Building a Composite Satisfaction Index

1. From Many Outcomes to a Single Index: The Rationale

Before diving into the "how," let's establish the "why." Why consolidate multiple satisfaction scores into one composite index?

There are three primary benefits:

1. Conceptual Clarity: A single index provides a holistic, system-level measure. It allows us to ask questions like, "Does living arrangement X improve the overall satisfaction of the triad?" This is often more conceptually meaningful than asking about its effect on six different, interdependent satisfaction scores simultaneously.
2. Statistical Power & Parsimony: Using a single primary outcome variable simplifies our final causal model. Instead of testing the effect of our intervention on multiple outcomes (which introduces issues of multiple comparisons and can dilute statistical power), we focus our inferential power on one, well-defined construct.
3. Meaningful Weighting: A simple, unweighted average of all satisfaction scores would implicitly assume that each person's and each dyad's satisfaction contributes equally to the system's overall health. This is a strong and likely incorrect assumption. A more sophisticated approach, which we will develop, allows the data to tell us which components are more or less central to the triad's global satisfaction.

To begin thinking about this, it's helpful to understand how variance in a measure like satisfaction is partitioned within a group. The Intraclass Correlation Coefficient (ICC) is the key metric for this.

2. Decomposing Satisfaction: Actor, Partner, and Relationship Effects

If we accept that individual satisfaction scores are not independent, we need a framework to describe the sources of that interdependence. While our last lesson focused on APIM, another highly relevant framework is the Social Relations Model (SRM). SRM decomposes the variance in a person's rating into three components:

• Actor Effect: The rater's general tendency to be satisfied. How satisfied is Person A in general, across all their relationships?
• Partner Effect: The target's general tendency to elicit satisfaction. How satisfied do people tend to be with Person B?
• Relationship Effect: The unique, dyadic component of satisfaction that isn't explained by the individuals involved. How uniquely satisfying is the A-B relationship, above and beyond A's general satisfaction and B's general elicitation of satisfaction?

Understanding this decomposition is crucial because it empirically demonstrates that the dyad is more than the sum of its parts.

The key takeaway from the SRM is that the unique relationship component is often substantial. This provides strong justification for building our composite index not from individual satisfaction scores, but from measures that capture the health of the dyads within the triad.

3. Constructing the Index: A Second-Order Factor Model

Given your background in SEM, the most rigorous and appropriate method for us is to use Confirmatory Factor Analysis (CFA) to construct a second-order latent variable. This approach formalizes the idea that "Triadic Satisfaction" is a higher-order construct that is expressed through the satisfaction of the underlying dyads.

Here is the hierarchical structure we will model:

• Level 0 (Observed Variables): The individual items from our satisfaction questionnaire. These are the raw responses from the pivot and partners A and B about their feelings toward each dyadic relationship.
• Level 1 (First-Order Latent Factors): These represent the satisfaction of each of the three dyads within the triad.
  • Dyadic_Sat_Pivot_A: A latent factor indicated by items from the pivot about partner A, and from partner A about the pivot.
  • Dyadic_Sat_Pivot_B: A latent factor indicated by items from the pivot about partner B, and from partner B about the pivot.
  • Dyadic_Sat_A_B: A latent factor indicated by items from partner A about partner B, and from partner B about partner A. This captures their unique dynamic (e.g., compersion, friendship, or friction).
• Level 2 (Second-Order Latent Factor): A single latent factor, which we will call Triadic_Satisfaction. This higher-order factor is indicated by the three first-order dyadic satisfaction factors.

The structure looks something like this:

The factor loadings from the dyadic factors to the triadic factor are not assumed to be equal. They are estimated from the data, essentially providing data-driven weights. If the Pivot-A relationship is most central to the triad's health, it will have the highest loading on the Triadic_Satisfaction factor.

A paper by Alfieri & Lanz provides a perfect walkthrough of this exact methodology.

4. Application to Our Vee Triad Experiment

Let's operationalize this for our N-of-1 study. After collecting sufficient EMA data within an experimental period, we would perform the following analysis:

1. Specify the Measurement Model: Using a structural equation modeling software package, we would define the second-order CFA model described above. The observed variables would be the items from our custom satisfaction scale, collected repeatedly over time.
2. Estimate the Model: We would fit this model to the data. The software would estimate all the parameters, including the crucial factor loadings.
3. Evaluate Model Fit: We would examine goodness-of-fit indices (e.g., CFI > .90, RMSEA < .08) to confirm that our theoretical model provides a good representation of the observed data.
4. Extract Factor Scores: Once the model is validated, we can compute the factor scores for the second-order latent variable, Triadic_Satisfaction, for each time point.

This time series of factor scores becomes our new, composite outcome variable. It is a single variable that represents the overall health of the triad at each moment, with the contributions of each dyad appropriately weighted based on the empirical data. This index can then be used as the dependent variable in the Multilevel Actor-Partner Interdependence Model we designed in the previous lesson to estimate the causal effects of our experimental conditions.

Conclusion

In this lesson, we have designed a methodologically rigorous approach to synthesize multiple, interdependent satisfaction measures into a single, composite index of triadic satisfaction. This process is not a mere convenience; it is a critical step in building a valid and powerful causal model of the triadic system.

Key Takeaways:

• A composite index provides conceptual clarity, improves statistical power, and avoids the naive assumption that all satisfaction components are equally important.
• Frameworks like SRM demonstrate that a significant portion of satisfaction variance lies at the unique relationship level, justifying a focus on dyadic measures as the building blocks of our index.
• A second-order Confirmatory Factor Analysis (CFA) is the ideal method for constructing the index. It models Triadic_Satisfaction as a higher-order latent variable indicated by latent dyadic satisfaction factors.
• This approach provides data-driven weights for each dyad's contribution to the whole and can be empirically validated using model fit statistics. The resulting factor score is a robust, theoretically grounded primary outcome for our experiment.

Now that we have defined precisely what we are measuring—both at the item level and the composite index level—we must turn our attention to the quality of the data collection itself. In our next lesson, we will design protocols to mitigate social desirability and observer biases during outcome reporting and assessment, ensuring that the data we feed into these sophisticated models is as accurate and unbiased as possible.