Lecture 18 Closing Back Doors: Controlling

Recap

• We discussed how to draw a causal diagram
• How to identify the front and back door paths
• And how we can close those back door paths by controlling/adjusting in order to identify the front-door paths we want!
• And so we get our causal effect

Today

• Today we’re going to be going a little deeper into what it means to actually control/adjust for things
• And we’re also going to talk about times when controlling/adjusting makes things WORSE - collider bias!
• I’m going to just start saying “controlling”, by the way - “adjusting” is a little more accurate, but “controlling” is more common

Controlling

• Up to now, here’s how we’ve been getting the relationship between X and Y while controlling for W:

1. See what part of X is explained by W, and subtract it out. Call the result the residual part of X.
2. See what part of Y is explained by W, and subtract it out. Call the result the residual part of Y.
3. Get the relationship between the residual part of X and the residual part of Y.

• With the last step including things like getting the correlation, plotting the relationship, calculating the variance explained, or comparing mean Y across values of X

In code

df <- tibble(w = rnorm(100)) %>%
  mutate(x = 2*w + rnorm(100)) %>%
  mutate(y = 1*x + 4*w + rnorm(100))
cor(df$x,df$y)

## [1] 0.9450687

df <- df %>% group_by(cut(w,breaks=5)) %>%
  mutate(x.resid = x - mean(x),
         y.resid = y - mean(y))
cor(df$x.resid,df$y.resid)

## [1] 0.7768234

In Diagrams

• The relationship between X and Y reflects both X->Y and X<-W->Y
• We remove the part of X and Y that W explains to get rid of X<-W and W->Y, blocking X<-W->Y and leaving X->Y

More than One Variable

• It’s quite possible to control for more than one variable at a time
• Although we won’t be doing it much in this class
• A common way to do this is called multiple regression
• You can do it with our method too, but it gets tedious pretty quickly

More than One Variable

df <- tibble(w = rnorm(100),v=rnorm(100)) %>%
  mutate(x = 2*w + 3*v + rnorm(100)) %>%
  mutate(y = 1*x + 4*w + 1.5*v + rnorm(100))
cor(df$x,df$y)

## [1] 0.9286176

df <- df %>% group_by(cut(w,breaks=5)) %>%
  mutate(x.resid = x - mean(x),
         y.resid = y - mean(y)) %>%
  group_by(cut(v,breaks=5)) %>%
  mutate(x.resid2 = x.resid - mean(x.resid),
         y.resid2 = y.resid - mean(y.resid))
cor(df$x.resid2,df$y.resid2)

## [1] 0.7785035

Graphically