Quantifying Emotional Influence: Variance Components in Cross-Classified Models

The Social Relations Model (SRM) as an Interpretive Framework

The cross-classified model we specified is the statistical engine that powers a well-established framework in social psychology called the Social Relations Model (SRM). Before we dive into the numbers, understanding the SRM's conceptual language will give us a powerful lens for interpretation. Your background in experimental design, which emphasizes the decomposition of variance (e.g., in ANOVA), will make this framework intuitive.

The SRM decomposes any interpersonal perception or behavior into three primary sources of variance:

1. Actor Effect: The tendency of an individual to feel or act in a certain way across all partners. For example, a husband's general disposition to be jealous. This is a property of the individual.
2. Partner Effect: The tendency of an individual to elicit a certain feeling or action from all partners. For example, a wife's tendency to consistently elicit jealousy from her partners. This is also a property of the individual, but from the "target" perspective.
3. Relationship Effect: The unique dynamic between a specific actor and a specific partner that cannot be explained by their general tendencies. For example, the unique level of jealousy a specific husband feels in relation to his specific wife, above and beyond his general jealousy and her general tendency to elicit it. This is a property of the dyad.

The following resource provides an excellent introduction to these concepts and the types of questions the SRM can answer.

When we estimate a cross-classified model, the variance components we extract correspond directly to these SRM effects.

• Variance in Actor Effects (${\sigma }_{\alpha }^2$): How much do individuals differ in their general emotional tendencies?
• Variance in Partner Effects (${\sigma }_{\beta }^2$): How much do individuals differ in the emotions they elicit from others?
• Variance in Relationship Effects (${\sigma }_{\gamma }^2$): How much do specific relationships vary in their unique emotional dynamics?

Estimating and Partitioning Variance

With the conceptual framework in place, let's turn to estimation. The goal is to obtain numerical estimates for ${\sigma }_{\alpha }^2$, ${\sigma }_{\beta }^2$, and ${\sigma }_{\gamma }^2$, plus the residual or error variance, ${\sigma }_{ϵ}^2$. We can then calculate the proportion of the total variance attributable to each component. This is often called the Intraclass Correlation (ICC) for a given component.

For example, the proportion of variance due to actor effects is:

This calculation tells you what percentage of the total emotional variation is "explained" by stable differences between actors. The following short video provides a helpful visual recap of how variance is partitioned in mixed models.

This partitioning is the quantitative answer to questions like: "Is the wife's guilt primarily driven by her own personality, or is it a feature of her specific relationships?"

A Worked Example: Interpreting the Output

Let's now walk through how to interpret the output of a cross-classified model fitted to social interaction data. Since your work involves continuous emotional outcomes (e.g., on a validated scale) and you have a strong statistical background, we will examine an advanced application.

The following resources detail how to implement and interpret these models. The first, from Eastwick et al., shows the practical side using R's glmmTMB package. The second, from Gill and Swartz, presents a formal Bayesian model for triadic data, which aligns with your expertise in Bayesian methods.

Now, let's consider a model specifically for triadic data, using a Bayesian approach.

Application to Your Research Question

Let's synthesize this for your study. Imagine you have collected repeated measures of guilt from the wife on a 1-10 scale. You fit a cross-classified model and obtain the following (hypothetical) variance components:

• Wife's Actor Variance (${\sigma }_{\alpha \_wife}^2$): 3.5
• Partner Variance (elicited from Husband, ${\sigma }_{\beta \_husband}^2$): 1.0
• Partner Variance (elicited from Bull, ${\sigma }_{\beta \_bull}^2$): 0.5
• Husband-Wife Relationship Variance (${\sigma }_{\gamma \_HW}^2$): 2.5
• Wife-Bull Relationship Variance (${\sigma }_{\gamma \_WB}^2$): 2.0
• Residual/Error Variance (${\sigma }_{ϵ}^2$): 1.5

Interpretation:

1. Calculate Total Variance:
   ${\sigma }_{Total}^2=3.5+1.0+0.5+2.5+2.0+1.5=11.0$

2. Calculate Proportions (ICCs):

   • Actor (Wife's trait): $3.5/11.0\approx 32\%$. A substantial portion of the variation in the wife's guilt is due to her stable, individual tendencies.
   • H-W Relationship: $2.5/11.0\approx 23\%$. The unique dynamic of the marriage is a major contributor to her guilt, separate from her own traits or her husband's.
   • W-B Relationship: $2.0/11.0\approx 18\%$. The dynamic with the bull is also a significant, independent source of guilt.
   • Partner (Husband's effect): $1.0/11.0\approx 9\%$. Some variance is explained by the husband's general tendency to elicit guilt.
   • Partner (Bull's effect): $0.5/11.0\approx 5\%$. The bull's general tendency to elicit guilt is a minor factor.
   • Residual: $1.5/11.0\approx 14\%$. This is the unexplained, moment-to-moment fluctuation.

Substantive Conclusion: The wife's guilt appears to be primarily a function of her own disposition (32%) and the specific dynamics of her two key relationships (23% + 18% = 41%). The people themselves (as partners) play a smaller role. This suggests that interventions aimed at managing guilt should focus less on the people and more on the dynamics within each dyad.

Conclusion

In this lesson, we moved from the abstract specification of a cross-classified model to the concrete task of estimation and interpretation. You learned how to use the language of the Social Relations Model (actor, partner, relationship) to give theoretical meaning to the model's variance components. By calculating the proportion of total variance accounted for by each component, you can now quantitatively assess whether emotions in the triad are driven more by stable individual differences or by the unique chemistry of specific dyadic pairings.

Key Takeaways:

• A cross-classified model's primary output for this purpose is its variance components, which correspond to actor, partner, and relationship effects in the Social Relations Model.
• Standardizing these components into percentages of total variance (ICCs) is essential for interpreting their relative importance.
• The magnitude of each variance component provides a quantitative answer to whether emotional variation is primarily individual-driven or relationship-driven.
• This decomposition allows for targeted insights; for instance, discovering that jealousy is mostly a relationship effect suggests focusing on dyadic communication rather than individual therapy.

This lesson concludes our deep dive into advanced multilevel models. We now have a powerful tool for understanding the static structure of emotional influence. In our next module, we will formally adopt the Social Relations Model framework to explore more nuanced dynamic processes, such as reciprocity. We will move beyond asking "how much variance" to asking "is the unique feeling between two people mutual?" This will allow us to investigate the symmetry and mutuality of emotions within the triad's dyads.