March 19, 2018

Midterm Grades

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: \(70.9\)%

Homework Scaling:

It is possible

From Correlation to Causality

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 \(X\):

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

Why?

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?

What is the problem?

What is the problem?

Is peace keeping effective?

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

  • Can address confounding
  • 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

  1. adjustment-based
    • 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: Experiment

Solutions: Examples

Adjustment-based:

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

How to think about adjustment/conditioning:

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?

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

Unadjusted correlation BIASED because…

  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

Is there an effect after conditioning?

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

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
  • Adjust for possible confounders:
    • Region, urban/rural, population, poverty, ethnic diversity, unemployment, gender, poverty
    • "Holding these factors constant", "ceteris parabis"

Conditioning in Action: