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]]>A colleague alerted me to the fact that “Dynamic treatment” has two interpretations. 1. A sequence of

actions do(X1 =x1), do(X2=x2)….do(Xk = xk) and,

2. a sequence of conditional actions, for example,

do(X1=x1 if Z1=z1, do X1=0 otherwise), do(X2=x2 if Z2….) etc…or, in general,do (Xi=gi(z) whenever Z=z), i = 1,2,3….

The second interpreation poses no special problem to

those concerned with the denominator of the post-treatment expression. Again, the rule is: decompose into a product and replace each pre-treatment mechanism by

its post-treatment expression. Thus, whereas in the case

of fixed treatment we had P'(X=1|Z=z)= 1, this term will become P*(X=1|Z=z), where P* is the specified treatment selection strategy in the post-treatment

world. This means that every treatment term in the original distribution will give rise to a fraction

P(X=1|Z=z)/P*(X=1|Z=z) in the denominator of the post-treatment distribution.

An example taken from process control is given on page 75

of Causality (2009). Again, the content of all denominators is dictated by quantity we wish to estimate,

and not by the technique we adapt to estimate it.

Identification conditions for partially observed confounders are discussed on page 354 of Causality.

Request to future bloggers: please use the term

“conditional plan” or “conditional treatments” when referring to this interpretation

of “dynamic treatment”.

add Nz to the right hand side of th second displayed

equation.

Likewise,add Z1 to the secon term in the denominator

of the 3rd displayed equation.

Also, change P(X=1, Z=z) into P(X=1 | Z=z). ]]>