Power Analysis for Triadic Designs

Why Standard Power Calculators Fall Short

In a simple experiment, power analysis often involves plugging an expected effect size, alpha level, and desired power (typically 80%) into a formula or software to get a required sample size. This works when observations are independent. In your triadic research, this assumption is fundamentally violated. A husband's emotional state is not independent of his wife's, which is not independent of her interactions with a bull.

The power to detect an effect in such a clustered design depends not just on the overall sample size, but on the number of clusters (triads) and the degree of non-independence within those clusters, often quantified by the Intraclass Correlation Coefficient (ICC). The effects you want to detect—actor, partner, and relationship effects—are themselves variance components that describe this intricate dependency structure. Therefore, estimating power requires a method that can account for this entire system.

The Parameters of Interest: Actor, Partner, and Relationship Effects

Before we can power a study to find these effects, let's formally revisit what they are. The Social Relations Model (SRM) provides a precise framework for this.

Your power analysis must be designed to ensure you have a large enough sample to reliably estimate these specific variance components and declare them statistically different from zero.

The Modern Solution: Simulation-Based Power Analysis

Since no simple formula can capture the complexities of an SRM or triadic APIM, the standard and most flexible approach is simulation-based power analysis, sometimes called a Monte Carlo power analysis. Your background in simulation makes the logic here straightforward: if you can write down a model of the world, you can simulate data from it to see how your statistical methods would perform.

The process is a computational experiment:

1. Define the Data Generating Process: Specify the full statistical model you intend to use for analysis (e.g., an SRM or APIM). This includes defining all the parameters: means, variances for actor, partner, and relationship effects, and any covariances (reciprocity).
2. Hypothesize a "True" Effect Size: This is the most critical step. You must make an educated guess about the magnitude of the effects you expect to find in the population. For instance, what percentage of the total variance in husband jealousy do you believe is due to the unique relationship effect (${\sigma }_{\gamma }^2$)? These assumptions can be based on prior research, theoretical importance, or a small pilot study.
3. Simulate a Dataset: For a given sample size (e.g., $N=50$ triads), generate one complete dataset according to the model and parameters you defined.
4. Analyze the Simulated Dataset: Fit your intended model to this single, simulated dataset and test for the significance of your effect of interest (e.g., is the estimate for ${\sigma }_{\gamma }^2$ significantly different from zero?).
5. Repeat and Tally: Repeat steps 3 and 4 thousands of times (e.g., 1000 or 5000 replications).
6. Calculate Power: The proportion of these replications in which the effect was found to be statistically significant is your estimated statistical power for that specific sample size and effect size.
7. Iterate: Repeat the entire process for different sample sizes ($N=60,70,\mathrm{80...}$) to generate a power curve, showing how power increases with sample size. You can then identify the sample size that yields your desired power (e.g., 80%).

This method is computationally intensive but has the immense advantage of being tailored perfectly to your specific design, model, and research question. It is widely used in complex modeling scenarios. The following paper on a triadic APIM study explicitly mentions using this approach.

A Guiding Principle for Triadic Designs

Before we walk through an example, there's a crucial insight for power in designs like yours. When dealing with many small groups (like triads), where does power come from? More people within a group, or more groups?

Conceptual Walkthrough: Powering a Study on Jealousy

Let's imagine you want to conduct your between-triad experiment, where one group of triads uses a "Structured Debrief" communication protocol and a control group does not. You want to know if this intervention reduces the husband's jealousy, and you hypothesize it does so by affecting the husband-wife relationship effect. You want to ensure you have enough power to detect this change.

Target: 80% power to detect a "medium" relationship variance component (${\sigma }_{{\gamma }_{HW}}^2$) for husband's jealousy, assuming it accounts for 10% of the total variance in the outcome.

Simulation Steps:

1. Define Model & Parameters: You decide the total variance of the husband's jealousy score is 1.0. You hypothesize the following variance components:

   • Husband Actor Variance (${\sigma }_{{\alpha }_H}^2$): 0.20
   • Wife Partner Variance (${\sigma }_{{\beta }_W}^2$): 0.15 (the wife's general effect on her partners' jealousy)
   • Husband-Wife Relationship Variance (${\sigma }_{{\gamma }_{HW}}^2$): 0.10 (This is your target effect)
   • Residual/Error Variance: 0.55
     You would specify similar parameters for all other perspectives (wife's guilt, bull's emotions, etc.).

2. Set Sample Size: Start with a guess, perhaps $N=60$ triads (30 per group).

3. Simulate: Write a script (e.g., in R using lme4 or brms, as shown in the SRM paper) to generate data for 60 triads based on the full parameter set defined in step 1.

4. Analyze: Fit an SRM to the simulated data and perform a likelihood ratio test (or use Bayesian methods) to see if the estimated relationship variance (${\hat{\sigma }}_{{\gamma }_{HW}}^2$) is statistically greater than zero.

5. Repeat: Run this simulation 1,000 times. Suppose the effect is significant in 680 of those runs.

6. Calculate Power: Your estimated power for $N=60$ is 680/1000 = 68%. This is below your 80% target.

7. Iterate: You increase the sample size to $N=80$ triads and re-run the entire simulation. This time, you find the effect is significant in 815 of 1000 runs. Your power is now 81.5%.

Conclusion: You would conclude that to have adequate power to detect the hypothesized relationship effect, you need a sample of approximately 80 triads.

Conclusion

You now possess the conceptual framework for tackling one of the most critical planning stages of your research. While designing the experiment and its randomization scheme ensures internal validity, a proper power analysis ensures your study is a worthwhile investment of time and resources.

Key Takeaways:

• Power analysis for triadic experiments must account for the non-independence of data, making simple calculators insufficient.
• The primary method is simulation-based power analysis, which involves repeatedly simulating and analyzing data from your hypothesized model to estimate power.
• The targets of your power analysis are the specific variance components of your model: actor, partner, and relationship effects.
• The most crucial input for a power simulation is your hypothesized effect sizes for these components.
• In designs composed of many small groups like triads, the most effective way to increase statistical power is to increase the number of triads.

We have established how to design the experiment and how to ensure it is sufficiently powered. However, even with a perfectly randomized and powered study, the interconnected nature of the triad introduces profound challenges for causal interpretation. In our next module, we will delve into the topic of Causal Inference Under Interference, exploring what happens when one person's "treatment" inevitably spills over to affect another's outcome—the very definition of a triadic interaction.