### Is the do(x) operator universal?

From **Bill Shipley, Universite de Sherbrooke, (Quebec) CANADA: **

In most experiments, the external manipulation consists of adding (or subtracting) some amount from *X* without removing pre-existing causes of *X*. For example, adding 5 *kg*/*h* of fertilizer to a field, adding 5 *mg/l* of insulin to subjects etc. Here, the pre-existing causes of the manipulated variable still exert effects but a new variable (*M*) is added.

… The problem that I see with the *do*(*x*) operator as a general operator of external manipulation is that it requires two things: (1) removing any pre-existing causes of *x* and (2) setting *x* to some value. This corresponds to some types of external manipulations, but not all (or even most) external manipulations. I would introduce an *add*(*x=n*) operator, meaning "*add*, external to the pre-existing causal process, an amount '*n*' of *x*''. Graphically, this consists of augmenting the pre-existing causal graph with a new edge, namely *M-n*–>*X*. Algebraically, this would consist of adding a new term –*n*– as a cause of *X*.

In many cases, your "additive intervention" represents indeed the only way we can intervene on a variable

X. In fact, the general notion of intervention (Causality, page 113) involves replacing the equation ofXby any other equation that fits the circumstances, not necessarily a constantX=x.What you are proposing corresponds to replacing the old equation of

X,x=f(pa_{X}) by a new equation:x=f(pa) + 1 This replacement is usually treated under the heading "instrumental variables", since it is equivalent to writing_{X}x=f(pa) +_{X}I(whereIis an instrument) and varyingIfrom 0 to 1.There are three points to notice: 1. The additive manipulation CAN be represented in the

do( ) framework — we merely apply thedo( ) operator to the instrumentI, and not toXitself. This is a different kind of manipulation that needs to be distinguished fromdo(x) because, as you noticed, the effect on y would be different.2. Scientists working with instrumental variables (e.g., epidemiologists) are not satisfied with estimating the effect of the instrument on

Y, but are trying hard to estimate the effect ofXitself. The former is known as "the effect of intention to treat" the latter "the effect of treatment" (seeCausality, page 261).3. Consider the loopy example where LISREL fails

y=bx+e_{1}+I,x=ay+e_{2}. If we interpret "total effects" as the response ofYto a unit change of the instrumentI, then LISREL's formula obtains: The effect ofIonYisb/(1-ab) However, if we adhere to the notion of "per unit change inX", as opposed to "per unit change in an instrument ofX", we get back thedo-formula. The effect ofXonYisb, notb/(1-ab), even though the manipulation is done through an instrument. In other words, we changeIfrom 0 to 1 and observe the changes inXand inY; if we divide the change inYby the change inX, we getb, notb/(1-ab).To summarize: Yes, additive manipulation is sometimes useful to model, normally it is done through instrumental variables, and we still need to distinguish between the effect of the instrument and the effect of

X. The former is not stable (Causality, page 261) the latter is. Lisrel's formula corresponds to the effect of an instrument, not to the effect ofX.Best wishes,

========Judea Pearl

Comment by judea — December 20, 2000 @ 12:00 pm