March 19, 2018

## Midterm performance:

Original Average $$\approx 60$$%

Similar across tutorial sections

## Midterm scaling:

$ScaledPoints = (MidtermPoints - 27.4) \cdot 0.715 + 32.6$ Original Average: $$59.9$$% New Average: $$70.9$$%

It is possible

## What is the problem?

### Correlation as clue to Causality

If $$X \rightarrow Y$$:

• Values of $$X$$ and $$Y$$ should move together
• Values of $$X$$ and $$Y$$ should be correlated
• This correlation should not be by chance

### A hiccup:

It is possible that $$X,Y$$ are correlated, without causation

## What is the problem?

i $$Y_i^0$$ $$Y_i^1$$ $$Y_i^1 - Y_i^0$$ $$X_i$$ $$W_i$$
1 5 9 4 1 1
2 4 8 4 1 1
3 3 7 4 0 0
4 2 6 4 0 0

## What is the problem?

True causal effect of $$X$$: $$4$$.

Apparent causal effect of $$X$$:

$\frac{9+8}{2} - \frac{3+2}{2} = \frac{12}{2} \neq 4$

## What is the problem?

i $$Y_i^0$$ $$Y_i^1$$ $$Y_i^1 - Y_i^0$$ $$X_i$$ $$W_i$$
1 5 9 4 1 1
2 4 8 4 1 1
3 3 7 4 0 0
4 2 6 4 0 0

## What is the problem?

### $$W$$ is related to $$X$$ and $$Y$$

1. $$X = 0$$ when $$W = 0$$. $$X = 1$$ when $$W = 1$$
• perfect correlation
2. $$Y \approx 4.5$$ when $$W = 0$$. $$Y \approx 6.5$$ when $$W = 1$$
• partial correlation

### $$X$$ causes $$Y$$…

but $$W$$ makes effect of $$X$$ look too big

$$W$$ is a confounding variable

## What is the problem?

### Restatement:

We observe $$X,Y$$ correlated…

• … but additional factors $$W_1 \ldots W_n$$ affect $$X$$ and $$Y$$
• So correlation is biased.
• Could be entirely spurious (no true causal effect of $$X$$ on $$Y$$)
• Or different from true effect of $$X$$. Observed effect too big/too small
• These other factors $$W_1 \ldots W_n$$ are called confounding variables

## What can we do?

Comparative Method:

If causal claim that $$X \rightarrow Y$$, then:

We generate empirical prediction:

If we observe two cases to be the same in all relevant respects except for value of $$X$$, then we should observe that the two cases differ in the value of $$Y$$

1. We don't need cases to be identical on all attributes
2. They need to be the identical on attributes related to both cause ($$X$$) and outcome ($$Y$$)
• In short: identical on factors that affect $$Y$$ and are related to or affect $$X$$

## What does comparative method do?

1. Examines whether $$Y$$ changes when $$X$$ changes
2. Within pairs of cases where confounding variables held constant

### Can we generalize this?

Comparative method

• But works with only a few cases, patterns by chance?

Correlation

• Correlation uses many cases, can judge statistical significance
• But does not address confounding/bias

## Solutions

Two approaches to make correlation work like the comparative method

• Identify possible confounding variables (e.g. $$W$$)
• Measure these variables
• explicitly adjust our correlation between $$X$$ and $$Y$$
• "conditioning" on confounding variables
2. design-based
• Carefully choose cases for comparison
• Structure of comparison accounts for many or all confounding variables
• Kind of comparison permits unconditional comparison, given assumptions.

## Solutions: Examples

### Design-based:

An experiment is design-based solution:

• We compare a "treated" group to an "untreated" group
• We test using unadjusted correlation of treatment and outcome in these groups
• We have good reason to believe that correlation implies causation:
• Not because we explicitly identified and accounted for confounding variables
• Because design (random assignment of treatment) eliminates confounding
• Wouldn't work if we compared randomly assigned treatment group non-random control

## Solutions: Examples

This extrapolates comparative method to correlation.

The goal is to:

1. observe confounding variables
2. examine the correlation of $$X$$ and $$Y$$…
3. holding confounding variables constant
• ceteris parabis, "all else being equal"

conditioning: the process of holding other variables constant while looking at relationship between $$X$$ and $$Y$$.

• Remaining correlation between $$X$$ and $$Y$$ after conditioning on other variables cannot result from confounding by those variables

## Solutions: Examples

It is like:

• many applications of comparative method simultaneously
• correlation of $$X$$ and $$Y$$ within groups where other variables $$W_1 \ldots W_k$$ are held constant
• Change in $$X$$ and $$Y$$ cannot be due to change in variables we condition on because these other variables are held constant

It is not this process EXACTLY, but similar

## Conditioning: Example

### Question

Does UN peace-keeping prevent re-occurrence of civil war?

### Causal Theory:

Exposure to UN peace-keeping at the end of a conflict causes countries to experience longer periods of peace

### Test?

Correlation is insufficient. Many variables may cause UN intervention AND ability to have durable peace

• Intensity of conflict (Deaths), Length of conflict, ethnic diversity, population, mountainous terrain, size of armed forces, democracy

## Conditioning: Example

### Test

Look at correlation between Peacekeeping and Peace within groups of cases that have same values of:

• Violence, War length, Diversity, Terrain, Army Size, Democracy

DOES NOT eliminate the causal link between confounding variables and peacekeeping by design. We adjust our correlation to account for confounders.

Example (1)

$$Peace_i^{UN}$$ $$Peace_i^{noUN}$$ $$UN_i$$ $$WarIntense_i$$ $$Peace_i^{Obs}$$ $$Effect_i$$
treated 20 10 1 0 $$\mathbf{20}$$ 10
treated 20 10 1 0 $$\mathbf{20}$$ 10
treated 15 5 1 1 $$\mathbf{15}$$ 10
untreated 20 10 0 0 $$\mathbf{10}$$ 10
untreated 15 5 0 1 $$\mathbf{5}$$ 10
untreated 15 5 0 1 $$\mathbf{5}$$ 10

## Conditioning: Example

What is the true effect of peace-keeping?

$\frac{10 + 10 + 10 + 10 + 10 + 10}{6} = 10$

$$10$$ years more peace.

What is the effect we see with unadjusted or unconditional correlation?

(Average in peacekeeping - Average in no-peacekeeping)

Example (1)

$$Peace_i^{UN}$$ $$Peace_i^{noUN}$$ $$UN_i$$ $$WarIntense_i$$ $$Peace_i^{Obs}$$ $$Effect_i$$
treated 20 10 1 0 $$\mathbf{20}$$ 10
treated 20 10 1 0 $$\mathbf{20}$$ 10
treated 15 5 1 1 $$\mathbf{15}$$ 10
untreated 20 10 0 0 $$\mathbf{10}$$ 10
untreated 15 5 0 1 $$\mathbf{5}$$ 10
untreated 15 5 0 1 $$\mathbf{5}$$ 10

## Conditioning: Example

$\frac{20 + 20 + 15}{3} - \frac{10 + 5 + 5}{3} = \frac{35}{3} \neq 10$

We have upward bias: it looks like peace-keeping causes more peace than it does

### WHY?

$$Peace_i^{UN}$$ $$Peace_i^{noUN}$$ $$UN_i$$ $$WarIntense_i$$ $$Peace_i^{Obs}$$ $$Effect_i$$
treated 20 10 $$\mathbf{1}$$ $$\mathbf{0}$$ 20 10
treated 20 10 $$\mathbf{1}$$ $$\mathbf{0}$$ 20 10
treated 15 5 1 1 15 10
untreated 20 10 $$\mathbf{0}$$ $$\mathbf{0}$$ 10 10
untreated 15 5 0 1 5 10
untreated 15 5 0 1 5 10
$$Peace_i^{UN}$$ $$Peace_i^{noUN}$$ $$UN_i$$ $$WarIntense_i$$ $$Peace_i^{Obs}$$ $$Effect_i$$
treated $$\mathbf{20}$$ $$\mathbf{10}$$ 1 $$\mathbf{0}$$ 20 10
treated $$\mathbf{20}$$ $$\mathbf{10}$$ 1 $$\mathbf{0}$$ 20 10
treated 15 5 1 1 15 10
untreated $$\mathbf{20}$$ $$\mathbf{10}$$ 0 $$\mathbf{0}$$ 10 10
untreated 15 5 0 1 5 10
untreated 15 5 0 1 5 10

## Conditioning: Example

1. Intense wars $$\xrightarrow{reduces}$$ Peacekeeping
2. Intense wars $$\xrightarrow{reduces}$$ Durable Peace

Result?

• Upward bias in effect of peacekeeping on durable peace

### Solution:

Conditioning! Calculate effect of peacekeeping where Intense War is $$1$$

$$Peace_i^{UN}$$ $$Peace_i^{noUN}$$ $$UN_i$$ $$WarIntense_i$$ $$Peace_i^{Obs}$$ $$Effect_i$$
treated 20 10 1 0 20 10
treated 20 10 1 0 20 10
untreated 20 10 0 0 10 10
treated $$\mathbf{15}$$ $$\mathbf{5}$$ $$\mathbf{1}$$ 1 $$\mathbf{15}$$ 10
untreated $$\mathbf{15}$$ $$\mathbf{5}$$ $$\mathbf{0}$$ 1 $$\mathbf{5}$$ 10
untreated $$\mathbf{15}$$ $$\mathbf{5}$$ $$\mathbf{0}$$ 1 $$\mathbf{5}$$ 10

## Conditioning: Examples

Effect of peacekeeping where war intensity is high (1):

$\frac{15}{1} - \frac{5 + 5}{2} = \frac{10}{1} = 10$

… and where war intensity is low (0)?

$$Peace_i^{UN}$$ $$Peace_i^{noUN}$$ $$UN_i$$ $$WarIntense_i$$ $$Peace_i^{Obs}$$ $$Effect_i$$
treated $$\mathbf{20}$$ $$\mathbf{10}$$ $$\mathbf{1}$$ 0 $$\mathbf{20}$$ 10
treated $$\mathbf{20}$$ $$\mathbf{10}$$ $$\mathbf{1}$$ 0 $$\mathbf{20}$$ 10
untreated $$\mathbf{20}$$ $$\mathbf{10}$$ $$\mathbf{0}$$ 0 $$\mathbf{10}$$ 10
treated 15 5 1 1 15 10
untreated 15 5 0 1 5 10
untreated 15 5 0 1 5 10

## Conditioning: Examples

Effect of peacekeeping where war intensity is low (0):

$\frac{20 + 20}{2} - \frac{10}{1} = \frac{10}{1} = 10$

## Conditioning: Examples

Within the groups of cases defined by values of war intensity:

• Estimated effect of peacekeeping (compare cases with to without) retrieves the true effect
• Conditioning solved our problem of bias

Why does this work?

1. Within groups of cases with same values on confounder (war intensity …
2. Potential outcomes (years of peace) with and without UN peacekeepers are same for "treated" and "untreated"
3. Same as: no other factor affects both UN peacekeeping ($$X$$) and peace ($$Y$$)
$$Peace_i^{UN}$$ $$Peace_i^{noUN}$$ $$UN_i$$ $$WarIntense_i$$ $$Peace_i^{Obs}$$ $$Effect_i$$
treated $$\mathbf{20}$$ $$\mathbf{10}$$ $$\mathbf{1}$$ 0 20 10
treated $$\mathbf{20}$$ $$\mathbf{10}$$ $$\mathbf{1}$$ 0 20 10
treated 15 5 1 1 15 10
untreated $$\mathbf{20}$$ $$\mathbf{10}$$ $$\mathbf{0}$$ 0 10 10
untreated 15 5 0 1 5 10
untreated 15 5 0 1 5 10
$$Peace_i^{UN}$$ $$Peace_i^{noUN}$$ $$UN_i$$ $$WarIntense_i$$ $$Peace_i^{Obs}$$ $$Effect_i$$
treated 20 10 1 0 20 10
treated 20 10 1 0 20 10
treated $$\mathbf{15}$$ $$\mathbf{5}$$ $$\mathbf{1}$$ 1 15 10
untreated 20 10 0 0 10 10
untreated $$\mathbf{15}$$ $$\mathbf{5}$$ $$\mathbf{0}$$ 1 5 10
untreated $$\mathbf{15}$$ $$\mathbf{5}$$ $$\mathbf{0}$$ 1 5 10

## Conditioning in Action:

### Peace-keeping and peace

(Gilligan & Sergenti 2008)

Look at correlation of peacekeeping and peace after conflict:

• Matching countries on: intensity of conflict (Deaths), length of conflict, ethnic diversity, population, mountainous terrain, size of armed forces, democracy

## Conditioning in Action:

(Lower proportional hazards => war less likely)

## Conditioning in Action:

(Gilligan & Sergenti 2008)

Conditioning on other factors (holding them constant):

Peacekeeping makes war 85% less likely to re-emerge

## Conditioning in Action:

### Causal Claim

The presence of guns causes an increase in violent crame

### Causal Theory:

The firearm ownership rate causes an increase in the violent crime rate

## Conditioning in Action:

### Test

Correlation between gun ownership and homicides is nearly $$0$$.

• Does this mean no causal relationship?
• Lots of possible sources of bias.

## Conditioning in Action:

### (Moore and Bergner 2016)

• Look at US counties
• Use "regression" to adjust correlation between guns and violent crime