Beyond Raw Percentages: Mastering Social Network Composition with Compositional SEM

TL;DR

Analyzing social networks often involves looking at proportions—what percentage of your support comes from family versus friends? However, standard statistics break when applied to these "proportions of a total." This paper introduces a robust Compositional Structural Equation Modeling (SEM) framework. By using isometric log-ratio (ilr) transformations and Multitrait-Multimethod (MTMM) designs, the authors provide a way to analyze network changes that is mathematically sound, accounts for survey measurement errors, and delivers highly interpretable results.

The "Sum-to-One" Trap: Why Standard Analysis Fails

Most researchers treat the "percentage of friends" in a network as an independent variable. This is a mathematical fallacy. If the percentage of friends increases, the percentage of another group (like family) must decrease. This creates several "pathological" statistical issues:

1. Spurious Correlations: Negative correlations appear simply because of the sum-to-one constraint.
2. Restricted Space: Data is bounded between 0 and 1, violating the normality assumptions of OLS regression and standard SEM.
3. Measurement Error: Survey data is notoriously "noisy," yet most compositional models assume the reported percentages are perfectly accurate.

The Methodology: Isometric Log-Ratios (ilr)

To solve this, the authors move away from raw proportions and toward log-ratios. Instead of looking at "percentage of partner," they look at the ratio of "partner vs. other family members."

Step 1: Transformation

The paper uses the ilr transformation, which creates a set of coordinates that are independent and lie in an unconstrained real space. For a network with four components (Partner, Family, Friends, Others), they construct three meaningful ratios:

• t1: Partner vs. Other Family
• t2: Friends vs. Other Non-family
• t3: Family vs. Non-family (Geometric means)

Step 2: The Structural Model

The authors embed these ratios into an SEM that includes:

• Predictors: Gender, Age, Education, and Extraversion.
• Outcomes: Tie Multiplexity (number of support functions) and Tie Strength (closeness).
• Measurement Error Correction: Using MTMM (Face-to-face vs. Telephone surveys) to strip away "method noise."

Figure 1: The conceptual SEM path diagram showing how covariates affect log-ratios, which in turn affect network outcomes.

Key Insights and Experimental Results

Using data from a representative sample of 1,033 inhabitants in Ljubljana, the study validated several long-standing sociological hypotheses with new precision.

1. The Predictors of Composition

• Gender: Women have a significantly lower partner-to-family ratio than men (Confirming that men rely more heavily on partners for support).
• Education: Higher education shifts the composition toward friends relative to other non-family members.
• Extraversion: Highly extraverted individuals naturally maintain a higher ratio of friends.

2. Composition and Life Outcomes

The study proved that how your network is composed directly impacts the quality of support:

• Multiplexity: A higher ratio of family to non-family (t3) and partner to family (t1) leads to "multiplex" ties—people who help you in multiple ways (Instrumental, Emotional, etc.).
• Closeness: The family-to-non-family ratio was the strongest predictor of overall tie strength (Standardized Coeff = 0.372).

Table 1: Standardized estimates showing the impact of network ratios on multiplexity and strength.

Critical Analysis: Why This Matters

The true brilliance of this paper is the Interpretability-Geometric Harmony. Usually, log-ratio transformations are mathematically "pure" but impossible for a sociologist to explain. By carefully choosing the components of the ratios (e.g., comparing "Family" clusters vs. "Non-family" clusters), the authors made the math serve the theory.

Limitations

• Zeros: Dealing with "essential zeros" (e.g., people without partners) remains a challenge, often requiring researchers to drop those cases or use imputation.
• Granularity: The study didn't distinguish between "close friends" and "distant friends," which might have diluted the findings for certain hypotheses.

Future Outlook

This approach is a wake-up call for any researcher working with Compositional Data—whether it is time-use diaries, budget allocations, or social networks. Moving forward, the integration of these models into software like Mplus or R (compositions package) will allow for more nuanced, error-free social science.

Takeaway: Don't analyze the slice; analyze the ratio of the slices.