Corrected Treatment Effect – To Adjust for multiple measured confounders

Adjusting in SWoP via the Corrected Treatment Effect

Single Confounder Corrected Treatment Effect

Corrected Treatment Effect (CTE) is an alternative way for adjusting for confounders without using multivariable regression. Here confounders are variables that predict both the predictor variable x and the outcome variable y. Multivariable regression for this purpose is to be avoided as the treatment effect is conditioned by the multivariable fit of that particular sample. This is problematic when the sample is not representative of the population.

CTE on the other hand calculates a factor for each confounder separately. This factor when multiplied by the treatment effect gives the CTE.

CTE is to be interpreted as the effect of x on y had the confounder been balanced for all values of X. When the confounder is balanced for all values of X then Z is no longer a confounder and this is akin to balancing confounding in a randomised control trial or a causal analysis.

The CTE is the treatment effect adjusted for that confounder. These factors of confounding for particular samples are actually values that can be used in other datasets where that confounder has not been measured. These factors can be compared between different datasets to create a relative measure for confounding between datasets. This is where CTE is particularly useful and valuable.

The CTE can also be used in other paradigms of statistics, including frequentist, Bayesian, bootstrapping based statistics. All that is required is to calculate a standard error for the CTE and a distribution with hypothesis testing.

\displaystyle y=m_xx+b_x

 

\displaystyle \hat{X}=m_zz+b_z

 

\displaystyle y=m_x\hat{X}+b_x

 

Here we use x-hat to approximate x. If z is not a predictor of x then we do not need to adjust for it. If the relationship of z onto x is not linear as per equation (2) then adaptation of the Corrected Treatment Effect technique is required and can be generalised with simple algebraic manipulation.

\displaystyle y=m_x(m_zz+b_z) + b_x

 

\displaystyle \frac{y-b_x}{m_zz+b_z}=m_x

To calculate the effect of predictor x (independent of confounder z) on outcome y,

we force mz=0 on the left-hand side and to balance this effect we multiply mx with az which is a correcting factor for the confounding imparted by z on x.

Then we have the effect of X on Y adjusted for Z:

 

\displaystyle \frac{(y-b_x)}{b_z}= a_zm_x : y_{x\perp z} = a_zm_x(X) + b_x _9b01817d6e3c5537_Documents_Office_Lens_7118_0628_Office_Lens

Multiple Confounders Product Method: Corrected Treatment Effect

To generalize this to the case of multiple confounders.

 

\displaystyle \hat{X} = m_qq+bq

 

\displaystyle y_{x\perp z}=a_zm_x(m_qq+b_q) + b_x

 

\displaystyle \frac{y_{x\perp z}-b_x}{m_qq+b_q}=a_zm_x

To calculate the effect of predictor x (independent of confounder z and q) on outcome y, we force mq=0 on the left hand side and to balance this effect we multiply mx with az to correct confounding by z on x (as above) and aq to correct confounding by as above.

Then we have the effect of X on Y adzusted for z and q as:

 

\displaystyle \frac{y_{x\perp z}-b_x}{b_q}=a_{q}a_{z}m_x

 

\displaystyle y_{x\perp z\perp q} = a_{qz}(X) + b_x

Similarly upon simplification it can be shown that :

\displaystyle a_z=\frac{X}{b_z}

 

and after adjusting for z, a_q is still also like a_z:

\displaystyle a_q=\frac{X}{b_q}

For estimation purposes:

\displaystyle y_{x\perp z\perp q} =\frac{\overline{(y-b_x)}^2}{m_xb_zb_q}X + b_x

Therefore the product method of the corrected treatment effect is symmetric regardless of which confounder you adjust first or second or in any order.

 

Multivariable Regression Method: Corrected Treatment Effect

There is a multivariable regression version of the Corrected Treatment Effect, which is surprisingly simple. The method regresses the x the dependent variable on all confounders. It can strictly be argued that any variable that predicts x can be used as a confounder in this setting as we are just getting the best prediction of x via X-hat.

 

\displaystyle y=m_xx+b_x

 

\displaystyle \hat{X}=m_{z1}z_1+m_{z2}z_2+m_{z3}{z3} + b_{z123}

 

\displaystyle y=m_x\hat{X}+b_x

 

\displaystyle \frac{y-b_x}{\hat{X}}=m_x

 

\displaystyle \frac{m_xx}{\hat{X}}=m_x

 

\displaystyle \frac{m_xx}{m_{z1}z_1+m_{z2}z_2+m_{z3}{z3} + b_{z123}}=m_x

In the above equation we force m{z1} and m{z2} and m{z3} etc… to equal 0 including any interaction variable coefficients on the left-hand side. To balance the equation we multiple the right-hand side with a{z123}

\displaystyle \frac{m_xx}{b_{z123}}=a_{z123}m_x

 

\displaystyle \frac{x}{b_{z123}}=a_{z123}

 

Therefore the corrected treatment effect for the dependent variable x onto y is a{z123}.

The crude effect of x onto y is mx from equation (10). The unconfounded effect of x onto y correcting for the confounders in equation (11) is a{z123} multiplied by mx.

\displaystyle y_{x\perp {z1,z2,z3}} = a_{z123}m_xx+b_x

 

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