Conditional Interpretation Across Hierarchical Levels

Why the Level of Interpretation Matters

In a model with several classifications, a coefficient has meaning only after specifying which outcome units are being compared and what is held fixed. A fixed coefficient may describe a contrast conditional on the random effects for a particular school, clinic, or assessor. A random coefficient describes how a group’s conditional relationship differs from the model’s fixed relationship. Neither should be read as an unconditional effect for every observation or as a causal effect without an appropriate study design.

The distinction is especially important in a generalized linear mixed model (GLMM), where the response mean is linked nonlinearly to a linear predictor. A coefficient on the logit scale, for example, is a conditional log-odds contrast; it is not generally a constant difference in response probabilities. The relevant conditioning includes the observed predictors and the values of the random effects.

A Conditional Model and Its Assumptions

Definition: A GLMM specifies the distribution of each response conditional on its predictors and random effects. Write \(Y_r\) for response \(r\), \(x_r\) for its predictor vector, and \(b\) for all relevant random effects. The conditional mean is \(E(Y_r\mid x_r,b)=\mu_r\), with \(g(\mu_r)=x_r^\mathsf{T}\beta+z_r^\mathsf{T}b\). Here \(g\) is the link function, \(\beta\) is the vector of fixed coefficients, and \(z_r\) describes which random coefficients apply to response \(r\). Random effects are modeled as draws from a specified distribution, commonly a mean-zero multivariate normal distribution with covariance matrix \(G\). Conditional on the random effects and predictors, responses are assumed independent, with a specified response distribution and link.

The indices in a model identify the units to which terms apply. A school random intercept is shared by observations in that school; a classroom-within-school effect is shared only by observations in that classroom. A crossed assessor effect is shared by observations assessed by the same assessor even if they come from different schools. Thus, the membership structure determines which random effects enter each response’s linear predictor.

A fixed coefficient is interpreted in the model’s chosen coding and conditional on the other predictors and applicable random effects. A random intercept shifts the linear predictor for its unit. A random slope changes the conditional coefficient of its predictor for that unit. In particular, if a fixed slope is \(\beta_x\) and unit \(j\) has random slope \(b_j\), its conditional slope is \(\beta_x+b_j\). With a nonlinear link, translating that slope to the response scale also depends on the other terms in the linear predictor.

These interpretations assume that the model’s link and response distribution are suitable, that the random-effects structure represents the relevant memberships, and that the stated random-effects distribution is a reasonable description of variation across units. They also require enough design information to distinguish the coefficients being interpreted. A coefficient is not meaningful apart from these model and design conditions.

Fixed Effects Conditional on Several Classifications

Consider a binary response for person \(i\), in school \(k\), evaluated using test form \(l\). Suppose schools and test forms are crossed: not every form must occur in every school, but a form can be used in more than one school. Let \(T_{ikl}\) indicate whether the person received a program, with \(T=0\) as the reference condition. A random-intercept logit model is

Y_{ikl}\mid u_k,v_l \sim \operatorname{Bernoulli}(p_{ikl}),\qquad \operatorname{logit}(p_{ikl})=\beta_0+\beta_T T_{ikl}+u_k+v_l.

For a fixed school and test form, changing \(T\) from 0 to 1 changes the conditional log odds by \(\beta_T\). The conditional odds ratio is \(\exp(\beta_T)\). The school and form intercepts cancel in this contrast because they are the same for both conditions. This is a comparison at the same values of the other predictors and the same random-effect values—not a comparison that averages across schools and forms.

Worked Example: A Conditional Program Contrast in a Crossed Design

Suppose a hypothetical fitted model has \(\beta_0=-1.00\), \(\beta_T=0.70\), a school effect \(u_k=0.40\), and a test-form effect \(v_l=-0.20\). The outcome is whether a person meets a binary performance criterion. Interpret the program coefficient for this school and form, and calculate the two conditional probabilities.

For a person in the reference condition, the linear predictor is \(\eta_0=-1.00+0.40-0.20=-0.80\). The conditional probability is \(p_0=1/(1+\exp(0.80))=0.3100\), to four significant figures.

For a person receiving the program under the same school and form effects, \(\eta_1=-1.00+0.70+0.40-0.20=-0.10\), so \(p_1=1/(1+\exp(0.10))=0.4750\). The conditional odds ratio is \(\exp(\beta_T)=\exp(0.70)=2.014\). Thus, at the same school and form effects, the modeled odds of meeting the criterion are about 2.014 times as large in the program condition as in the reference condition. The corresponding modeled probability difference for these particular random effects is \(0.4750-0.3100=0.1650\), or 16.50 percentage points. That difference is specific to this linear-predictor setting; it is not the coefficient and need not be the same in other schools or forms.

The conditional odds ratio is obtained directly by taking the ratio of the modeled odds:

\frac{p(T=1\mid u_k,v_l)/[1-p(T=1\mid u_k,v_l)]} {p(T=0\mid u_k,v_l)/[1-p(T=0\mid u_k,v_l)]} = \frac{\exp(\beta_0+\beta_T+u_k+v_l)} {\exp(\beta_0+u_k+v_l)} =\exp(\beta_T).

This cancellation justifies the odds-ratio interpretation under the stated model. It does not imply that averaging probabilities across the random-effects distribution produces the same odds ratio. In general, applying the inverse link after averaging the linear predictor is not equivalent to averaging the inverse-link probabilities.

Random Slopes and Unit-Specific Conditional Effects

A random slope makes a predictor’s conditional effect vary across units. For example, let count \(Y_{ijk}\) be observed in classroom \(j\) nested within school \(k\), and let \(x_{ijk}\) be study hours. Suppose the conditional mean count has a log link, with an exposure offset \(e_{ijk}\):

Y_{ijk}\mid u_k,v_{jk},b_{jk}\sim\operatorname{Poisson}(\mu_{ijk}),\qquad \log(\mu_{ijk})=\log(e_{ijk})+\beta_0+\beta_x x_{ijk}+u_k+v_{jk}+b_{jk}x_{ijk}.

Here \(u_k\) is a school intercept, \(v_{jk}\) a classroom-within-school intercept, and \(b_{jk}\) a classroom-specific deviation in the study-hours slope. Holding the random effects fixed, the conditional log rate changes by \(\beta_x+b_{jk}\) per additional study hour. Its corresponding rate ratio per hour is \(\exp(\beta_x+b_{jk})\). The fixed slope \(\beta_x\) is the slope when the random slope deviation is zero; it is not the slope for every classroom.

Worked Example: A Classroom-Specific Rate Ratio

For a hypothetical count model, take \(\beta_0=-0.40\), \(\beta_x=0.20\), \(u_k=0.10\), \(v_{jk}=-0.20\), and \(b_{jk}=-0.05\). For an observation with \(x=3\) study hours and exposure \(e=2\), calculate its conditional mean and interpret the study-hours slope for this classroom.

The classroom-specific slope on the log scale is \(\beta_x+b_{jk}=0.20-0.05=0.15\) per hour. Therefore the rate ratio per additional hour, holding the school and classroom effects fixed, is \(\exp(0.15)=1.162\). In this classroom, the model associates one additional study hour with a 16.2% higher conditional expected count rate, other modeled terms held fixed. This is a model-based conditional association, not by itself evidence of a causal effect.

For the specified observation, the linear predictor including the offset is \(\log(2)-0.40+0.20(3)+0.10-0.20-0.05(3)=0.6431\), approximately. Hence \(\mu=\exp(0.6431)=1.902\). This is the conditional expected count at exposure 2 for the stated predictor and random-effect values. The rate ratio describes a change per study hour; it is not a fixed difference in expected counts across all exposure levels.

Combining Nested and Crossed Random Slopes

When two classifications contribute random slopes for the same predictor, the conditional slope includes both deviations. Consider a hypothetical count response with clinics nested in regions and assessors crossed with clinics. Let \(x\) be a measurement in millimeters. A possible log-link predictor is

\log(\mu_{ijk})=\log(e_{ijk})+\beta_0+\beta_x x_{ijk} +u_k+v_{jk}+w_l+b_{jk}x_{ijk}+c_lx_{ijk},

where \(u_k\) is a region intercept, \(v_{jk}\) a clinic-within-region intercept, \(w_l\) an assessor intercept, \(b_{jk}\) a clinic slope deviation, and \(c_l\) an assessor slope deviation. For an observation in clinic \(j\), region \(k\), and assessor \(l\), the conditional slope is \(\beta_x+b_{jk}+c_l\). This interpretation is conditional on the particular clinic and assessor together; a different assessor can change the slope even for the same clinic.

Worked Example: A Slope for One Clinic–Assessor Combination

Suppose the fixed slope is \(\beta_x=0.12\) per millimeter. A hypothetical clinic has \(b_{jk}=-0.04\), and a crossed assessor has \(c_l=0.03\). Calculate the conditional rate ratio for a one-millimeter increase and compare it with the fixed-slope rate ratio.

For this clinic–assessor combination, the slope is \(0.12-0.04+0.03=0.11\) per millimeter. The conditional rate ratio is \(\exp(0.11)=1.116\), so the modeled expected rate is 11.6% higher for a one-millimeter increase, holding exposure and the other terms fixed. The rate ratio from the fixed slope alone is \(\exp(0.12)=1.127\), or 12.7% higher. That fixed-slope value is not the rate ratio for this particular combination because both random slope deviations contribute to its conditional relationship.

The clinic and assessor intercepts affect the conditional mean level but not this rate ratio: they do not multiply \(x\). If the model included an interaction or additional random slope terms, those terms would also have to be included when forming the relevant conditional contrast.

Conditional and Population-Averaged Interpretations

A GLMM coefficient is usually called conditional or unit-specific because it describes the response distribution conditional on the random effects. A population-averaged quantity instead averages predictions over the random-effects distribution. For a nonlinear link, these are distinct calculations. In a logit model, for instance, the conditional probability is \(\operatorname{logit}^{-1}(x^\mathsf{T}\beta+z^\mathsf{T}b)\); a population-averaged probability involves integrating that inverse-link value over the distribution of \(b\). It is generally incorrect to set \(b=0\) and call the resulting prediction the average over units.

The same distinction applies to slopes. A fixed slope in a random-slope model is the slope at zero random-slope deviations on the link scale. It may summarize the center of the modeled slope distribution, but nonlinear response-scale contrasts can vary across units. A population-averaged response contrast must be defined and calculated on the response scale; it cannot generally be read directly from a conditional link-scale coefficient.

A model with group indicators treated as fixed effects is another possible approach when inference is intended to concern only the observed groups and the design supports estimating their separate coefficients. A random-effects model instead represents group variation through a distribution and can express a conditional relationship for the modeled population of groups. A model with random intercepts only imposes a common link-scale slope; adding random slopes allows that slope to vary. These approaches answer different questions, so the choice and interpretation should be tied to the target units and estimand.

Checks, Limitations, and Common Pitfalls

Before interpreting a coefficient, verify the outcome distribution and link, the reference coding, the units of the predictor, and which memberships apply to the observation. For a random slope, explicitly add its deviation to the fixed slope. For a contrast, state which random effects are held equal. When reporting a response-scale result, specify whether it is conditional for stated random effects or averaged over their distribution.

Interpretations also depend on whether the data contain information about the terms in the model. Sparse or disconnected membership patterns can leave some contrasts weakly supported or not separately identified. Large uncertainty, unstable estimates, convergence warnings, or estimated random-slope variation near a boundary can make a unit-specific interpretation unreliable. These are reasons to examine model fit and design support, not reasons to silently discard the conditioning in the interpretation.

Pitfall: treating a conditional coefficient as an unconditional probability effect. A logit coefficient is a change in conditional log odds, not a probability-point change. Translate it using the inverse link at specified covariate and random-effect values, or calculate a defined marginal contrast.
Pitfall: reporting only the fixed slope when random slopes are present. For a particular unit, use the fixed slope plus all applicable random slope deviations.
Pitfall: confusing crossed membership with nesting. A crossed assessor effect applies wherever that assessor appears; it is not a separate assessor effect within each clinic unless the model explicitly defines it that way.
Pitfall: interpreting a coefficient without its coding or units. State the reference category or unit increase and the conditions held fixed. The coefficient’s numerical value alone is not a complete interpretation.

Key takeaway: A GLMM coefficient is always tied to a scale, a set of units, and a conditioning statement. Identify the random effects that apply to the observation, combine their slope deviations where needed, and distinguish conditional effects from quantities averaged over groups.

Check Your Understanding

Answer each question, then compare with the solutions below.

A logit GLMM has a treatment coefficient of \(0.50\) and random intercepts for hospital and physician. What is the treatment odds ratio conditional on the same hospital and physician effects?
A log-link model has a fixed slope of \(-0.10\) and a group-specific random slope deviation of \(0.06\). What is the group’s conditional slope and rate ratio per one-unit increase?
In a model with a school random intercept and a crossed test-form random intercept, which effects are held fixed in a conditional comparison of two treatment values for the same school and form?
Why is the inverse link evaluated at the fixed linear predictor with random effects set to zero not generally a population-averaged probability?

Solutions. (1) The conditional odds ratio is \(\exp(0.50)=1.649\), comparing treatment values at the same hospital and physician effects. (2) The conditional slope is \(-0.10+0.06=-0.04\); the rate ratio is \(\exp(-0.04)=0.9608\) per one-unit increase, a 3.92% lower modeled rate. (3) Hold the school and test-form random intercepts fixed, as well as the other modeled predictors; both values are compared within the same school and form. (4) A population-averaged probability integrates the inverse-link probability over the random-effects distribution. In general, that integral is not equal to the inverse link evaluated at the mean random effect, because the inverse link is nonlinear.