Building Models With Mixed Nested and Crossed Effects

Why a Model May Need Both Structures

A study can contain more than one kind of grouping. Patients may belong to a single clinic, so patients are nested within clinics. Clinicians, however, may see patients in several clinics, so clinician identity crosses clinic membership. A model that treats every grouping factor as nested can split one clinician into several artificial clinic-specific identities. A model that omits patient or clinic grouping can overlook repeated or shared structure in the observations.

The key is to describe the data-generating design before writing the random-effects terms. Identify the unit attached to each observation, which group each unit belongs to, and whether a factor’s levels occur in only one level or in multiple levels of another factor. The model then encodes these relationships while allowing the outcome distribution and link to suit the response.

Index the Units Before Writing the Model

Suppose clinics are indexed by \(k\), patients within clinic \(k\) by \(j\), and repeated observations on a patient by \(t\). Let \(l(k,j,t)\) denote the clinician who provides the care recorded at that observation. The notation matters: \(j\) is a within-clinic index, whereas \(l\) is a clinician’s global identifier. If clinician 12 works at two clinics, the clinician is still clinician 12 at both.

For a binary outcome, such as whether a hypothetical patient has a specified outcome by a follow-up visit, one possible model is:

Here \(u_k\) is a clinic random intercept, \(v_{jk}\) is a patient random intercept nested within clinic, and \(a_l\) is a clinician random intercept crossed with clinic. The fixed effects \(\beta_0\) and \(\beta_1\) describe the population-level intercept and the association with covariate \(x\). The logit link is \(\operatorname{logit}(p)=\log\{p/(1-p)\}\).

For this example, assume \(u_k\sim N(0,\sigma^2_{\text{clinic}})\), \(v_{jk}\sim N(0,\sigma^2_{\text{patient}})\), and \(a_l\sim N(0,\sigma^2_{\text{clinician}})\). Assume the three sets of random effects are mutually independent, and that observations are conditionally independent given the random effects and covariates. These are modeling assumptions, not consequences of nesting or crossing. Another application may require a different distribution, link, random-effects dependence structure, or within-unit conditional distribution.

The term \(v_{jk}\) uses the patient’s identity within clinic. It must not be replaced by an effect indexed only by \(j\) if patient labels are reused across clinics. Conversely, the clinician term uses the same \(a_l\) for every observation involving clinician \(l\), regardless of clinic. This distinction is the core of the specification.

Worked Specifications and Calculations

Why the Terms Enter Together

A mixed-effects model combines contributions from the observation’s relevant groups in one conditional linear predictor. In the binary example, that predictor contains a clinic contribution, a patient contribution, and a clinician contribution, as well as fixed effects. Conditional on those group effects, the outcome follows its chosen response distribution. The probability model for the observed data is obtained by averaging the conditional likelihood over the random-effects distributions.

Here \(f\) is the conditional response density or probability mass function, \(\beta\) denotes the fixed-effect parameters, and \(\theta\) denotes random-effects distribution parameters. The product expresses conditional independence of observations given all included effects; the integration accounts for uncertainty and variation in those effects. Importantly, two observations can share one group effect without sharing every other effect. For example, observations from different clinics by C7 share the clinician effect but not the clinic effect.

This justification also explains why a random-effect term must match the identity that is actually shared. If a clinician’s effect is mistakenly split into clinic-specific effects, the model no longer represents one persistent clinician contribution across clinics. If patient identities are mistakenly shared across clinics when patient labels are only locally unique, the model instead imposes sharing where the design has none.

Extensions, Diagnostics, and Alternatives

The same design logic applies to other conditional response distributions and links. A count outcome might use a log link and a count distribution; a continuous outcome might use a Gaussian conditional distribution. The nesting and crossing structure is determined by which observations share units, not by whether the response is binary, a count, or continuous.

Random slopes can be included when a covariate’s association is expected to vary across a grouping unit and the data support that additional variation. For example, if \(x\) is measured repeatedly within clinics, a clinic-specific slope could be added as \(u_{1k}x_{kjt}\), alongside the clinic intercept \(u_{0k}\). A random slope should be tied to a clearly identified unit and a meaningful varying covariate; it is not a generic remedy for lack of fit. Additional random terms require enough informative replication and may make estimation difficult.

Several checks help identify a specification that is not working as intended:

• Audit identifiers. Verify that a patient’s clinic is unique and that a clinician who appears in multiple clinics retains one global identifier. Check for accidental reuse of local labels.
• Check the observed overlap. Count how many clinicians appear in multiple clinics and how many patients have repeated observations. If the supposed crossing or repetition is absent, the corresponding interpretation may not be supported by the data.
• Inspect fitting warnings. Convergence problems, very large uncertainty in variance estimates, or estimates near a boundary can signal sparse information, an overcomplicated structure, or numerical difficulty. They do not by themselves prove which term should be removed.
• Assess conditional fit. Look for systematic residual patterns by clinic, patient, clinician, time, or covariate. A persistent pattern may indicate a missing fixed effect, an unsuitable response distribution, or an inadequate random-effects structure.
• Check the assumptions against the design. The illustrative model assumes independent random-effect sets and conditional independence. Shared time shocks, clinician-specific slopes, or other dependencies may make those assumptions implausible.

A nested-only model is appropriate when each clinician belongs to one clinic, but it is inadequate for representing the same clinicians working across clinics if it assigns a separate clinician effect within each clinic. A crossed-only model that includes clinicians but omits clinic or patient effects discards the nesting and repetition in the design. Treating all group factors as fixed effects is another possible analysis, especially for a small set of groups that are themselves the main targets of inference, but it changes the inferential target and does not substitute automatically for a random-effects model. Model choice should follow the scientific question, the sampling design, and the available replication.

The model has limits. Random effects do not correct confounding, guarantee causal interpretation, or compensate for incorrect identifiers. A random-intercept specification also assumes a particular form of group-level variation; it may not capture changing effects over time or dependence not represented by the terms. Sparse crossing can make it difficult to estimate the intended components. A defensible formula is therefore a design-based starting point, followed by checks of fit and estimation—not a guarantee that every assumption is satisfied.

Common Mistake