Modifying disease by changing exposure or confounding.

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.

\displaystyle y_{x\perp z} = a_zm_xx+b_x


\displaystyle a_z = \frac{\bar{x}}{b_z}

The above formula can be rewritten as:

\displaystyle \hat{y_{x\perp z}} = \frac{(m_xx)(m_zz)}{b_z} + \hat{y}

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:


Below is a reference proof for deriving a_z.