Conditional and Marginal Effects in Logistic Mixed Models

Why Conditional and Population-Averaged Effects Differ

In a binary PK/PD analysis, the probability of response may vary with drug concentration and with stable differences among subjects. A subject-specific random effect represents some of that heterogeneity. The model then gives a response probability conditional on a particular random effect, while a population prediction averages over the random-effects distribution. These predictions answer different questions: one describes a subject at a specified latent propensity, and the other describes the expected response in the modeled population.

The distinction matters because the logistic function is nonlinear. Averaging probabilities over heterogeneous subjects is not generally the same as evaluating the logistic function at the average random effect. Likewise, a conditional odds ratio from a mixed model is not generally the odds ratio calculated from population-averaged probabilities.

Definition: Let $Y_{ij}$ be a binary response for subject $i$ at observation $j$, and let $x_{ij}$ denote a specified exposure or treatment predictor. A random-intercept logistic model is $$ Y_{ij}\mid b_i \sim \operatorname{Bernoulli}(p_{ij}(b_i)), \qquad \operatorname{logit}\{p_{ij}(b_i)\}=\alpha+\beta x_{ij}+b_i, \qquad b_i\sim N(0,\sigma_b^2). $$ Here $\operatorname{logit}(p)=\log\{p/(1-p)\}$, and $b_i$ is a subject-specific random intercept. The conditional probability is $p_{ij}(b_i)$, given $b_i$. The population-averaged probability at exposure $x$ is $$ \bar p(x)=E_b\left[\operatorname{logit}^{-1}(\alpha+\beta x+b)\right], $$ where the expectation is over the specified random-effects distribution.

This model assumes that, conditional on the random effect and predictors, the response follows a Bernoulli distribution with the stated logit mean. It also assumes the random intercepts follow the specified normal distribution, are independent across subjects, and are independent of the predictors. In repeated-response data, conditional independence of observations given the random effect is a further usual assumption. If random effects and exposure are associated, averaging over the unconditional distribution of $b_i$ need not produce the appropriate population prediction at that exposure.

What the Conditional Odds Ratio Means

For a one-unit increase in $x$, holding the subject’s random intercept fixed, the conditional log odds increase by $\beta$. The conditional odds ratio is therefore $\exp(\beta)$. It is the same for every $b_i$ in this random-intercept model because $b_i$ shifts both exposure conditions by the same amount. This does not mean that every subject has the same probability change: probability changes depend on the subject’s baseline risk.

A population-averaged contrast is calculated from $\bar p(x)$. For example, a probability difference between exposures $x_1$ and $x_0$ is $\bar p(x_1)-\bar p(x_0)$. The corresponding marginal odds ratio is $$ \frac{\bar p(x_1)/\{1-\bar p(x_1)\}}{\bar p(x_0)/\{1-\bar p(x_0)\}}. $$ Neither quantity is generally obtained by simply applying $\exp(\beta)$ to the average probability. They must be calculated from the marginal probabilities.

Worked Example: Integrating Predictions Over Subject Heterogeneity

Suppose a hypothetical PK/PD model is $$ \operatorname{logit}\{p(x\mid b)\}=-2+1.2x+b,\qquad b\sim N(0,0.8^2). $$ Compare $x=0$ and $x=1$. The conditional odds ratio for the one-unit increase is $\exp(1.2)=3.320$, for any fixed $b$. To obtain population-averaged probabilities, integrate the inverse-logit over the random-intercept distribution.

For a transparent numerical approximation, use five-point Gauss–Hermite quadrature expressed as standard-normal nodes $z_k$ and weights $w_k$. The nodes are $-2.856970,-1.355626,0,1.355626,2.856970$, with respective weights approximately $0.0112574,0.222076,0.533333,0.222076,0.0112574$. Set $b=0.8z_k$, calculate a conditional probability at each node, and take the weighted sum.

At $x=1$, the linear predictor is $-0.8+0.8z_k$. The node-specific probabilities, rounded, are $0.0437,0.1319,0.3100,0.5706,0.8154$. Thus, $$ \bar p(1)\approx 0.0112574(0.0437)+0.222076(0.1319)+0.533333(0.3100) +0.222076(0.5706)+0.0112574(0.8154) \approx 0.33103. $$ The unrounded quadrature values give approximately $0.331032$.

At $x=0$, the linear predictor is $-2+0.8z_k$. The corresponding probabilities are approximately $0.0136,0.0437,0.1192,0.2859,0.5709$. Applying the same weights gives $\bar p(0)\approx0.143358$. The population-averaged probability difference is therefore $$ \bar p(1)-\bar p(0)\approx0.331032-0.143358=0.187674. $$ In this hypothetical population, the modeled response probability is about 18.8 percentage points higher at $x=1$ than at $x=0$.

The marginal odds ratio is approximately $$ \frac{0.331032/(1-0.331032)}{0.143358/(1-0.143358)} \approx \frac{0.4948}{0.1673}\approx2.96. $$ It is not the conditional odds ratio of $3.320$. Both are valid summaries, but they describe different contrasts.

Why Logistic Effects Are Not Collapsible

Even when the random effect is independent of exposure, averaging across subjects generally changes the odds ratio. This is a feature of the odds scale, not necessarily evidence of confounding or model error. A short derivation shows how heterogeneity changes the marginal slope in the random-intercept model.

Differentiate the marginal probability.
Write $p_b(x)=\operatorname{logit}^{-1}(\alpha+\beta x+b)$. Since $d p_b(x)/dx=\beta p_b(x)\{1-p_b(x)\}$, differentiation under the expectation gives $d\bar p(x)/dx=\beta E_b[p_b(x)\{1-p_b(x)\}]$, provided the expectation and derivative can be interchanged. The integrand is bounded, so this condition holds here.

Convert the probability slope to a marginal logit slope.
The derivative of $\operatorname{logit}\{\bar p(x)\}$ is $\{d\bar p(x)/dx\}/[\bar p(x)\{1-\bar p(x)\}]$. Substituting the first step gives $\beta E[p_b(1-p_b)]/[\bar p(1-\bar p)]$.

Relate the slope to between-subject variation.
Because $E[p_b(1-p_b)]=\bar p(1-\bar p)-\operatorname{Var}_b(p_b)$, the marginal logit slope equals $\beta\left[1-\operatorname{Var}_b(p_b)/\{\bar p(1-\bar p)\}\right]$. If there is heterogeneity in the conditional probabilities, this slope differs from $\beta$; for positive $\beta$, it is smaller than $\beta$.

This derivative result concerns the local marginal logit slope at a given $x$, not a claim that every finite-contrast marginal odds ratio has one constant value. The marginal logit curve is generally not linear in $x$, even though the conditional logit is linear. With no random-effect variation, $\operatorname{Var}_b(p_b)=0$, and the marginal and conditional slopes coincide.

Worked Example: Same Conditional Odds Ratio, Different Subject Risks

Use the hypothetical model $\operatorname{logit}\{p(x\mid b)\}=-2+1.2x+b$. Compare $x=0$ with $x=1$ for subjects with $b=-0.8,0,$ and $0.8$. For each subject, the conditional odds ratio is $\exp(1.2)=3.320$.

For $b=-0.8$, the linear predictors are $-2.8$ and $-1.6$, giving probabilities $\operatorname{logit}^{-1}(-2.8)=0.0573$ and $\operatorname{logit}^{-1}(-1.6)=0.1680$. The probability increase is $0.1107$.

For $b=0$, the predictors are $-2$ and $-0.8$, giving probabilities $0.1192$ and $0.3100$, an increase of $0.1908$. For $b=0.8$, the predictors are $-1.2$ and $0$, giving probabilities $0.2315$ and $0.5000$, an increase of $0.2685$.

Thus the conditional odds ratio is identical across these subjects, while their absolute probability changes differ substantially. A conditional odds ratio alone does not specify a single clinical probability benefit; the baseline risk distribution also matters.

Population Predictions and Alternative Summaries

A population-averaged prediction integrates over the random effects for a new or representative subject drawn from the modeled population. It is distinct from a prediction for an observed subject using that subject’s estimated random effect. The latter is conditional on the subject’s data and can be useful for individual-level prediction; the former is appropriate when the question concerns an average response across the population.

A conventional logistic model without random effects gives a probability at a reference covariate pattern but does not represent persistent subject heterogeneity. A logistic mixed model estimates conditional effects and can account for within-subject dependence; its population prediction is obtained by integration or numerical averaging. A population-average model, such as one fit with generalized estimating equations, targets a population-level relationship directly under its own modeling and dependence assumptions. Its coefficient need not equal the mixed-model conditional coefficient. These approaches answer related, but not identical, questions.

Worked Example: How Heterogeneity Changes the Population Contrast

Return to the hypothetical model in the first example. If $\sigma_b=0$, there is no random-intercept variation, so the population probabilities are simply the inverse-logit at $b=0$: $p(0)=\operatorname{logit}^{-1}(-2)=0.1192$ and $p(1)=\operatorname{logit}^{-1}(-0.8)=0.3100$. The probability difference is $0.3100-0.1192=0.1908$, and the odds ratio is $\exp(1.2)=3.320$.

With $\sigma_b=0.8$, the integrated probabilities are approximately $\bar p(0)=0.143358$ and $\bar p(1)=0.331032$. The difference is $0.187674$, while the marginal odds ratio is approximately $2.96$. Heterogeneity raises these averaged probabilities relative to the zero-random-effect predictions in this example and changes the marginal odds ratio. The direction and size of changes in absolute probability depend on the exposure and model parameters; the key point is to integrate rather than substitute $b=0$ when a population average is intended.

Assumptions, Diagnostics, and Common Pitfalls

The interpretation of an integrated prediction depends on the fitted model and on the target population represented by the random-effects distribution. Check that the exposure scale and contrast are clinically meaningful, that the random-effect distribution is plausible for the application, and that the random-intercept structure matches the repeated-measure design. If treatment or exposure varies within a subject, clarify whether the target is a within-subject contrast, a between-subject contrast, or a specified combination.

Signs of a poor random-effects specification include substantial unexplained clustering, predictions that poorly reflect observed response frequencies across subjects, or estimated heterogeneity that is unstable under reasonable modeling choices.
Random slopes may be needed when subjects plausibly differ in their exposure response. Then the conditional slope, and thus the conditional odds ratio, varies by subject; the simple common conditional odds ratio derived above no longer applies.
Integration must match the target. Averaging over $b\sim N(0,\sigma_b^2)$ is appropriate under the stated model for a population draw. For prediction conditional on an observed subject’s history, use the relevant conditional distribution of that subject’s random effect instead.

A common mistake is to report $\exp(\beta)$ as the population-averaged odds ratio because it is the model’s treatment odds ratio. The correction is to label $\exp(\beta)$ as the conditional, subject-specific odds ratio, calculate $\bar p(x)$ at each exposure by integrating over the random effects, and then form the marginal probability difference or marginal odds ratio from those probabilities. Also avoid treating $\bar p(x)$ as the probability for every subject: it is an average over modeled heterogeneity.

Key takeaway: In a logistic mixed model, the conditional coefficient describes a contrast holding random effects fixed. Population-averaged probabilities require integrating conditional probabilities over the random-effects distribution, and population-average odds contrasts must be calculated from those integrated probabilities.

Check Your Understanding

Answer each question using the stated random-intercept logistic model and distinguish conditional from marginal quantities.

If $\beta=0.7$, what is the conditional odds ratio for a one-unit increase in exposure, holding the random intercept fixed?
Why is $\operatorname{logit}^{-1}(\alpha+\beta x)$ generally not the population-averaged probability when $\sigma_b>0$?
In the first worked example, what is the approximate population-averaged probability difference between $x=1$ and $x=0$?
What additional modeling feature would allow the exposure effect to vary among subjects?

Solutions. (1) The conditional odds ratio is $\exp(0.7)\approx2.014$. (2) The inverse-logit is nonlinear, so $E_b[\operatorname{logit}^{-1}(\alpha+\beta x+b)]$ generally differs from $\operatorname{logit}^{-1}(\alpha+\beta x+E[b])$; the average must be calculated over the random-effects distribution. (3) The difference is $0.331032-0.143358=0.187674$, or about 18.8 percentage points. (4) Include a subject-specific random slope for exposure; then the conditional exposure coefficient varies across subjects.