What you are proposing corresponds to replacing the old equation of *X*, * x* = *f*(*pa*_{X}) by a new equation: * x* = *f*(*pa _{X}*) + 1 This replacement is usually treated under the heading "instrumental variables", since it is equivalent to writing

There are three points to notice: 1. The additive manipulation CAN be represented in the *do*( ) framework — we merely apply the *do*( ) operator to the instrument *I*, and not to * X* itself. This is a different kind of manipulation that needs to be distinguished from *do*(*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 of *X* itself. The former is known as "the effect of intention to treat" the latter "the effect of treatment" (see *Causality*, 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 of *Y* to a unit change of the instrument *I*, then LISREL's formula obtains: The effect of * I* on * Y* is *b*/(1*-ab*) However, if we adhere to the notion of "per unit change in *X*", as opposed to "per unit change in an instrument of *X*", we get back the *do*-formula. The effect of * X* on * Y* is *b*, not *b*/(1-*ab*), even though the manipulation is done through an instrument. In other words, we change * I* from 0 to 1 and observe the changes in * X* and in *Y*; if we divide the change in * Y* by the change in *X*, we get *b*, not *b*/(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 of *X*.

Best wishes,

========Judea Pearl