Example of Causality

Defining Notation for Determining Causality

• As an example, let's say we want to know if providing schools with laptops will cause their students to receive better SAT scores

  • The control group will include schools without laptops
  • The treatment group will include schools with laptops

• Each variable has the following notation:

  • $i$ represents a given school

  • $t$ represents the group that school $i$ falls into

    • $t=0$ if the $i^{th}$ school isn't provided laptops
    • $t=1$ if the $i^{th}$ school is provided laptops

  • $Y$ represents the average SAT scores for school $i$

    • $Y^{0}$ are the average SAT scores for the control group
    • $Y^{1}$ are the average SAT scores for the treatment group

Illustrating $ATE$ with SAT Scores

• Theoretically, let's just imagine we're able to somehow capture data for each $i^{th}$ school in both a treatment and control group
• Again, this wouldn't be possible, but our data could look like this:

$i$	$t$	$Y$	$Y^{0}$	$Y^{1}$	$\delta$
School 1	$0$	$800$	$800$	$850$	$50$
School 2	$0$	$900$	$900$	$870$	$-30$
School 3	$1$	$1150$	$1100$	$1150$	$50$
School 4	$1$	$1400$	$1300$	$1400$	$100$

• In this case, we can calculate the population parameter $ATE$
• Specifically, it would be the following:

$$ATE = \frac{850+870+1150+1400}{4} - \frac{800+900+1100+1300}{4} = 1067 - 1025 = 42$$

• This indicates providing laptops to schools would increase SAT scores by $42$ points on average
• Thus, we see there isn't any real significant difference between the control group and treatment group

Illustrating $SDO$ with Test Scores

• In reality, customers can be in either the control or treatment group

  • For example, school $1$ can't actually participate in both groups at the same time
  • For school 1, $Y^{1}$ is a counterfactual
  • For school 4, $Y^{0}$ is his counterfactual
• As a result, our data actually looks like the following:

$i$	$t$	$Y$	$Y^{0}$	$Y^{1}$	$\delta$
School 1	$0$	$800$	$800$	null	null
School 2	$0$	$900$	$900$	null	null
School 3	$1$	$1150$	null	$1150$	null
School 4	$1$	$1450$	null	$1450$	null

• Notice, in almost all cases we can't calculate $ATE$
• Instead, we must estimate $ATE$ by calculating $SDO$
• Specifically, $SDO$ would be the following:

$$SDO = \frac{1150+1450}{2} - \frac{800+900}{2} = 1300 - 850 = 450$$

• Notice, this indicates providing laptops to schools would increase SAT scores by 450 points on average
• In this case, making an assumption about causality based on $SDO$ would be a mistake
• Again, $SDO$ is an estimator, so this difference is due to selection bias

Motivating Selection Bias in Causality

• Bias is what separates association from causation

• Selection bias could exist if the treatment group contains schools with more resources

  • For example, maybe the treatment group contains private schools
  • And, maybe the control group are underfunded
• Since bias exists, we can't draw any conclusions about causal effects using $SDO$, $ATU$, or $ATT$
• If there wasn't any bias, we could see the treatment group has much better SAT scores compared to the control group
• Specifically, we could see this by computing and comparing $ATU$ and $ATT$
• We can see $ATU$ and $ATT$ are the following:

$$ATT = \frac{100+50}{2} = 75$$ $$ATU = \frac{50-30}{2} = 10$$

• Notice, the $ATE$ is just a weighted average of $ATT$ and $ATU$

References

• Causal Inference Textbook
• Python Causality Handbook
• Comprehensive Causal Inference Textbook