Marginal Proportional-Odds Failure Under Ordinal Random Intercepts

Why Conditional Proportional Odds May Not Be Marginal Proportional Odds

A proportional-odds assumption describes effects conditional on the random effects in a model. It does not guarantee that the same proportionality holds after those random effects are averaged out. For ordinal outcomes, the averaging acts on each cumulative probability; taking a logit after averaging is generally not the same as averaging the conditional logits. Consequently, a predictor with one common conditional effect can have different marginal effects at different outcome thresholds.

This distinction matters when the scientific target is a population-averaged comparison. The conditional coefficient describes comparisons at the same random-intercept value, whereas marginal cumulative logits compare probabilities averaged over the random-intercept distribution. The latter may not be summarized by one common odds ratio.

Model and Assumptions

Let $Y_{ij}$ be an ordinal outcome for observation $j$ in cluster $i$, with ordered categories $1,\ldots,K$. Let $x_{ij}$ be a scalar predictor and $b_i$ a cluster-specific random intercept. Use the cumulative probability $P(Y_{ij}\leq k)$, for $k=1,\ldots,K-1$, and write $L(u)=1/(1+\exp(-u))$.

Definition: A conditional proportional-odds random-intercept model has $$ \operatorname{logit}\{P(Y_{ij}\leq k\mid x_{ij},b_i)\} =\alpha_k-\beta x_{ij}-b_i,\qquad k=1,\ldots,K-1. $$ The cutpoints satisfy $\alpha_1<\cdots<\alpha_{K-1}$. The random intercepts are assumed $b_i\sim N(0,\sigma_b^2)$, independent across clusters and independent of the covariates. Conditional on the random effects and covariates, outcomes are independent. The common coefficient $\beta$ is the conditional proportional-odds effect.

With this sign convention, increasing $x$ by one unit multiplies the conditional odds of being at or below any cutpoint, versus being above it, by $\exp(-\beta)$, holding $b_i$ fixed. The model is coherent because the ordered cutpoints give ordered conditional cumulative probabilities. The marginal cumulative probability is obtained by integrating the conditional probability over the random-intercept distribution:

q_k(x)=P(Y_{ij}\leq k\mid x_{ij}=x) =E_{b}\left[L(\alpha_k-\beta x-b)\right].

In general, $\operatorname{logit}\{q_k(x)\}$ is not equal to $\alpha_k-\beta x$ plus a constant that is the same for every $k$. The expectation is taken on the probability scale; the logit is applied afterward. This order of operations is central to the result.

Why the Marginal Slopes Change

A short derivative argument shows how random-intercept variation modifies the marginal slope. For a fixed cutpoint, set $\eta=\alpha_k-\beta x$, let $B$ have the random-intercept distribution, and write $p_B=L(\eta-B)$. Then $q=E(p_B)$. Differentiating under the expectation gives $dq/d\eta=E\{p_B(1-p_B)\}$, since $L'(u)=L(u)\{1-L(u)\}$.

\frac{d}{d\eta}\operatorname{logit}(q) =\frac{E\{p_B(1-p_B)\}}{q(1-q)} =\frac{E\{p_B(1-p_B)\}} {E\{p_B(1-p_B)\}+\operatorname{Var}(p_B)}.

The second equality follows by expanding $q(1-q)$: since $q=E(p_B)$, $q(1-q)=E(p_B)-\{E(p_B)\}^2=E\{p_B(1-p_B)\}+\operatorname{Var}(p_B)$. The derivative is therefore at most 1, and is strictly less than 1 when $p_B$ varies across random-intercept values. By the chain rule,

\frac{d}{dx}\operatorname{logit}\{q_k(x)\} =-\beta\, \frac{E\{p_B(1-p_B)\}} {E\{p_B(1-p_B)\}+\operatorname{Var}(p_B)}.

The multiplier of $-\beta$ depends on the distribution of $p_B$, which changes with $\eta=\alpha_k-\beta x$. Since each cutpoint has a different $\alpha_k$, its marginal slope will generally differ. The word “generally” is important: special cases, including $\sigma_b^2=0$ or $\beta=0$, do not exhibit threshold-specific slopes. For nonzero random-intercept variation and a nonzero predictor effect, marginal proportional odds are not guaranteed.

Worked Example: Averaging One Cumulative Probability

Consider a hypothetical ordinal outcome with $\alpha_1=-3$, $\beta=1$, $x=0$, and $b_i\sim N(0,1)$. The conditional cumulative probability at the first cutpoint is $L(-3-b_i)$. To illustrate the integration, use the three-point standard-normal quadrature approximation with nodes $0,-\sqrt{3},\sqrt{3}$ and respective weights $2/3,1/6,1/6$. This approximation is used for calculation; it is not an exact identity for a logistic function.

Substitution gives

q_1(0)\approx \frac{2}{3}L(-3) +\frac{1}{6}L(-3-\sqrt{3}) +\frac{1}{6}L(-3+\sqrt{3}).

The three logistic values are approximately $0.0474259$, $0.0087311$, and $0.219609$. Thus

q_1(0)\approx \frac{2}{3}(0.0474259) +\frac{1}{6}(0.0087311) +\frac{1}{6}(0.219609) =0.069674.

The corresponding marginal cumulative logit is $\log\{0.069674/(1-0.069674)\}\approx-2.592$. The conditional logit at $b_i=0$ is $-3$; the marginal logit is not obtained by simply retaining that conditional value.

Threshold-Specific Marginal Effects in Numbers

The next example uses two cutpoints and a binary predictor to compare marginal logits directly. The quadrature values are illustrative approximations to the normal integrals, and the same nodes and weights are used at both predictor values and both cutpoints.

Worked Example: Comparing Marginal Slopes Across Two Cutpoints

Suppose $K=3$, $\alpha_1=-3$, $\alpha_2=-1$, $\beta=1$, and $\sigma_b=1$. Compare $x=0$ with $x=1$. At $x=0$, the linear predictors before subtracting $b_i$ are $-3$ and $-1$; at $x=1$, they are $-4$ and $-2$.

At the second cutpoint and $x=0$, the quadrature calculation is

q_2(0)\approx \frac{2}{3}L(-1)+\frac{1}{6}L(-1-\sqrt{3}) +\frac{1}{6}L(-1+\sqrt{3}) \approx 0.3020.

For the first cutpoint at $x=1$, the same calculation gives

q_1(1)\approx \frac{2}{3}L(-4)+\frac{1}{6}L(-4-\sqrt{3}) +\frac{1}{6}L(-4+\sqrt{3}) \approx 0.02816.

At the second cutpoint and $x=1$, it gives

q_2(1)\approx \frac{2}{3}L(-2)+\frac{1}{6}L(-2-\sqrt{3}) +\frac{1}{6}L(-2+\sqrt{3}) \approx 0.1556.

The marginal logits and their changes are therefore approximately

\begin{aligned} \operatorname{logit}\{q_1(0)\}&\approx -2.592, &\quad \operatorname{logit}\{q_1(1)\}&\approx -3.541, &\quad \Delta_1&\approx -0.949,\\ \operatorname{logit}\{q_2(0)\}&\approx -0.838, &\quad \operatorname{logit}\{q_2(1)\}&\approx -1.691, &\quad \Delta_2&\approx -0.853. \end{aligned}

The approximate marginal changes differ across cutpoints. In contrast, the conditional logit changes by exactly $-1$ at both cutpoints, so the conditional odds ratio for $x=1$ versus $x=0$ is $\exp(-1)\approx0.3679$ at each cutpoint. The calculation demonstrates why that common conditional odds ratio should not be reported as a common marginal odds ratio.

From Cumulative Probabilities to Category Probabilities

For an ordinal outcome, cumulative probabilities also determine the probabilities of individual categories. For three categories, these are $P(Y=1)=q_1$, $P(Y=2)=q_2-q_1$, and $P(Y=3)=1-q_2$. Integrating each threshold probability in the same way preserves their ordering and yields valid category probabilities.

Worked Example: Checking the Marginal Category Probabilities

Use the hypothetical model and quadrature results from the preceding example at $x=0$: $q_1(0)\approx0.069674$ and $q_2(0)\approx0.3020$. The implied marginal category probabilities are

\begin{aligned} P(Y=1\mid x=0)&\approx 0.069674,\\ P(Y=2\mid x=0)&\approx 0.3020-0.069674=0.2323,\\ P(Y=3\mid x=0)&\approx 1-0.3020=0.6980. \end{aligned}

They are nonnegative because $0.069674\leq0.3020$, and they sum to approximately $0.069674+0.2323+0.6980=1.000$, subject to rounding. This check verifies that the two marginal cumulative probabilities have been combined consistently. It does not make their marginal cumulative logits proportional.

Interpretation, Diagnostics, and Alternatives

The conditional proportional-odds model and a marginal ordinal model answer different questions. A conditional model compares observations at the same random-intercept value; a marginal model compares probabilities averaged over the random-intercept distribution. The distinction between conditional and population-averaged targets is also important in GLMM-versus-GEE choices. A direct marginal ordinal analysis may be appropriate when population-averaged cumulative effects are the target, but it should not be assumed to have the same coefficient as a conditional GLMM. In contrast to the special probit-normal marginalization result, logistic-normal averaging generally has no simple exact logistic form.

Useful checks focus on the implied probabilities, not only on whether the conditional coefficient is common across thresholds:

For each covariate pattern, check that the fitted cumulative probabilities are ordered and that the implied category probabilities are nonnegative and sum to one.
Compare fitted and observed cumulative proportions across predictor values and thresholds. Systematic threshold-specific discrepancies can signal that the conditional model, random-effects distribution, or other model components are inadequate.
When a marginal effect is the target, calculate or predict marginal cumulative probabilities and inspect threshold-specific contrasts. Do not infer marginal proportional odds solely from a conditional proportional-odds specification.
For numerical integration, check that the chosen approximation is sufficiently accurate for the probabilities and contrasts being reported; small errors in probabilities can affect logits, especially near zero or one.

These checks have limits. Similar marginal slopes at a few observed predictor values do not prove proportional odds over the full covariate range. Conversely, threshold-specific marginal slopes do not by themselves show that the conditional proportional-odds assumption is false: averaging can create the difference even when the conditional model is correctly specified. A different random-effects distribution or a conditional model with threshold-specific effects may change the results, but each is a modeling choice requiring its own justification.

Common mistake: “The conditional model uses one $\beta$, so its marginal cumulative logit must also have one slope.” This overlooks that the model first averages $L(\alpha_k-\beta x-b)$ over $b$ and only then takes a logit. Because the averaging and logit operations do not commute, the marginal slope can depend on the cutpoint.

Check Your Understanding

Answer each question, then compare with the solutions.

In the stated model, what is the conditional odds ratio for a one-unit increase in $x$, at any fixed cutpoint?
Why does averaging the conditional cumulative probabilities not generally preserve a common marginal slope?
For three ordinal categories, express the three category probabilities using $q_1$ and $q_2$.
If $\sigma_b^2=0$, what happens to the marginal cumulative logit?

Solutions. (1) The conditional odds ratio for being at or below a fixed cutpoint, versus above it, is $\exp(-\beta)$, holding the random intercept fixed. (2) Averaging is applied on the probability scale, and the logit is then a nonlinear transformation; the derivative of the marginal logit includes a cutpoint-dependent factor. (3) The probabilities are $q_1$, $q_2-q_1$, and $1-q_2$. (4) There is no random-intercept variation to average over, so $q_k(x)=L(\alpha_k-\beta x)$ and the marginal cumulative logit equals $\alpha_k-\beta x$; proportional odds holds both conditionally and marginally.