Point Estimation and adjusting using Logistic Regression

Logistic regression is used when the outcome is dichotomous or binary (1 or 0):

Logistic regression is just a logit transformation of the outcome variable and the resulting regression remains linear so we can still use least squares estimation.

When adjusting for confounder in logistic regression an interesting concept arises. The regression equation has a logit outcome and if the predictor is binary the adjustment equation from the corrected treatment effect also has a logit outcome.

It can be derived that the Corrected Treatment Effect as an odds ratio (OR_CTE) is as follows:

OR_CTE = Exp(Coef* x/inverse-logit(b_z))

The inverse-logit function is (exp(b_z))/(1+exp(b_z)).

Exp means the term in brackets is exponentiated.

Coef means coefficient. The coefficient here is the coefficient of the predictor onto the outcome in the regression equation logit(y) = m_x*x + b_x where m is the coefficient.

x is the prevalence of exposure, which is merely the average of all x-values.

b_z is the constant of the adjustment equation:  logit(x) = m_z*z + b_z


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