes by $5(0.08)=0.40$. The odds of being above any specified category cutpoint are multiplied by $\exp(0.40)=1.492$, approximately. Thus, conditional on the subject effect and other covariates, each cutpoint has 49.2% higher odds of being above it after five additional months. This does not mean the probability of a higher category rises by 49.2%; odds and probabilities are different quantities.

The model’s predicted probabilities can also be calculated. At time zero, $\eta=0$, giving cumulative probabilities $\operatorname{logit}^{-1}(-1)=0.2689$, $\operatorname{logit}^{-1}(0.5)=0.6225$, and $\operatorname{logit}^{-1}(1.5)=0.8176$. The category probabilities are $0.2689$, $0.6225-0.2689=0.3536$, $0.8176-0.6225=0.1951$, and $1-0.8176=0.1824$.

At five months, $\eta=0.40$, so the cumulative probabilities are $\operatorname{logit}^{-1}(-1.4)=0.1978$, $\operatorname{logit}^{-1}(0.1)=0.5250$, and $\operatorname{logit}^{-1}(1.1)=0.7503$. The category probabilities are $0.1978$, $0.5250-0.1978=0.3272$, $0.7503-0.5250=0.2253$, and $1-0.7503=0.2497$. The probability in the highest category rises in this illustration, but the direction of change in every middle-category probability need not be the same.

Specification Choices, Diagnostics, and Alternatives

A time predictor may be linear, or it may use a basis such as a spline when the expected trajectory is nonlinear. The cumulative-logit structure still uses one set of ordered thresholds and, under proportional odds, a common coefficient for each time-basis component across cutpoints. In a longitudinal model, a random intercept can capture persistent subject differences; random slopes or other structured subject effects may be considered when scientifically plausible and supported by the data. Complexity should be limited when the number of subjects or repeated visits is small.

Several checks help identify a poor specification:

Check category frequencies. Very rare or empty categories can make threshold estimates unstable or prevent estimation. Combining categories is only defensible when the resulting categories remain scientifically meaningful.
Assess proportional odds. Compare observed patterns across cutpoints and use suitable model diagnostics or tests. Strongly different predictor associations at different cutpoints suggest that a single common coefficient is inadequate.
Check predicted probabilities. For relevant predictor combinations, confirm that probabilities lie between zero and one, are nonnegative, and sum to one. Ordered thresholds guarantee ordered cumulative probabilities under the model; numerical estimates that violate threshold ordering indicate a fitting or parameterization problem.
Assess subject-level variation and fit. Compare observed within-subject patterns with model-based predictions. A random-intercept-only model may be insufficient if subjects have distinctly different time trends; a complicated random-effects structure can instead be weakly identified or unstable.
Inspect sensitivity to sparse data and influential subjects. Separation-like patterns, near-empty cells, or a few subjects with unusual category sequences can produce large standard errors or extreme estimates.

The cumulative-logit model is not the only ordinal approach. A cumulative probit model uses the normal distribution function in place of the logistic function; the two links often give similar fitted patterns but use different coefficient scales. Adjacent-category models compare neighboring categories, while continuation-ratio models represent sequential transitions through categories. These alternatives answer different conditional questions and do not share the same proportional-odds interpretation. Treating the response scores as continuous is simpler but adds an equal-spacing assumption and may poorly represent category probabilities. A binary model is appropriate only when the scientific question genuinely reduces the categories to two groups.

The main limitations are the proportional-odds restriction, the assumed random-effects distribution, and reliance on the chosen time and predictor forms. Random effects induce a particular structure of within-subject dependence; they do not automatically accommodate every possible pattern. In a joint longitudinal and event analysis, subject-level effects can be part of a larger shared-effects construction, but the ordinal response model alone does not determine how the event process should be specified. That connection requires its own assumptions and interpretation.

A Short Justification of the Probability Construction

The category-probability formula is not an additional modeling assumption; it follows from the cumulative probabilities. For $2\leq k\leq K-1$, the event $Y_{ij}=k$ is exactly the difference between the nested events $Y_{ij}\leq k$ and $Y_{ij}\leq k-1$. Hence

P(Y_{ij}=k\mid b_i,x_{ij}) =P(Y_{ij}\leq k\mid b_i,x_{ij}) -P(Y_{ij}\leq k-1\mid b_i,x_{ij}) =F_{ij,k}-F_{ij,k-1}.

Because $\alpha_k>\alpha_{k-1}$ and the inverse-logit is increasing, $\alpha_k-\eta_{ij}>\alpha_{k-1}-\eta_{ij}$, which implies $F_{ij,k}>F_{ij,k-1}$. Thus the difference is nonnegative. Summing all category probabilities gives $F_{ij,1}+\sum_{k=2}^{K-1}(F_{ij,k}-F_{ij,k-1})+(1-F_{ij,K-1})=1$. This verifies both nonnegativity and total probability for each fixed set of predictors and subject effects.

Common Pitfall

Common mistake: “A positive coefficient means the probability of every higher category increases by the same amount.” This is incorrect. A positive coefficient multiplies the conditional odds of being above each cutpoint by the same factor under proportional odds. It does not multiply probabilities by that factor, and the probability of a middle category can decrease even while the distribution shifts upward. Calculate category probabilities when a probability-scale statement is needed.

A related error is to treat the thresholds as freely unordered cutpoints. Without $\alpha_1<\cdots<\alpha_{K-1}$, cumulative probabilities could decrease as $k$ increases, producing negative category probabilities. The threshold order is a structural requirement of the model.

Check Your Understanding

Answer each question, then compare with the solutions below.

For a four-category outcome, the cumulative probabilities at a particular visit are $0.20$, $0.55$, and $0.80$. Find the four category probabilities.
In the stated parameterization, what does a positive coefficient $\beta_r$ imply about the odds of being above a given cutpoint for a one-unit increase in $x_r$?
Why are ordered thresholds necessary in a cumulative-logit model?
What does a subject-level random intercept contribute to a repeated ordinal biomarker model?
Name one sign that the proportional-odds assumption may be inadequate.

Solutions. (1) The probabilities are $0.20$, $0.55-0.20=0.35$, $0.80-0.55=0.25$, and $1-0.80=0.20$; they sum to 1. (2) The odds of being above the cutpoint are multiplied by $\exp(\beta_r)$, conditional on the other predictors and subject effect. (3) Ordered thresholds make cumulative probabilities increase with the category boundary, so the differences defining category probabilities are nonnegative. (4) It shifts that subject’s cumulative logits across visits and induces dependence among the subject’s repeated outcomes. (5) For example, predictor associations that differ substantially across cutpoints, based on diagnostics or comparisons of cutpoint-specific patterns, conflict with a common proportional-odds effect.