Hypothesized mechanisms can be converted into hypothesized statistical models (probability distributions) by first translating the mechanism into a causal graph (a directed acyclic graph (DAG) in this chapter) and then translating the causal graph into the probability distribution that is generated from this graph. The operation of ‘d-separation’ in the causal graph is key. Given d-separation, we can use the same inferential logic as used in the controlled experiment to test hypothesized mechanisms using only observational data.

A dsep test uses the notion of d-separation to produce an inferential statistical test comparing observed data to the hypothesized mechanism (piecewise SEM). This involves obtaining a subset of d-separation claims that logically imply all others (the union basis set), obtaining the null probability of each of the (conditional) independence claims implied by the d-separation claims in this basis set, and combining them. Two ways of obtaining the null probabilities of each of these d-separation claims are explained: using regression slopes and using the generalized covariance statistic. These are implemented in the ‘piecewiseSEM’ and ‘pwSEM’ packages of R respectively. The rules for interpreting and manipulating the resulting path coefficients are explained.

This chapter introduces directed acyclic graphs (DAGs) as a way to represent multivariate probability distributions. DAGs help clarify the structure of probabilistic models and the dependencies among their variables and serve as a central tool in later chapters. Every DAG corresponds to a specific factorisation of a joint mass or density function into a product of conditional distributions. While a DAG encodes how the distribution breaks down into conditionals, it does not fully determine the distribution itself. Instead, it implies certain dependency constraints among variables. These constraints can be examined using the concept of d-separation, which allows us to infer conditional independence relationships directly from the graph.