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Fix cite typo
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guides/research-questions/effect-types_en.qmd

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$$\Delta = log\frac{E(Y_i(1))}{1-E(Y_i(1))} - log\frac{E(Y_i(0))}{1-E(Y_i(0))}$$
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@freemand_2008b shows that the coefficient from a logistic regression adjusting for covariates in a randomized experiments produces biased estimates of this causal effect. The basic intuition for Freedman’s argument comes from the fact that taking the log of averages is not the same as taking the average of logs and so the treatment coefficient estimated from a logistic regression conditioning on covariates will not provide a consistent estimator of log-odds of success. Instead, Freedman recommends taking the predicted probabilities varying subjects’ treatment status but maintaining their observed covariate profiles to produce a consistent estimator of the log-odds.
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@freedman_2008b shows that the coefficient from a logistic regression adjusting for covariates in a randomized experiments produces biased estimates of this causal effect. The basic intuition for Freedman’s argument comes from the fact that taking the log of averages is not the same as taking the average of logs and so the treatment coefficient estimated from a logistic regression conditioning on covariates will not provide a consistent estimator of log-odds of success. Instead, Freedman recommends taking the predicted probabilities varying subjects’ treatment status but maintaining their observed covariate profiles to produce a consistent estimator of the log-odds.
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## Attributable Effects
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