Causation- Possible elimination

Possible elimination will have the same graphical forms as exacerbating factor and definite elimination. A possible eliminator (Z) will possible eliminate an effect.

P(YX and Z) < P(YX)
As a reminder, a definite eliminator (Z) is when P(YX and Z) = 0 and 0<P(YX) <=1.
To clarify and distinguish an exacerbating factor from a possible eliminator, an exacerbating factor (Z) is when: P(YX and Z) > P(YX)
Ok, back to a possible eliminator (Z): P(YX and Z) < P(YX)
A condition that is necessary for a possible eliminator is: 0<P(YX) <=1 (that is X is a possible or definite cause of Y). If P(YX)=0 then there is nothing for Z to eliminate.

The first graphical form simply requires the above basic condition for a possible eliminator:

ExacFactor3

P(YX and Z) < P(YX)

and

0<P(YX) <=1

The second graphical form again requires the basic conditions above

ExacFactor1

P(YX and Z) < P(YX)
0<P(YX) <=1
I also think we need the added condition that P(ZX) < 1
If P(ZX) = 1 then whenever X we would have Z so Z would always impact Y and I think that would boil down to : P(YX and Z) = P(YX) because P(YX) would be impossible to separate from P(YX and Z) due to the definite cause X on Z. If anyone disagrees, please let me (on any of these as they are all a work in progress).
The third graphic (classical confounding) once again requires the basic condition for a possible ExacFactor2eliminator P(YX and Z) < P(YX)   and     0<P(YX) <=1
But we also need to consider P(XZ). There are three possibilities, of the three it seems as though if P(XZ) = 0 makes things interesting but I think Z remains a possible eliminator. If P(YZ) = .2 and the P(YX) = .8, and the P(YX and Z) = .5 then Z is a possible eliminator. If P(XZ) = o then Z blocks X and then the probability of Y when Z is alone is equal to when X and Z are present: P(YX and Z) = P(YZ) (Since Z blocks X). But, since P(YX and Z) < P(YX), and since technically 0<P(YX)<=1, Z remains a possible eliminator even if P(XZ) = 0

The fourth graphic used for definite elimination is therefore not warranted for possible elimination.

In summary, when considering cause and effect and limiting yourself to two variables (cause X and effect Y)(factors, states, conditions, etc) there are basically two possibilities: possible cause or definite cause. When just adding a single variable (Z) we then immediately increase by 3 possibilities for that 3rd variable: exacerbating factor, definite eliminating factor, possible eliminating factor. Whether the factor Z is exacerbating or eliminating has to do with what happens to the effect (Y). Exacerbation means increasing the probability of Y, whereas eliminating means decreasing the probability of Y. I would consider this understanding, and the use of the graphics to reason through these possible situations to be foundational to the use of DAGs as causal models to understand the causal networks at work for clinical practice. As you can imagine, things get complicated quickly, and what we know about the causal networks is encoded in a causal model and that is used in practice. Getting to know the causal network is what we are trying to learn when doing research - be is discovering the causal connections or integrating existing knowledge about connections.

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