Cross-Classified Multilevel Models for Dyadic Relationships

Welcome to the next lesson in our module on advanced multilevel modeling. In our previous sessions, we've built a solid foundation by specifying a three-level hierarchical model (observations nested in individuals, in triads) and then enhancing it with random slopes to capture individual differences in emotional reactivity. This allowed us to model how the effect of a lifestyle activity on an emotion like jealousy might vary from person to person.

However, a purely hierarchical structure, akin to a set of Russian dolls, doesn't fully capture the intricate web of relationships in your triadic research context. For instance, the wife is simultaneously a member of the husband-wife dyad and the wife-bull dyad. These are not nested groupings; they are overlapping. An emotional observation from the wife is influenced by both relational contexts simultaneously. Ignoring this complex structure can lead to confounded results, where effects from one dyad are incorrectly attributed to another.

This lesson directly addresses that complexity. Our goal is to specify a cross-classified multilevel model to represent emotional observations that are nested within individuals, who are in turn cross-classified by the different dyadic relationships they participate in. This will allow us to more accurately partition emotional variance and attribute it to the correct sources—the individual, a specific dyadic relationship, or the interaction between them.

Distinguishing Hierarchical and Crossed Structures

Before we build the model, it's crucial to solidify the distinction between nested (hierarchical) and crossed data structures. In a nested structure, each lower-level unit belongs to one and only one higher-level unit. For example, in our previous model, each observation belongs to only one individual, and each individual belongs to only one triad.

In a crossed structure, lower-level units can belong to multiple higher-level units that are not nested within each other. The classic example is students being classified by both the school they attend and the neighborhood they live in. A student belongs to one school and one neighborhood, but schools and neighborhoods are not nested—students from many neighborhoods can attend one school, and students from one neighborhood can attend many schools.

The following video provides a concise visual explanation of nested versus crossed random effects.

Linear mixed effects models

This video from Matthew E. Clapham on linear mixed effects models clearly illustrates the difference between nested and crossed designs.

Watch the segment from "nested which means". Focus on the diagrams that visually contrast a nested structure with a fully crossed design. This will help you conceptualize how your triadic data differs from a simple hierarchy.

In your research, an emotional observation is nested within the individual reporting it. However, that individual is simultaneously a member of multiple dyads. For example, the wife is a member of the {Husband, Wife} dyad and the {Wife, Bull} dyad. These two dyads act as two distinct "classifications," much like school and neighborhood in the example. An emotional response from the wife is therefore influenced by her individual tendencies, the dynamics of her marital relationship, and the dynamics of her relationship with the bull. A cross-classified model is required to disentangle these influences.

Why a Cross-Classified Model is Essential

Your background in experimental design highlights the importance of controlling for confounding variables. Ignoring a relevant data structure is a primary source of such confounding. If you were to analyze the wife's emotional data using a model that only recognizes her as an individual within a triad, you would be unable to separate the variance due to the husband-wife dynamic from the variance due to the wife-bull dynamic.

The following resource provides a foundational explanation of cross-classified data and the problems that arise when they are not modeled correctly.

Cross-classified and Multiple Membership Structures in ...

This review paper by Fielding and Goldstein is a cornerstone text on complex data structures. It clearly explains the nature of cross-classifications and the rationale for using them.

First, read the section "The Nature Of Cross-Classifications And Their Effects" (pp. 20-22). Pay close attention to the school-and-neighborhood example and the discussion on how an unspecified effect (e.g., neighborhood) can be wrongly attributed to the specified one (e.g., school). This is directly analogous to disentangling dyadic effects in your work. Next, read the section "Some Objectives Of Analysing Cross-Classified Multilevel Models" (pp. 23-24). This outlines the key benefits, including improved precision and, most importantly, the ability to correctly partition variance among different sources.

For your research, the main objectives of using a cross-classified model are:

To Isolate Dyadic Effects: To separately estimate the amount of emotional variance attributable to the husband-wife dyad versus the wife-bull dyad versus the husband-bull dyad.
To Avoid Confounding: To ensure that the estimated effect of an activity within one dyad (e.g., a wife-bull date) is not contaminated by unmodeled dynamics from another dyad (e.g., a recent argument between the husband and wife).
To Increase Precision: To get more accurate estimates of the effects of your experimental manipulations by correctly specifying the random effects structure.

Specifying the Cross-Classified Model

Let's now specify a cross-classified model for your triadic data. The core idea is that an emotional observation is nested within a person, but it is simultaneously cross-classified by the dyadic relationships that person is involved in.

Consider modeling an emotion like guilt, measured repeatedly over time. Let $y_{i p d_{1} d_{2}}$ be a guilt observation $i$ for person $p$ , where person $p$ is a member of dyad $d_{1}$ and dyad $d_{2}$ . For example, if the person is the wife, $p$ would be the wife's ID, $d_{1}$ would be the ID for the husband-wife dyad, and $d_{2}$ would be the ID for the wife-bull dyad.

A cross-classified random intercept model for this observation would be:

y_{i p d_{1} d_{2}} = β_{0} + u_{p} + v_{d_{1}} + w_{d_{2}} + e_{i p d_{1} d_{2}}

Let's break this down:

$y_{i p d_{1} d_{2}}$ is the $i$ -th guilt observation.
$β_{0}$ is the fixed intercept, representing the grand mean of guilt across all observations, people, and dyads.
$u_{p} \sim N (0, σ_{p}^{2})$ is the random effect for the person. This captures how person $p$ 's average guilt level deviates from the grand mean. This is the compositional or individual-level effect.
$v_{d_{1}} \sim N (0, σ_{d_{1}}^{2})$ is the random effect for the first dyad. This captures how the average guilt within dyad $d_{1}$ deviates from the grand mean, after accounting for the person effect.
$w_{d_{2}} \sim N (0, σ_{d_{2}}^{2})$ is the random effect for the second dyad.
$e_{i p d_{1} d_{2}} \sim N (0, σ_{e}^{2})$ is the Level-1 residual, representing the within-person fluctuation at that specific moment.

The power of this model lies in its variance components:

$σ_{p}^{2}$ : Variance attributable to stable differences between individuals.
$σ_{d_{1}}^{2}$ : Variance attributable to the unique dynamics of the first dyad type (e.g., husband-wife relationships).
$σ_{d_{2}}^{2}$ : Variance attributable to the unique dynamics of the second dyad type (e.g., wife-bull relationships).
$σ_{e}^{2}$ : Within-person, moment-to-moment variance.

By comparing the magnitudes of these variance components, you can quantitatively answer questions like: "Is the variation in the wife's guilt driven more by her individual traits ( $σ_{p}^{2}$ ), her marital dynamic ( $σ_{d_{1}}^{2}$ ), or her dynamic with the bull ( $σ_{d_{2}}^{2}$ )?"

To see how this framework is used in practice across various domains, the following article provides an excellent overview. The application to social networks is particularly relevant.

Modern applications of cross-classified random effects ...

This paper frames many different statistical models as special cases of Cross-Classified Random Effects Models (CCREMs), demonstrating the framework's versatility.

Start by reviewing the model definition (sections 1 and 2), which reinforces the core concepts and provides the general mathematical formulation. Then, explore the application in the section on "Social network analysis" (pages 9-10). The example of modeling resource sharing between households, cross-classified by "giver" and "receiver," is a powerful analogy for modeling emotional dynamics between members of a triad.

This model can be extended, just as we did in the last lesson, by adding predictors and random slopes. For instance, you could add an Activity predictor and test whether its effect on guilt varies not just by person, but also by the dyadic context.

Conclusion

Today, we have made a critical transition from purely hierarchical models to the more flexible and realistic cross-classified framework. This approach is purpose-built for data where units of analysis belong to multiple, non-nested groups—a perfect description of individuals operating within the overlapping dyadic relationships of a triad.

Key Takeaways:

Cross-classified structures arise when lower-level units are classified by two or more higher-level factors that are not nested within each other.
Ignoring a cross-classification leads to confounded variance components and potentially biased estimates, undermining causal claims.
A cross-classified multilevel model for your triadic data allows you to specify separate random effects for individuals and for the dyads they participate in.
The resulting variance components ( $σ_{p er so n}^{2}$ , $σ_{d y a d 1}^{2}$ , $σ_{d y a d 2}^{2}$ , etc.) provide a quantitative decomposition of emotional variation, attributing it to individual versus relational sources.

In our previous module, we touched upon the Social Relations Model (SRM) for decomposing dyadic perceptions. You may now see that the cross-classified model we've specified today is the fundamental engine behind the SRM. Our next module, "Deconstructing Group Dynamics with the Social Relations Model," will build directly on this foundation. We will formalize the concepts of perceiver, target, and relationship effects and learn how to estimate and interpret them to gain an even deeper understanding of the reciprocal dynamics within the triad.

Recommended resources

Distinguishing Hierarchical and Crossed Structures

Why a Cross-Classified Model is Essential

Specifying the Cross-Classified Model

Conclusion