Causal Analysis in Theory and Practice

From Dennis Lindley 

Equations (1.42) and (1.43) and the general issue of description by equations, still perplex me. It is incoherent to state both p(x|y) and p(y|x). (Try it with x and y binary, when these statements describe 4 values, whereas we know only 3 are needed for the joint distribution of x and y.) There are special cases, as with normal, linear regression, where the coherence is avoided. Generally I do not see how there can be two links between x and y.

Apart from the innate symmetricity of structural equation systems, the very definition for the conditional probability as well as Bayes's law of inverse probability unambiguously suggest the reversibility of cause-effect relationships. …
“A is a cause of B only when B can cause A.'' Amen.
(Quoted from Mr. Asha review of Causality, amazon.com, August 29, 2000)

(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.