Computing direct and total effects of variables

Abstract

One of the lessons to be drawn from structural equation models and path analysis is that an independent variable can have both direct and indirect effects on a dependent variable. In other words, some of the effects of an independent variable on a dependent variable may be transmitted through intervening variables. Consequently, the partial regression coefficient for an independent variable in a multiple regression equation may not capture all of its effects on the dependent variable. In order to understand the difference between direct and total effects, let us consider the example of the simple structural equation model that explains income in terms of education and occupation. The relationship between these three variables was represented by the following set of structural equations, where the variables are expressed in standard form:

$$ \begin{gathered} z_2 = {\mathbf{ }}z_3 b_{23}^* {\mathbf{ }} + {\mathbf{ }}e_2 \hfill \\ z_1 = {\mathbf{ }}z_3 b_{13.2}^* {\mathbf{ }} + {\mathbf{ }}z_2 b_{12.3}^* + {\mathbf{ }}e_1 \hfill \\ \end{gathered} $$

where variable Z₃ is education, variable Z₂ is occupation, and variable zi is education. It is evident from these structural equations that a one unit change in variable Z₃ will produce a change in variable Z₂ and a change in variable Z₃. In turn, the change in variable Z₂ produced by a change in variable Z₃ will produce a further change in variable zi.