March 20, 2019

## Plan for Today:

### (1) Recap

• correlation
• two problems:
• confounding/bias
• random association

### (2) Solutions for Bias

• design-based vs. adjustment-based
• conditioning
• what is it?
• how does it work?

## Testing Causal Claims

1. We start with a causal claim

2. Turn claim into a causal theory
• causal logic, independent \(X\)/dependent \(Y\) variables
3. Turn causal logic and \(X\)/\(Y\) into hypotheses
• expect that potential outcomes of \(Y\) change with \(X\)
4. B/C of FPCI: imperfectly test hypotheses using correlation of observed values of \(X\) and \(Y\)

5. Infer causality if:
• assumptions about cases we compare let us ignore confounding/bias
• correlation unlikely to have occured by chance

## Problems with Correlation

#### Bias/Confounding

Equivalently:

1. Observed cases with different values of \(X\) have different potential outcomes of \(Y\)
2. Observed cases with different values of \(X\) have different values of \(W\), which is related to \(Y\)
3. There is a "backdoor" causal path from \(X\) to \(Y\)

#### Random Error

1. Correlation between \(X\) and \(Y\) is by chance and reflects no change in \(Y\) due to \(X\).

## Solutions to Problems

#### Bias/Confounding

2. design-based solutions

#### Random Error

1. Increase number of independent cases
2. Take courses in statistics

## Solutions to Bias

• Identify possible confounding variables (e.g. \(W, Z, V, U\))
• Measure these variables
• adjust correlation of \(X\) and \(Y\) by "conditioning" on confounding variables

### design-based

• Compare cases that, by assumption, are
• similar in terms of confounding variables \(W\)/ potential outcomes of \(Y\)
• exposed to \(X\) in a manner unrelated to \(W\)/potential outcomes of \(Y\)

Adjustment-based approaches start from the…

### Comparative Method:

Sometimes called the "method of difference" (via John Stuart Mill), this assesses whether \(X\) causes \(Y\) by…

• comparing two cases that are the same in all relevant respects, except for value of \(X\)
• assess for these two cases whether \(Y\) changes when \(X\) changes (correlation)

### Question:

What causes the spread of cholera?

### Causal claim:

Contaminated water causes cholera outbreaks

### Obvious?

19th Century London saw repeated outbreaks of cholera, with mass death

• Dominant view was that "miasmas" or bad air caused diseases like cholera