Lecture 16 Back Doors

Recap

• We’ve now covered how to create causal diagrams
• (aka Directed Acyclic Graphs, if you’re curious what “dag”itty means)
• We simply write out the list of the important variables, and draw causal arrows indicating what causes what
• This allows us to figure out what we need to do to identify our effect of interest

Today

• But HOW? How does it know?
• Today we’ll be covering the process that lets you figure out whether you can identify your effect of interest, and how
• It turns out, once we have our diagram, to be pretty straightforward
• So easy a computer can do it!

The Back Door and the Front Door

• The basic way we’re going to be thinking about this is with a metaphor
• When you do data analysis, it’s like observing that someone left their house for the day
• When you do causal inference, it’s like asking how they left their house
• You want to make sure that they came out the front door, and not out the back door, not out the window, not out the chimney

The Back Door and the Front Door

• Let’s go back to this example

The Back Door and the Front Door

• We’re interested in the effect of IP spend on profits. That means that our front door is the ways in which IP spend causally affects profits
• Our back door is any other thing that might drive a correlation between the two - the way that tech affects both

Paths

• In order to formalize this a little more, we need to think about the various paths
• We observe that you got out of your house, but we want to know the paths you might have walked to get there
• So, what are the paths we can walk to get from IP.spend to profits?

Paths

• We can go Ip.spend -> profit
• Or IP.spend <- tech -> profit

The Back Door and the Front Door

• One of these paths is the one we’re interested in!
• Ip.spend -> profit is a front door path
• One of them is not!
• IP.spend <- tech -> profit is a back door path

Now what?

• Now, it’s pretty simple!
• In order to make sure you came through the front door…
• We must close the back door
• We can do this by controlling/adjusting for things that will block that door!
• We can close IP.spend <- tech -> profit by adjusting for tech

So?

• We already knew that we could get our desired effect in this case by controlling for tech.
• But this process lets us figure out what we need to do in a much wider range of situations
• All we need to do is follow the steps!
  • List all the paths
  • See which are back doors
  • Adjust for a set of variables that closes all the back doors!

Example

• How does wine affect your lifespan?

Paths

• Paths from wine to life:
• wine -> life
• wine -> drugs -> life
• wine <- health -> life
• wine <- income -> life
• wine <- health <- U1 -> income -> life
• wine <- income <- U1 -> health -> life
• Don’t leave any out, even the ones that seem redundant!

Paths

• Front doors/Back doors
• wine -> life
• wine -> drugs -> life
• wine <- health -> life
• wine <- income -> life
• wine <- health <- U1 -> income -> life
• wine <- income <- U1 -> health -> life

Adjusting

• By adjusting for variables we close these back doors
• If an adjusted variable appears anywhere along the path, we can close that path off
• Once ALL the back door paths are closed, we have blocked all the other ways that a correlation COULD appear except through the front door! We’ve identified the causal effect!
• This is “the back door method” for identifying the effect. There are other methods; we’ll get to them.

Adjusting for Health

• Front doors/Open back doors/Closed back doors
• wine -> life
• wine -> drugs -> life
• wine <- health -> life
• wine <- income -> life
• wine <- health <- U1 -> income -> life
• wine <- income <- U1 -> health -> life

Adjusting for Health

• Clearly, adjusting for health isn’t ENOUGH to identify
• We need to adjust for health AND income
• We haven’t covered how to actually control for multiple variables
• We won’t be focusing on it, but it’s important for us to be able to know what needs to be controlled for

Adjusting for Health and Income

• Front doors/Open back doors/Closed back doors
• wine -> life
• wine -> drugs -> life
• wine <- health -> life
• wine <- income -> life
• wine <- health <- U1 -> income -> life
• wine <- income <- U1 -> health -> life

How about Drugs?

• Should we adjust for drugs?
• No! This whole procedure makes that clear
• It’s on a front door path
• If we adjusted for that, that’s shutting out part of the way that wine DOES affect life

The Front Door

• In fact, remember, our real goal isn’t necessarily to close the back doors
• It’s to make sure you came through the front door!
• Sometimes (rarely), we can actually isolate the front door ourselves

The Front Door

• Imagine this version

The Front Door

• This makes it real clear that you shouldn’t control for drugs - that shuts the FRONT door! There’s no way to get out of your house EXCEPT through the back door!
• Note in this case that there’s no back door from wine to drugs
• And if we control for wine, no back door from drugs to life (let’s check this by hand)
• So we can identify wine -> drugs and we can identify drugs -> life, and combine them to get wine -> life!

The Front Door

• This is called the “front door method”
• Much less common than the back door method, but actually older
• So we’ll only cover it briefly
• Historical relevance! This is similar to how they proved that cigarettes cause cancer

Cigarettes and Cancer

Paths

• Front doors/Back doors
• cigs -> tar -> cancer
• cigs <- income -> cancer
• cigs <- health -> cancer

Paths

• Closing these back doors is the problem that epidemiologists faced
• They can’t just run an experiment!
• Problem: there are actually MANY back doors we’re not listing
• And sometimes we can’t observe/control for these things
• How can you possibly measure “health” well enough to actually control for it?

The Front Door Method

• So, noting that there’s no back door from cigs to tar, and then controlling for cigs no back door from tar to cancer, they combined these two effects to get the causal effect of cigs on life
• This is how we established this causal effect!
• Doctors had a decent idea that cigs caused cancer before, but some doctors disagreed
• And they had good reason to disagree! The back doors were VERY plausible reasons to see a cigs/cancer correlation other than the front door

Practice

• We want to know how X affects Y. Find all paths and make a list of what to adjust for to close all back doors

Practice Answers

• Front door paths: X -> Y, X -> E -> Y
• Back doors: X <- A -> Y, X <- B -> Y, X <- A <- U1 -> B -> Y, X <- B <- U1 -> A -> Y, X <- A <- C -> Y, X <- B <- U1 -> A <- C -> Y
• (that last back door is actually pre-closed, we’ll get to that later)
• We can close all back doors by adjusting for A and B.