Overview

In clinical research, sample sizes are often small due to cost, recruitment, or ethical constraints.
Traditional (frequentist) structural equation models (SEM) and confirmatory factor analysis (CFA) often perform poorly in such settings, producing unstable estimates or convergence issues.

This case Study explores how Bayesian SEM / CFA can provide more stable and reliable parameter estimation under small-sample conditions.

Data and Context

  • The dataset represents a real or simulated clinical study with multiple observed and latent variables.
  • The small sample size increases the risk of weak identification, overfitting, and unstable inference.
  • The model includes both latent constructs and structural paths between them.

Methodological Approach

1. Bayesian Model Specification

  • Bayesian priors are assigned to all parameters, including factor loadings, variances, and path coefficients.
  • Priors are weakly informative or domain-informed, stabilizing parameter estimation when data are sparse.

2. Model Estimation

  • Model estimation is performed using Markov Chain Monte Carlo (MCMC) methods (e.g., Stan or JAGS).
  • Convergence diagnostics such as trace plots, R-hat statistics, and effective sample size are used to verify stability.

3. Posterior Summaries

  • Posterior means, medians, and credible intervals are computed for all parameters.
  • Bayesian model fit is evaluated using posterior predictive checks.
  • Sensitivity analyses are conducted to test robustness against prior assumptions.

4. Comparison to Frequentist SEM

  • Results are compared with those obtained from a classical SEM (where estimable).
  • The comparison highlights differences in parameter estimates, uncertainty intervals, and convergence stability.

5. Interpretation and Application

  • The Bayesian approach is evaluated for its ability to recover true parameter values under small-sample constraints.
  • The analysis identifies situations where Bayesian inference performs better or differently than the classical approach.

Key Findings

  • Stability: Bayesian SEM produces more stable parameter estimates and fewer convergence issues.
  • Regularization: Priors serve as a natural form of shrinkage, preventing extreme estimates.
  • Credible Intervals: Although wider than classical confidence intervals, they provide a more realistic representation of uncertainty.
  • Model Reliability: Bayesian inference remains feasible even when the frequentist approach fails.
  • Limitations: Priors must be carefully chosen, and diagnostic checking is essential.

Practical Implications

  • Bayesian SEM is particularly useful for clinical studies with small sample sizes.
  • Weakly informative priors help incorporate expert knowledge without overpowering the data.
  • Posterior predictive checks should always be used to assess model fit.
  • This approach bridges exploratory and confirmatory analysis, providing flexible inference under limited data conditions.

Summary

Bayesian Structural Equation Modeling provides a promising framework for small-sample clinical studies, offering improved robustness and interpretability.
While it does not eliminate uncertainty, it formalizes it — producing results that are both statistically stable and clinically meaningful.

Resources

🔗 Download both the notebook and dataset from our GitHub repository: Here