From Conditional Probabilities to Marginal Cumulative Odds
An ordinal GLMM describes ordered outcomes such as mild, moderate, and severe symptoms. Its cumulative probabilities are typically specified conditional on a cluster’s random effect. But a population-level question often asks for the probability of being in category \(k\) or below after averaging over the random effects. The corresponding cumulative odds must be calculated from that integrated probability—not by averaging the conditional odds.
This distinction matters because odds are nonlinear functions of probabilities. Also, ordinal outcomes have several cumulative cutpoints, and their probabilities must remain ordered so that the implied category probabilities are nonnegative. The calculation therefore has two linked parts: integrate the cumulative probabilities, then derive cumulative odds from those integrated probabilities.
For a cumulative-logit random-intercept model, one common convention is \(\operatorname{logit}\{F_k(x,b)\}=\alpha_k+x\beta+b\), where \(\operatorname{logit}(p)=\log\{p/(1-p)\}\). The cutpoints satisfy \(\alpha_1<\cdots<\alpha_{K-1}\); this ensures that, at any fixed \(x,b\), \(F_1(x,b)<\cdots<F_{K-1}(x,b)\). The random effect has a specified distribution, commonly a mean-zero normal distribution, and the response is conditionally distributed according to the ordinal model given \(x,b\). Integration must use the same random-effects distribution that defines the target population.
The marginal odds at a cutpoint are a population-averaged summary. They are distinct from the conditional odds at a specified value of \(b\), and distinct from the average of conditional odds. The conditional model may have a common odds ratio across cutpoints, but the corresponding marginal odds ratios need not share that property.
Why Integration Preserves Valid Category Probabilities
The category probabilities are obtained from cumulative probabilities by differencing adjacent cumulative probabilities. Define \(C_0(x)=0\) and \(C_K(x)=1\). Then the marginal probability of category \(k\) is \(P(Y=k\mid x)=C_k(x)-C_{k-1}(x)\). This construction is valid if the cumulative probabilities remain ordered from zero to one.
The justification is direct. For adjacent cutpoints, \(F_{k+1}(x,b)-F_k(x,b)\geq0\) for every \(b\). Taking expectations preserves the inequality:
Also \(0\leq F_1(x,b)\) and \(F_{K-1}(x,b)\leq1\), so their expectations are bounded by zero and one. Finally, the category probabilities telescope: \(\sum_{k=1}^{K}\{C_k(x)-C_{k-1}(x)\}=C_K(x)-C_0(x)=1\). Thus integrating valid cumulative probabilities preserves validity; integrating category probabilities directly is possible too, but they must all be evaluated under the same random-effects distribution.
Worked Example: Integrating Cumulative Probabilities at Two Cutpoints
Consider a hypothetical three-category ordinal outcome with categories 1, 2, and 3. At a particular covariate value, suppose the random intercept takes values \(b=0.5893,\ 0.0553,\ -0.8107\), with respective probabilities \(1/6,\ 2/3,\ 1/6\). These masses give a mean-zero random effect up to rounding. Let the conditional cumulative-logit model be \(\operatorname{logit}\{F_k(x,b)\}=\alpha_k-0.5x+b\), with \(\alpha_1=-0.2553\) and \(\alpha_2=0.9447\), so the cutpoints are ordered. Evaluate at \(x=1\).
For cutpoint 1, the linear predictors at the three random-effect values are approximately \(-0.166,\ -0.700,\ -1.566\). Applying the inverse-logit function gives conditional cumulative probabilities \(0.4586,\ 0.3318,\ 0.1727\), respectively. Therefore,
At cutpoint 2, each linear predictor is 1.2 greater because \(\alpha_2-\alpha_1=1.2\). The three conditional cumulative probabilities are therefore approximately \(0.7377,\ 0.6225,\ 0.4095\). Integrating gives
The marginal category probabilities are \(P(Y=1\mid x=1)=C_1(1)=0.3264\), \(P(Y=2\mid x=1)=C_2(1)-C_1(1)=0.2798\), and \(P(Y=3\mid x=1)=1-C_2(1)=0.3938\). They are nonnegative and sum to 1.0000. The marginal cumulative odds at the two cutpoints are
These are odds of being at or below the respective cutpoint in the population represented by the random-effects distribution. They are not conditional odds for any one cluster.
Why Marginal Odds Are Not Average Conditional Odds
The cumulative odds transformation \(p\mapsto p/(1-p)\) is nonlinear. In general, transforming an average probability does not give the average of the transformed probabilities: \(\{E_b(F_k)\}/\{1-E_b(F_k)\}\ne E_b\{F_k/(1-F_k)\}\). The marginal cumulative odds are defined by the left-hand expression. The right-hand expression is an average of conditional odds and answers a different question.
Worked Example: A Marginal Odds Calculation Is Not an Odds Average
Suppose a hypothetical ordinal random-intercept model has \(b=-\log(4)\) or \(b=\log(4)\), each with probability \(1/2\). At the first cutpoint, let the conditional logit be \(b\). The conditional cumulative probabilities are \(0.2\) and \(0.8\), so the integrated cumulative probability is \(C_1=(0.2+0.8)/2=0.5\). The marginal cumulative odds are therefore \(0.5/(1-0.5)=1\).
The two conditional odds are \(0.2/0.8=0.25\) and \(0.8/0.2=4\). Their average is \((0.25+4)/2=2.125\), not 1. Thus averaging conditional odds would substantially overstate the marginal cumulative odds in this example.
Ordering is also preserved here. Set the second cutpoint’s fixed component to \(\log(4)\), greater than the first cutpoint’s value of zero. Then the conditional probabilities at cutpoint 2 are \(0.5\) and approximately \(0.9412\), both above their corresponding cutpoint-1 probabilities. Hence \(C_2=(0.5+0.9412)/2=0.7206\), and the category probabilities are \(0.5000,\ 0.2206,\ 0.2794\). The second marginal cumulative odds are \(0.7206/0.2794\approx2.579\).
Contrasting Conditional and Marginal Cumulative Odds Ratios
For the cumulative-logit random-intercept model, increasing \(x\) by one unit changes each conditional cumulative log-odds by \(\beta\), holding \(b\) fixed. The conditional cumulative odds ratio is therefore \(\exp(\beta)\) at every cutpoint. After integration, however, the change in cumulative odds depends on the integrated probabilities at both covariate values. In general, it is not equal to \(\exp(\beta)\).
Worked Example: Comparing Odds Ratios Across Scales
Continue the hypothetical model in the first example, with \(\beta=-0.5\) and the same random-effect distribution. At \(x=0\), the first-cutpoint linear predictors are approximately \(0.334,\ -0.200,\ -1.066\), giving conditional cumulative probabilities \(0.5827,\ 0.4502,\ 0.2562\). Their weighted average is
The marginal cumulative odds at \(x=0\) are \(0.4399/0.5601\approx0.7855\); at \(x=1\), the first example gave \(0.4846\). Thus the marginal cumulative odds ratio comparing \(x=1\) with \(x=0\) is approximately
By contrast, the conditional odds ratio is \(\exp(-0.5)=0.6065\). The two values are close here but are not identical. This is not a calculation error: averaging over random effects changes the odds scale. Interpret the marginal ratio as a comparison of population-averaged cumulative odds, and the conditional ratio as a comparison at a fixed random effect.
Numerical Integration, Diagnostics, and Limitations
If \(G\) is continuous, the integral \(C_k(x)=\int F_k(x,b)\,dG(b)\) will usually require numerical integration or simulation. For a normal random effect, Gaussian quadrature approximates the integral by a weighted sum of function evaluations. Monte Carlo integration instead draws random effects from the assumed distribution and averages the resulting conditional cumulative probabilities. The earlier material on marginal category probabilities in ordinal GLMMs covers the same integration target at the category-probability level; cumulative odds are then obtained from the integrated cumulative probabilities.
Check the calculation and model at several points:
- Cutpoint ordering: Confirm that the fitted conditional cumulative probabilities increase with \(k\) for every evaluated \(b\). A negative category probability after differencing signals a violation, a numerical problem, or an incorrectly specified computation.
- Integration accuracy: Increase the number of quadrature points or Monte Carlo draws and check whether \(C_k(x)\) and the resulting odds are stable. Numerical integration error can be consequential when \(C_k(x)\) is close to 0 or 1, because the odds then change rapidly.
- Common integration distribution: Use the same specified random-effects distribution for every cutpoint at a given covariate value. Integrating cutpoints under inconsistent distributions can destroy their ordering.
- Model adequacy: Ordered probabilities do not establish that the ordinal model fits well. Check whether the conditional link, cutpoint structure, random-effects distribution, and covariate specification are plausible for the application.
A frequent pitfall is to integrate conditional odds directly and call the result a marginal cumulative odds. That calculation estimates an average of cluster-specific odds, not odds formed from a population-averaged probability. Another is to integrate each cumulative probability correctly but fail to difference adjacent values when reporting category probabilities. Report both the cumulative estimand and, when useful, the implied category probabilities so readers can see the complete ordinal distribution.
The results are conditional on the specified model and random-effects distribution. A different distribution can produce different marginal probabilities even if some conditional model coefficients are unchanged. For inference, uncertainty in estimated model parameters and numerical integration should both be considered; for example, a suitable bootstrap can repeat model fitting and integration in each resample. A precise integral does not compensate for a poorly fitting model.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- If \(C_k=0.40\), calculate the marginal cumulative odds at cutpoint \(k\).
- For two ordered categories beyond the first cutpoint, suppose \(C_1=0.32\) and \(C_2=0.71\). Find the three marginal category probabilities.
- Conditional cumulative probabilities at one cutpoint are 0.1 and 0.7 in two equally weighted groups. Compare the odds from the integrated probability with the average conditional odds.
- Why does integrating ordered conditional cumulative probabilities preserve the order of the marginal cumulative probabilities?
- In a cumulative-logit random-intercept model, must the marginal cumulative odds ratio equal \(\exp(\beta)\)? Explain.
Solutions. (1) The odds are \(0.40/(1-0.40)=0.40/0.60=0.6667\). (2) The category probabilities are \(0.32,\ 0.71-0.32=0.39,\ 1-0.71=0.29\); they are nonnegative and sum to 1. (3) The integrated probability is \((0.1+0.7)/2=0.4\), giving odds \(0.4/0.6=0.6667\). The conditional odds are \(0.1/0.9=0.1111\) and \(0.7/0.3=2.333\); their average is approximately 1.222, not 0.6667. (4) For each \(b\), the difference between adjacent conditional cumulative probabilities is nonnegative. Its expectation is therefore nonnegative, so adjacent marginal cumulative probabilities remain ordered. (5) No. \(\exp(\beta)\) is the conditional odds ratio at fixed \(b\). The marginal odds ratio is calculated from integrated probabilities and generally differs because odds are nonlinear in probability.