# Causality - [[correlation|Correlation]] doesn't tell us enough about causality. - Solutions - Observe the same individual at different points in time. - Observe two nearly identical individuals and treat differently. $ \widehat{ATE} \approx \mathbb E_N[y_i \mid t_i = 1] - \mathbb E_N[y_i \mid t_i = 0] - \text{Selection Bias} $ $ \text{Selection Bias} = \mathbb E[Y(1) \mid T = 0] - \mathbb E[Y(0) \mid T = 0] $ - Randomized Controlled Experiment (RCE) $ Y_i \approx \hat\beta_0 + \hat\beta_1T_i $ - [[linear-regression|Regression model]] - $\hat\beta_0$: average outcome in the control group - $\hat\beta_0 + \hat\beta_1$: average outcome in the control group - $\hat\beta_1$: the _Average Treatment Effect_ $\widehat{\mathrm{ATE}}$ - _Propensity score_: represents the conditional probability of receiving a particular treatment given a set of observed covariates. ## Procedures - Compute a _propensity_ score model using [[logistic-regression|logistic regression]], which predicts receiving the treatment based on observed covariates. - Calculate propensity score of each observation, match the ones with similar scores, can be based on [[knn|KNN]] or similar methods. - Use the matched sample to make a fair comparison between treated and non-treated and examine the impact. -