Beyond logistic regression: structural equations modelling for binary variables and its application to investigating unobserved confounders

BACKGROUND: Structural equation modelling (SEM) has been increasingly used in medical statistics for solving a system of related regression equations. However, a great obstacle for its wider use has been its difficulty in handling categorical variables within the framework of generalised linear models. METHODS: A large data set with a known structure among two related outcomes and three independent variables was generated to investigate the use of Yule's transformation of odds ratio (OR) into Q-metric by (OR-1)/(OR+1) to approximate Pearson's correlation coefficients between binary variables whose covariance structure can be further analysed by SEM. Percent of correctly classified events and non-events was compared with the classification obtained by logistic regression. The performance of SEM based on Q-metric was also checked on a small (N = 100) random sample of the data generated and on a real data set. RESULTS: SEM successfully recovered the generated model structure. SEM of real data suggested a significant influence of a latent confounding variable which would have not been detectable by standard logistic regression. SEM classification performance was broadly similar to that of the logistic regression. CONCLUSION: The analysis of binary data can be greatly enhanced by Yule's transformation of odds ratios into estimated correlation matrix that can be further analysed by SEM. The interpretation of results is aided by expressing them as odds ratios which are the most frequently used measure of effect in medical statistics.