A tutorial for understanding SEM using R: Where do all the numbers come from?

Abstract

Structural equation modeling (SEM) is often seen as a complex and difficult method, especially for those who want to understand how the numbers in SEM software output are actually computed. Although many open-source SEM tools are now available-especially in the R programming environment-looking into their source code to understand the underlying calculations can still be overwhelming. This tutorial aims to provide a clear and accessible introduction to the basic computations behind standard SEM analyses. Using two well-known example datasets, we show how to manually reproduce key results such as parameter estimates, standard errors, and fit measures using simple R scripts. The focus is on clarity and understanding rather than speed or efficiency. We hope that by following this tutorial, readers will gain a better grasp of how SEM works "under the hood," and be able to apply similar ideas in their own research.