# Selection On Observables

``````df = data.table(W = as.integer(1:200>100))
df[, X := .5 + 2 * W + rnorm(.N)][,
Y := -.5*X + 4*W + 1 + rnorm(.N)][, time := "1"][,
`:=`(mean_X = mean(X), mean_Y = mean(Y)), by = W]

# df <- data.frame(W = as.integer((1:200>100))) %>%
#   mutate(X = .5+2*W + rnorm(200)) %>%
#   mutate(,time="1") %>%
#   group_by(W) %>%
#   mutate(mean_X=mean(X),mean_Y=mean(Y)) %>%
#   ungroup()
# %%
#Calculate correlations
before_cor <- paste("1. Start with raw data. Correlation between X and Y: ",round(cor(df\$X,df\$Y),3),sep='')
after_cor <- paste("6. Analyze what's left! Correlation between X and Y controlling for W: ",
round(cor(df\$X-df\$mean_X,df\$Y-df\$mean_Y),3),sep='')

#Add step 2 in which X is demeaned, and 3 in which both X and Y are, and 4 which just changes label
dffull <- rbind(
#Step 1: Raw data only
df %>% mutate(mean_X=NA,mean_Y=NA,time=before_cor),
df %>% mutate(mean_Y=NA,time='2. Figure out what differences in X are explained by W'),
#Step 3: X de-meaned
df %>% mutate(X = X - mean_X,mean_X=0,mean_Y=NA,time="3. Remove differences in X explained by W"),
#Step 4: Remove X lines, add Y
df %>% mutate(X = X - mean_X,mean_X=NA,time="4. Figure out what differences in Y are explained by W"),
#Step 5: Y de-meaned
df %>% mutate(X = X - mean_X,Y = Y - mean_Y,mean_X=NA,mean_Y=0,time="5. Remove differences in Y explained by W"),
#Step 6: Raw demeaned data only
df %>% mutate(X = X - mean_X,Y = Y - mean_Y,mean_X=NA,mean_Y=NA,time=after_cor))

p = ggplot(dffull,aes(y=Y,x=X,color=as.factor(W)))+geom_point()+
geom_vline(aes(xintercept=mean_X,color=as.factor(W)))+
geom_hline(aes(yintercept=mean_Y,color=as.factor(W)))+
guides(color=guide_legend(title="W"))+
labs(title = 'The Relationship between Y and X, Controlling for a Binary Variable W \n{next_state}')+
transition_states(time,
transition_length=c(6,16,6,16,6,6),
state_length=c(50,22,12,22,12,50),
wrap=FALSE)+
ease_aes('sine-in-out')+