Two Scales, Two Interpretations
A fixed-effect coefficient in a generalized linear mixed model (GLMM) describes a change in the linear predictor, holding the random effects and other predictors fixed. That is not usually the same as the change in the expected outcome after averaging over the random-effects distribution. The first change is on the link, or latent-predictor, scale; the second is on the response scale.
This distinction matters because the inverse link is generally nonlinear: applying it to a predictor and then averaging over random effects does not usually give the same result as applying it to an averaged predictor. The earlier material on integrated means establishes this distinction for marginal predictions. Here the focus is how to describe and calculate changes on the two scales, using binary and count outcomes.
For example, if a predictor \(x\) enters the linear predictor as \(\beta x\), changing \(x\) by \(\Delta x\) changes the predictor by \(\beta\Delta x\), provided the other terms are held fixed. This is a conditional statement: it applies at a fixed value of \(b_i\). The corresponding response-scale marginal change depends on the inverse link, the other predictor values, and the random-effects distribution. It may vary with the starting value of \(x\), even when \(\beta\) is constant.
Binary Outcomes: A Probit Random-Intercept Model
Suppose \(Y_{ij}\) is binary, \(Y_{ij}\mid b_i,x_{ij}\) is Bernoulli, and \(P(Y_{ij}=1\mid b_i,x_{ij})=\Phi(\alpha+\beta x_{ij}+b_i)\), where \(b_i\sim N(0,\tau^2)\). Here \(\Phi\) is the standard normal cumulative distribution function. Assume the random intercept is independent of the covariate and that the same random-effects distribution is used at each value of \(x\). The coefficient \(\beta\) is the change in the conditional probit linear predictor for a one-unit increase in \(x\). It is not, by itself, a change in marginal probability.
The integration behind this result is short but important. Introduce an independent standard normal variable \(\varepsilon\). A Bernoulli success with conditional probability \(\Phi(\eta+b_i)\) can be represented as the event \(\varepsilon\leq \eta+b_i\), where \(\eta=\alpha+\beta x_{ij}\). Rearranging gives \(Y_{ij}=1\) when \(\varepsilon-b_i\leq\eta\). Since \(\varepsilon-b_i\) is normal with mean zero and variance \(1+\tau^2\), standardizing this event gives the stated marginal probability. This calculation uses the normal random-intercept assumption; it is not a general formula for every GLMM.
Worked Example: A Probit Coefficient and a Marginal Probability Change
Suppose a hypothetical binary-outcome model has \(\alpha=0\), \(\beta=1\), and \(\tau^2=3\). Compare \(x=0\) with \(x=1\). The conditional response probability is \(\Phi(x+b_i)\), and the random intercept is normally distributed with mean zero and variance 3.
Latent scale. For a one-unit increase, the change in the probit linear predictor, at any fixed \(b_i\), is \(\beta(1-0)=1\) probit unit.
Marginal probabilities. The marginal model has denominator \(\sqrt{1+\tau^2}=\sqrt{4}=2\). Therefore \(P(Y=1\mid x=0)=\Phi(0/2)=\Phi(0)=0.5000\), and \(P(Y=1\mid x=1)=\Phi(1/2)=\Phi(0.5)\approx0.6915\).
Response scale. The marginal probability change is \(0.691462-0.500000=0.191462\), or about 0.1915. In this hypothetical population, increasing \(x\) by one unit corresponds to an estimated increase of about 19.15 percentage points in the probability of a positive outcome, under this model. It does not correspond to a one-unit increase in probability; probability is bounded between zero and one.
Count Outcomes: A Poisson Log-Link Model
For a count outcome, suppose \(Y_{ij}\mid b_i,x_{ij}\) follows a Poisson distribution with conditional mean \(\lambda_{ij}=\exp(\alpha+\beta x_{ij}+b_i)\), and \(b_i\sim N(0,\tau^2)\). Assume conditional independence of observations given the random effects and predictors, and assume the random intercept is independent of \(x\). These are assumptions of this example, not consequences of the log link. The coefficient \(\beta\) is a change in the conditional log mean per unit increase in \(x\).
To integrate the conditional mean, use the moment-generating function of a normal variable: if \(b_i\sim N(0,\tau^2)\), then \(E\{\exp(b_i)\}=\exp(\tau^2/2)\). Consequently, \(E(Y_{ij}\mid x_{ij})=\exp(\alpha+\beta x_{ij}+\tau^2/2)\). The marginal mean includes the multiplicative factor from random-effect variation. A finite one-unit change in \(x\) changes the log of the conditional mean by \(\beta\), but the absolute change in the marginal expected count depends on the starting mean.
Worked Example: Integrated Counts at Two Predictor Values
Consider a hypothetical Poisson random-intercept model with fixed part \(\alpha=0.18\), \(\beta=0.20\), and \(\tau^2=2\ln(2)\). Assume the Poisson conditional distribution, normally distributed random intercept independent of \(x\), and no offset. Compare \(x=0\) and \(x=1\), where \(x\) is measured in one-unit increments.
Integrate over the random intercept. Since \(\exp(\tau^2/2)=\exp(\ln 2)=2\), the marginal mean at \(x\) is \(m(x)=2\exp(0.18+0.20x)\). The latent-scale change from \(x=0\) to \(x=1\) is \(0.20(1-0)=0.20\) log-mean units.
Calculate the response-scale means and change. \(m(0)=2\exp(0.18)\approx2.3944347\) and \(m(1)=2\exp(0.38)\approx2.9245692\). Thus \(2\exp(0.38)-2\exp(0.18)\approx2.9245692-2.3944347=0.5301345\), or about 0.530 expected events per observation period. This is an absolute difference in marginal expected counts, not a change of 0.20 events.
At the next one-unit step, from \(x=1\) to \(x=2\), the latent-scale change is still 0.20. But the response-scale change is \(2\exp(0.58)-2\exp(0.38)\approx3.5720769-2.9245692=0.6475077\), or about 0.648 expected events per observation period. The absolute change is larger because the starting marginal mean is larger. The calculation illustrates why a constant log-mean coefficient need not imply a constant absolute change in expected counts.
Why the Response-Scale Change Depends on the Starting Point
The preceding examples show two common patterns. In the probit example, the marginal probability change depends on where the predictor starts because the normal cumulative distribution function has different slopes at different parts of its curve. In the Poisson log-link example, the marginal mean is exponential in the predictor, so equal changes in the log mean correspond to larger absolute changes at higher starting means.
A third calculation makes the first pattern explicit. Retain the probit model from the binary example: \(\alpha=0\), \(\beta=1\), and \(\tau^2=3\). Compare the same one-unit increment at two starting points. From \(x=-1\) to \(x=0\), the marginal probabilities are \(\Phi(-0.5)\approx0.308538\) and \(\Phi(0)=0.500000\), giving a change of \(0.191462\). From \(x=1\) to \(x=2\), they are \(\Phi(0.5)\approx0.691462\) and \(\Phi(1)\approx0.841345\), giving a change of \(0.149882\). Both predictor increments have latent-scale change \(1\), but their marginal probability changes differ.
Worked Example: Same Latent Change, Different Probability Changes
Using the probit random-intercept assumptions above, calculate and compare the marginal changes for \(x=-1\) to \(0\), and for \(x=1\) to \(2\). The marginal probability formula is \(P(Y=1\mid x)=\Phi(x/2)\).
For \(x=-1\) to \(0\), substitute the endpoints: \(\Phi(0/2)-\Phi(-1/2)=0.500000-0.308538=0.191462\). For \(x=1\) to \(2\), the calculation is \(\Phi(2/2)-\Phi(1/2)=0.841345-0.691462=0.149883\) (using unrounded values, approximately \(0.149882\)). Both are one-unit increases, so each has latent-scale change \(1\). The different probability changes arise from the different starting locations on the marginal probability curve.
The interpretation is conditional on this model and its random-effects distribution. These calculations are not evidence that an intervention causes either change; they quantify model-implied differences under the stated assumptions.
Assumptions, Diagnostics, and Alternatives
A scale comparison is only as credible as the model used to obtain it. Before interpreting an integrated effect, check that the outcome distribution, link, fixed-effects structure, and random-effects structure are defensible. The examples assumed normally distributed random intercepts independent of the predictor. A random-slope model, dependence between predictors and cluster effects, or a different random-effects distribution can change the integrated response mean and its changes. If exposure time varies for count outcomes, an appropriate offset may also be needed; omitting it changes what the mean represents.
In practice, inspect the fit on both scales. Check conditional residual or simulation-based diagnostics for outcome-distribution problems, examine whether cluster-specific patterns suggest unmodeled slope variation, and assess sensitivity to plausible random-effects specifications. For marginal predictions, confirm that the numerical integration is accurate, especially with several random effects. If simulation is used to integrate, use enough draws and check that repeated calculations are stable. A well-behaved conditional fit does not automatically guarantee that marginal predictions are insensitive to distributional assumptions.
The main alternatives answer different questions. A conditional GLMM coefficient describes the association on the link scale at fixed random effects. Integrating the fitted GLMM gives a population-averaged mean over its specified random-effects distribution. A model fit directly to a marginal mean can instead target population-average relationships, but its coefficients need not match conditional GLMM coefficients. These approaches should not be treated as interchangeable; choose the estimand that matches the scientific question.
| Quantity | What is held or averaged | Interpretation |
|---|---|---|
| Fixed coefficient \(\beta\) | Random effects held fixed | Change in the conditional linear predictor per unit of the predictor |
| Integrated mean at \(x\) | Random effects averaged over their distribution | Population-averaged expected outcome at that predictor value |
| Response-scale marginal change | Integrated means compared at two predictor values | Difference in expected outcomes, in response units |
| Directly fitted marginal-model coefficient | Model targets a marginal relationship | Population-average parameter under that marginal model |
A common mistake is to report a fixed coefficient as though it were an absolute response-scale change. For instance, the count-model coefficient \(0.20\) is a 0.20-unit change in the conditional log mean per unit of \(x\), not an increase of 0.20 expected events. The correction is to calculate the marginal means at the specified predictor values, integrating over the random-effects distribution, and then compare those means in the outcome’s units. The starting value and the model assumptions must be stated because they affect that comparison.
Check Your Understanding
Answer each question, then compare with the brief solutions below.
- In a probit random-intercept model, what does a fixed slope \(\beta\) describe when \(x\) increases by two units?
- For a Poisson log-link random-intercept model with \(b_i\sim N(0,\tau^2)\), what factor does integrating over \(b_i\) add to the conditional mean?
- In the worked Poisson example, why is the marginal count change from \(x=1\) to \(x=2\) larger than the change from \(x=0\) to \(x=1\), even though the predictor increment is the same?
- For the probit model with marginal probability \(\Phi(x/2)\), calculate the change from \(x=0\) to \(x=1\). How does it compare with the change from \(x=1\) to \(x=2\)?
- Name one assumption that should be checked before interpreting an integrated marginal change from a GLMM.
Solutions. (1) The latent-scale change is \(2\beta\), with random effects held fixed; it is not generally a probability change of \(2\beta\). (2) The factor is \(\exp(\tau^2/2)\), from the normal moment-generating function. (3) The integrated mean is exponential in \(x\), so an equal log-mean increment yields a larger absolute count change at a higher starting mean. (4) The change is \(\Phi(0.5)-\Phi(0)=0.691462-0.500000=0.191462\), larger than the \(x=1\) to \(x=2\) change of approximately \(0.149882\). (5) Examples include normality of the random effects, their independence from the predictor, or adequacy of the conditional outcome distribution and link.