April 3rd 2018: By Dr Mithilesh Dronavalli
Please read the section on corrected treatment effect before reading this section.
Let a disease be quantified by a continuous variable (y) and let it be predicted by exposure(x), where the relationship between exposure(x) and outcome(y) is governed by a confounder (z).
Then intervening on confounder(z) and exposure(x) or changing the relationship(m_x) between disease(y) and exposure(x) or changing the relationship(m_z) between exposure(x) and confounder(z) will change the level of disease (y) by the following formula.
The above formula can be rewritten as:
So by changing the level of the exposure (x) or the relationship(m_x) between the exposure(x) and the disease(y) or by changing the elvel of the confounder(z) or the relationship(m_z) between the confounder(z) and the exposure(x) we get a new level of disease (y) given by the driectly above equation.
Below is the proof for the above equation:ModifyingY
Below is a reference proof for deriving a_z._9b01817d6e3c5537_Documents_Office_Lens_7118_0628_Office_Lens