Fig. 1

Causal diagrams depicting different causal (or structural) relationships between the indicators of the latent class \(C^*\), and the “external” observed variables such as treatment (Z) and outcome (Y). In the left column (subfigures a and c), no covariates are predictors of the latent class, so \((X_1, \ldots , X_p)\) are indicators. In the right column (subfigures b and d), a subset of the covariates \((X_1, \ldots , X_q)\) are (explanatory) predictors of the latent class and its indicators \((X_{q+1}, \ldots , X_p)\). In the top row (subfigures a and b), the indicators are conditionally independent of all external variables given the latent class, as represented by the absence of arrows linking the indicators with any other observed variables. In the bottom row (subfigures c and d), the indicators are permitted to affect, or be affected by, external observed variables, as represented by the red arrows. Rectangular nodes denote observed variables, while round nodes denote latent variables