Blocking the Back Door

Suppose you want to estimate the causal effect of a marketing email (X) on purchase (Y), and you know a confounder Z (customer's prior engagement level) affects both whether they receive the email and whether they purchase, forming the backdoor path X <- Z -> Y. Which adjustment strategy correctly estimates the causal effect of X on Y from observational data?

Drop Z from the model entirely since including confounders biases the estimate

Replace X with a random binary indicator to remove all correlation with Z

Condition on Z (e.g., stratify or regress on prior engagement level) when estimating the effect of X on Y

Condition on Y itself to remove its variance before regressing on X

Only look at customers who received the email, ignoring the control group