Dependence Between Observations Sharing a Crossed Unit

Why a Shared Crossed Unit Creates Dependence

In a crossed design, each observation belongs to a unit from each of two classifications. For example, a response might be recorded for a particular subject and item, or for a particular target and rater. Two observations can involve the same unit from one classification while involving different units from the other. Even if the observations are independent after conditioning on all random effects, they can be dependent marginally because they share one random effect.

The focus here is a pair \(Y_{ij}\) and \(Y_{ik}\): both observations involve unit \(i\) from classification A, but \(j\ne k\), so they involve different units from classification B. The correlation depends on the model and response distribution. In a Gaussian random-intercept model, it is a simple variance-component ratio. In a generalized linear mixed model with a nonlinear link, the response-scale correlation generally requires integrating over the random effects.

Model and Assumptions

First consider a Gaussian crossed random-intercept model with an identity link:

Assume \(a_i\sim N(0,\sigma_A^2)\), \(b_j\sim N(0,\sigma_B^2)\), and \(\varepsilon_{ij}\sim N(0,\sigma^2)\). The A effects, B effects, and residuals are mutually independent, and residuals are independent across observations. The fixed-effect term is treated as nonrandom. These assumptions imply conditional independence of distinct responses given the random effects. For the pair \(Y_{ij},Y_{ik}\), the residuals and the B effects differ, while \(a_i\) is shared.

The same dependence logic applies to a GLMM with a non-Gaussian response: assume conditional independence of the response observations given all random effects, and let \(\mu_{ij}(a_i,b_j)=E(Y_{ij}\mid a_i,b_j)\) denote a conditional mean. If \(Y_{ij}\) and \(Y_{ik}\) are conditionally independent, their marginal covariance is the covariance of their conditional means. This is the key step for nonlinear-link models.

Gaussian Shared-Unit Correlation

For the Gaussian model, subtracting the fixed means leaves \(a_i+b_j+\varepsilon_{ij}\) and \(a_i+b_k+\varepsilon_{ik}\). Independence makes every cross-product expectation zero except the contribution from the shared \(a_i\). Thus the covariance is \(\sigma_A^2\). Each response has variance \(\sigma_A^2+\sigma_B^2+\sigma^2\), so the correlation is:

This expression assumes the same variance components for both observations. It does not depend on their fixed-effect means: adding constants changes neither covariance nor variance. The denominator includes variation from both crossed classifications and the observation-level residual. The numerator includes only the component shared by this pair.

Non-Gaussian Responses: The Poisson-Lognormal Case

For a nonlinear link, the shared variance component is not generally the response covariance. Consider a crossed Poisson-lognormal random-intercept model:

Here \(\lambda_{ij}\) is a positive fixed-effect mean factor, and the random effects are independent across classifications and units. Conditional on them, responses are independent. The marginal mean is \(m_{ij}=\lambda_{ij}\exp\{(\sigma_A^2+\sigma_B^2)/2\}\). For two observations sharing A but having independent B effects, their conditional means share the factor \(\exp(a_i)\). Since a normal variable \(a_i\) has \(E\{\exp(a_i)\}=\exp(\sigma_A^2/2)\) and \(E\{\exp(2a_i)\}=\exp(2\sigma_A^2)\), the covariance of those means is \(m_{ij}m_{ik}\{\exp(\sigma_A^2)-1\}\).

The marginal variance of one response includes Poisson sampling variation as well as variation in its random mean:

The first term is the conditional Poisson variance averaged over the random effects; the second is the variance of the conditional mean. Consequently the pairwise response correlation is the covariance above divided by the square root of the two marginal variances. Unlike the Gaussian case, it depends on the marginal means as well as the random-effect variances.

Interpretation, Diagnostics, and Alternatives

A useful check is to state exactly which unit is shared. In the Gaussian example, observations sharing A have covariance \(\sigma_A^2\); observations sharing B instead have covariance \(\sigma_B^2\), under the same independence assumptions. If a pair shares neither unit, its covariance is zero in this random-intercept model. These statements concern the model-implied marginal dependence, not necessarily the empirical correlation in a finite sample.

For a logistic GLMM, a commonly used latent-scale summary adds a logistic residual variance of \(\pi^2/3\) to the denominator. That yields a correlation summary on an assumed latent scale, not the observed binary-response correlation. The observed response correlation depends on the fixed predictors and on integration over the random effects. It should be obtained from the fitted model’s implied joint probabilities or another appropriate marginal calculation. The same caution applies to other non-Gaussian families: a variance-component fraction is not automatically the response-scale correlation.

Diagnostics should check whether the assumed dependence pattern is plausible. Compare observed response similarity for pairs sharing A with similarity for pairs sharing B or neither, while accounting for the response type and fixed predictors. Inspect conditional or simulated residual patterns for remaining clustering, and check whether the fitted variance component is near zero or is poorly determined. Sparse crossing, a small number of units, or strong imbalance can make these checks imprecise. Persistent dependence among pairs sharing neither unit can indicate omitted clustering or residual dependence; treating it as explained by the shared-A effect would be a model error.

The main alternative is to report variance components or a latent-scale intraclass correlation rather than a response-scale pair correlation. Variance components describe the model’s sources of variation; a response-scale correlation directly describes the association for a specified kind of pair, but may depend on its predictor values. A model with an explicit residual correlation or additional shared effects may be needed if conditional independence is not credible. These approaches answer different questions, and none repairs a design that does not contain enough replication to distinguish the relevant variance components.