### Indirect effects in nonlinear models

(Quoted from Jacques A. Hagenaars' comments on my SMR paper (Pearl, 1998a), dated February 24, 2000. Full text accessible through http://www.knaw.nl/09public/rm/koster1.pdf.)

In general, researchers are interested in the nature and sizes of direct, total and indirect effects. In a way (but see below), Pearl shows how to compute direct and total effects in the general (nonparametric) model, but is silent about indirect effects. …. indirect effects do occupy an important place in substantive theories. Many social science theories `agree' on the input (background characteristics) and output (behavioral) variables, but differ exactly with regard to the intervening mechanisms. To take a simple example, we know that the influence of Education on Political Preferences is mediated through `economic status' (higher educated people get the better jobs and earn more money) and through a `cultural mechanism' (having to do with the contents of the education and the accompanying socialization processes at school). It is important what the causal directions (signs) of these two processes are and which one is the dominant one (at least in The Netherlands they did tend to go into different directions, one leading to a right wing preference, the other to a left wing). We need to know and separate the nature and consequences of these two different processes, that is, we want to know the signs and the magnitudes of the indirect effects. In the parametric linear version of structural equation models, there exists a `calculus of path coefficients' in which we can write total effects in terms of direct and several indirect effects. But this is not possible in the general nonparametric cases and not, e.g., in the loglinear parametric version. For systems of logit models there does not exist a comparable `calculus of path coefficients' as has been remarked long ago. However, given its overriding theoretical importance, the issue of indirect effects cannot be simply neglected.

In line with my own proposals (Hagenaars, 1993, see below), maybe something might be derived from collapsing tables over one but not the other intervening variable; formulated in terms of the `*do*-operator', maybe some assessment of indirect effects might be obtained by setting not only the `causal factor' (here: Education) to a particular value, but also one of the two intervening variables. Or maybe we must simply conclude that only given particular definitions (parameterizations) of causal effects it makes sense to talk about indirect effects, e.g., only if we take differences between distributions as our causal measure, but not when we use ratios.

Reference: Hagenaars, Jacques A., *Loglinear Models with Latent Variables,* Sage University Papers Series, Newbury Park, CA: Sage, 49–50, 1993.

Stimulated by Jacques A. Hagenaars' comments, I have given further thought to the issue of indirect effects, and I no longer subscribe to the opinion stated on page 165 of CAUSALITY (last paragraph of Section 5.4.2) that indirect effects lack intrinsic operational meaning because they cannot be isolated using the

do(x) operator. I have found that it is possible to define partial effects, confined to any given set of paths without controlling variables on the remaining paths. This permits the assessment of indirect effects in nonparametric models, as well as in nonlinear parametric models (e.g., the logit variety). Details can be viewed in R-273-U [ps] [pdf]. Best wishes,

========Judea Pearl

Comment by judea — February 21, 2007 @ 11:41 pm