Power equivalence in structural equation modelling

Implementing large-scale empirical studies can be very expensive. Therefore, it is useful to optimize study designs without losing statistical power. In this paper, we show how study designs can be improved without changing statistical power by defining power equivalence, a relation between structural equation models (SEMs) that holds true if two SEMs have the same power on a likelihood ratio test to detect a given effect. We show systematic operations of SEMs that maintain power, and give an algorithm that efficiently reduces SEMs to power-equivalent models with a minimal number of observed parameters. In this way, optimal study designs can be found without reducing statistical power. Furthermore, the algorithm can be used to drastically increase the speed of power computations when using Monte Carlo simulations or approximation methods.