Merge pull request #219 from joonho112/master

Upload Causal IV case study in education folder

education/causal_iv_one-sided/references.bib

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+@book{imbens2015causal,
+  title={Causal inference in statistics, social, and biomedical sciences},
+  author={Imbens, Guido W and Rubin, Donald B},
+  year={2015},
+  publisher={Cambridge University Press}
+}
+
+@article{imbens1997bayesian,
+  title={Bayesian inference for causal effects in randomized experiments with noncompliance},
+  author={Imbens, Guido W and Rubin, Donald B},
+  journal={The annals of statistics},
+  pages={305--327},
+  year={1997},
+  publisher={JSTOR}
+}
+
+@article{sommer1991estimating,
+  title={On estimating efficacy from clinical trials},
+  author={Sommer, Alfred and Zeger, Scott L},
+  journal={Statistics in medicine},
+  volume={10},
+  number={1},
+  pages={45--52},
+  year={1991},
+  publisher={Wiley Online Library}
+}
+
+@article{feller2016compared,
+  title={Compared to what? Variation in the impacts of early childhood education by alternative care type},
+  author={Feller, Avi and Grindal, Todd and Miratrix, Luke and Page, Lindsay C and others},
+  journal={The Annals of Applied Statistics},
+  volume={10},
+  number={3},
+  pages={1245--1285},
+  year={2016},
+  publisher={Institute of Mathematical Statistics}
+}
+
+@article{page2015principal,
+  title={Principal stratification: A tool for understanding variation in program effects across endogenous subgroups},
+  author={Page, Lindsay C and Feller, Avi and Grindal, Todd and Miratrix, Luke and Somers, Marie-Andree},
+  journal={American Journal of Evaluation},
+  volume={36},
+  number={4},
+  pages={514--531},
+  year={2015},
+  publisher={Sage Publications Sage CA: Los Angeles, CA}
+}
+
+@article{gelman1992inference,
+  title={Inference from iterative simulation using multiple sequences},
+  author={Gelman, Andrew and Rubin, Donald B and others},
+  journal={Statistical science},
+  volume={7},
+  number={4},
+  pages={457--472},
+  year={1992},
+  publisher={Institute of Mathematical Statistics}
+}

education/causal_iv_one-sided/stan/cace_with_exclusion.stan

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+data {
+  int<lower=1> N;                    // Sample size N 
+  array[N] int<lower=0, upper=1> Z;  // Treatment assigned Z
+  array[N] int<lower=0, upper=1> W;  // Treatment received W
+  array[N] int<lower=0, upper=1> Y;  // Outcome Y
+}
+
+parameters {
+  // Population probability of being a complier
+  real<lower=0, upper=1> pi_c;
+
+  // Probabilities for the binomial outcome distributions
+  real<lower=0, upper=1> eta_c0;
+  real<lower=0, upper=1> eta_c1;
+  real<lower=0, upper=1> eta_n;
+}  
+
+model {
+  // Define local variables for efficiency
+  real log_pi_c = log(pi_c);
+  real log1m_pi_c = log1m(pi_c);
+
+  // Prior for Complier probability
+  // implicit prior: pi_c ~ Unif(0, 1)
+
+  // Priors for outcome model parameters
+  eta_c0 ~ beta(2, 2);  
+  eta_c1 ~ beta(2, 2);  
+  eta_n ~ beta(2, 2); 
+
+  // Likelihood
+  for(n in 1:N){
+
+    // Complier (assigned to treatment)
+    if (Z[n] == 1 && W[n] == 1){
+      target += log_pi_c + bernoulli_lpmf(Y[n] | eta_c1) ;
+    }
+
+    // Never-taker (assigned to treatment)
+    else if (Z[n] == 1 && W[n] == 0){
+      target +=  log1m_pi_c + bernoulli_lpmf(Y[n] | eta_n);
+    }
+
+    // Complier or Never-taker (assigned to control)
+    else if (Z[n] == 0 && W[n] == 0){
+      target += log_mix(
+        pi_c,                           // Complier probability
+        bernoulli_lpmf(Y[n] | eta_c0),  // Complier
+        bernoulli_lpmf(Y[n] | eta_n)    // Never-taker
+      );  
+    }
+  }
+}
+
+generated quantities {
+  // Superpopulation complier average causal effect (CACE)
+  // in per-1000 units
+  real CACE = (eta_c1 - eta_c0) * 10^3;
+}
+
+
+

education/causal_iv_one-sided/stan/cace_without_exclusion.stan

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+data {
+  int<lower=1> N;                    // Sample size N 
+  array[N] int<lower=0, upper=1> Z;  // Treatment assigned Z
+  array[N] int<lower=0, upper=1> W;  // Treatment received W
+  array[N] int<lower=0, upper=1> Y;  // Outcome Y
+}
+
+parameters {
+  // Population probability of being a complier
+  real<lower=0, upper=1> pi_c;
+
+  // Probabilities for the binomial outcome distributions
+  real<lower=0, upper=1> eta_c0;
+  real<lower=0, upper=1> eta_c1;
+  real<lower=0, upper=1> eta_n0;
+  real<lower=0, upper=1> eta_n1;
+} 
+
+model {
+  // Define local variables for efficiency
+  real log_pi_c = log(pi_c);
+  real log1m_pi_c = log1m(pi_c);
+
+  // Prior for Complier probability
+  // implicit prior: pi_c ~ Unif(0, 1)
+
+  // Priors for outcome model parameters
+  eta_c0 ~ beta(2, 2);  
+  eta_c1 ~ beta(2, 2);  
+  eta_n0 ~ beta(2, 2);
+  eta_n1 ~ beta(2, 2);
+
+  // Likelihood
+  for(n in 1:N){
+
+    // Complier (assigned to treatment)
+    if (Z[n] == 1 && W[n] == 1){
+      target += log_pi_c + bernoulli_lpmf(Y[n] | eta_c1) ;
+    }
+
+    // Never-taker (assigned to treatment)
+    else if (Z[n] == 1 && W[n] == 0){
+      target +=  log1m_pi_c + bernoulli_lpmf(Y[n] | eta_n1);
+    }
+
+    // Complier or Never-taker (assigned to control)
+    else if (Z[n] == 0 && W[n] == 0){
+      target += log_mix(
+        pi_c,                            // Complier probability
+        bernoulli_lpmf(Y[n] | eta_c0),   // Complier
+        bernoulli_lpmf(Y[n] | eta_n0)    // Never-taker
+      );
+    }
+  }
+}
+
+generated quantities {
+  // Super-population average causal effects 
+  real CACE = (eta_c1 - eta_c0) * 10^3;
+  real NACE = (eta_n1 - eta_n0) * 10^3;
+}
+
+