Causation- Definite elimination

For possible and definite elimination graphically we are using the same images as for exacerbating factor. The difference is in the relationships. I have decided to take definite elimination first, much as I took definite cause before possible cause.

For a factor, Z, to be a definite elimination factor the following must be true:

P(YX and Z) = 0 and 0<P(YX)<=1

In other words, the probability of Y given X and Z is 0 (certain to not happen - no effect) when the probability of Y given just X is at least a probable cause, but can also be a definite cause (without Z). Remember, when we discussed definite cause we were dealing with only X and Y.

Again there are three graphical representations of the possible causal models giving rise to definite elimination. They are the same as with an exacerbating factor.

The first form simply requires that P(YZ) = 0 and that this supersedes any effect of X on Y so that P(YX and Z) = 0ExacFactor3
For the second form we continue to require that P(YZ) = 0 and that this supersedes any effect of X on Y so that P(YX aExacFactor1nd Z) = 0; we add the additional requirement that 0<P(ZX)<1 (possible cause) because if P(ZX) = 1 or P(ZX)=0 then X would simply exert its influence on Y through the path X, Z, Y and since that path is definite and results in no effect (no Y) then the path X,Y is no longer a possibility (if you go back up the definition of definite elimination it will make sense why this causal model would, under those circumstances not depict Z as a definite elimination).
For the third form we are back to classical confounding. To be a definite elimination, P(YZ) = 0 and 0<P(YX)<=1, but what about the P(XZ)?   There are no limitations on P(XZ) for definite elimination.ExacFactor2 For example, if P(XZ)>0 then Z is  a definite eliminator that eliminates effect so long as P(YZ) = 0 and P(YX and Z) = 0,
If P(XZ)=0 then Z is still a definite eliminator, but the graph can be changed, the connection Z, Y can be removed as it is inconsequential. But we still have the situation where Z is eliminating the effect, it is doing so by eliminating the cause. This is depicted below. Of course this is what we hope for with treatments. We hope to identify the cause and then come up with a treatment that eliminates the cause, we then might measure the cause (as an effect) and the downstdef_elim4ream effect (Y).
The introduction of a 4th graphic raises the question in my mind now about whether this new graphic also can represent an exacerbating factor. My initial reaction is that it cannot because the impact of Z on X (if possible or definite causal) with  a possible or definite causal relationship between X and Y simply leads to a causal chain. The definition of an exacerbating factor is: P(YX and Z) > P(YX) and that would not be possible with the 4th graphic as Z does not directly cause Y so all of Z’s effects are through X.

In the final post on this 5 part series on the 5 types of cause I will take on possible elimination. This will require me to consider the probabilities I have been using thus far in order to distinguish possible elimination from exacerbation.

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