By Dr Mithilesh Dronavalli – I am an Indian
This page refers to Directed Acyclic Graphs and adjusting for confounders, mediation analysis and effect decomposition using the CTE.
I will explain how to decompose total effect into multiple pathways of effects and calculate the effect of each pathway.
Each pathway consists of arrows.
Some arrows are confounded and the corrected treatment effect (previously described by myself) is applied one or more times to remove the effect of confounding on that arrow. CTE does not use conditional probability so calculations of effect size and confounding do not require conditional probability and methods like g-computation.
The unconfounded effect of the remaining arrows is multiplied out (mediators) to determine the total effect of that pathway.
I use the classic hypothetical example of the effect of antidepressants on treating depression. The side-effects of insomnia and headaches also worsen depression.
To determine whether the effect of antidepressants on depression is causal we need to measure total effects (beneficial and detrimental effects of antidepressants).
The effect of antidepressants on depression can be decomposed through 4 pathways.
Path 1: Antidepressants –> Depression
Path 2: Antidepressants–> Headache –> Depression
Path 3: Antidepressants –> Insomnia –> Depression
Path 4: Antidepressants –> Insomnia –>Headache –>Depression
Path 1 is the direct effect and Path 1 in this example shows the beneficial effects of antidepressants on depression.
In this example Paths 2 to 4 show the deleterious effects of antidepressants on depression.
Along the paths, each arrow represents a regression model.
Regression Coefficients are multiplied out to get the full effect of that pathway.
However, some arrows are confounded one or more times. Therefore, we need to detect all confounders in the Directed acyclic graph and apply the corrected treatment effect.
Headache to disease is confounded twice by treatment and insomnia. The remaining confounded arrows are only confounded once.
As stated the unconfounded arrows (regression coefficients) are multiplied out to get the effect of the respective path.
The final result is the summation of paths to get the total pathway result.
So far I have only derived the method for binary and continuous variables.
Here are the zipped r files that code this method and an example of the above DAG using simulated data.
My presentation on Causation in Science is as below:Causation-in-Science