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Plan/Philosophy of the course

Causality, Potential Outcomes, Review

Allport and a large literature suggests that prejudice against groups can be reduced through contact

• Contact could create empathy, contradict negative stereotypes. But people could also build empathy without contact.
• Contact is often chosen… hard to test whether it is contact that actually shifts prejudice, rather than levels of prejudice driving contact

Broockman and Kalla replicated a fraudulent experiment that tested prejudice reduction against gays using gay vs. straight canvassers for gay rights.

• Trans-rights organization canvassed to reduce transphobia in Miama, FL
• Researchers worked with them to randomize which people received different "treatments"

Subjects

• 1825 registered voters
• Drawn from responders to invitation to join online survey

Treatments

Subjects either saw:

• Canvasser on law againts transgender discrimination who was cis-gender or transgender
• Canvasser for recycling program who was cis-gender or transgender
• Canvasser in treatment: asked people to consider time when they were judged negatively for being different

Survey

• Answered survey questions about discrimination law, transgender tolerance
• From 3 days after canvass to 3 months

Simple analysis, nothing fancy: difference in means

Design-based inference:

• causality through research design, not statistical tool
• analysis still depends on a statistical model (though simple, usually credible)

Contrast with model-based inference:

"The power of multiple regression analysis is that it allows us to do in non-experimental environments what natural scientists are able to do in a controlled laboratory setting: keep other factors fixed" (Wooldridge 2009: 77)

Strengths:

• Causal inference possible
• Statistical model simple, transparent, plausible

Limitations:

• analysis may not follow from the research design
• lack of external validity

(1) Counterfactuals

(2) Manipulation

• A counterfactual statement with a false (untrue about the world) premise, and an assertion of what would have happened had the premise been true:

"If the legal marijuana policy has not been adopted, then Y…"

• For cause C, causal effect of C is the difference between the two states of the world where C is present versus absent.

Not a sufficient criterion!

• mercury in barometer and storms

• Causation as forced action

Combination with counterfactuals gives us a strong criterion:

• counterfactuals without manipulation do not give us the direction of causality
• manipulation without counterfactuals may permit spurious results

Statistical approach to causality combines both:

Neyman (Rubin; Holland) model:

An experiment with two treatment conditions (treatment and control)

• potential outcome under treatment is \(Y_i^1\): what unit \(i\) would experience if assigned to treatment
• potential outcome under control is \(Y_i^0\): what unit \(i\) would experience if assigned to control.
• unit causal effect is the difference between these:

\[\tau_i = Y_i^1 - Y_i^0\]

Neyman model is a causal model

• Explicitly states how units behave under different counterfactual conditions
• These sets of potential outcomes or response schedules are crucial for using statistics to achieve causal inference.

TROUBLE

• Can never directly observe the unit causal effect: only ever see \(Y_i^1\) or \(Y_i^0\), not both.

Fundamental Problem of Causal Inference

fundamental problem of causal inference:

Because unit causal effects are fundamentally unobservable, focus has been on average causal effects (or average treatment effects)

• For units in \(i = 1 \ldots N\)

\[ACE = \bar{\tau} = \frac{1}{N}\sum\limits_{i=1}^N [Y_i^1 - Y_i^0]\]

• This parameter takes the difference between two counterfactuals:
  • average outcome of all units if assigned to treatment
  • average outcome of all units if assigned to control

We have a parameter (average causal effect) that we would like to know. But half the data we need is always missing.

Analogical problem to estimating, e.g., population mean (a parameter) without observing the entire population.

• Use random sampling and statistics to solve our problem

random variable: a chance procedure for generating a number

observed value (realization): value of a particular draw of a random variable.

Six-sided fair die

• Equal probability of landing on \(1,2,3,4,5,6\)

• Imagine the random variable as a box containing all possible values of a die roll on tickets

• A roll of the die would be a realization

(1) Arithmetic operations on random variables are new random variables

• E.g. \(Y * X = Z\); where \(Y\) is \(-1,1\) based on a coin flip and \(X\) is \(1,2,3,4,5,6\) based on die roll.

(2) Expected value of a random variable is a number.

• If \(X\) is a random draw from a box of numbered tickets, then \(E(X)\) is the mean of the tickets in the box
• Any realization of the random variable may be above or below the expected value

\(X\) and \(Y\) are random variables.

• If you know the value of \(X\) and the chances of \(Y\) taking on a specific value depends on that value of \(X\), \(X\) and \(Y\) are dependent
• If you know the value of \(X\) and the chances of \(Y\) taking on a specific value do not depend on that value of \(X\), \(X\) and \(Y\) are independent

• We often have data variables: lists of numbers

• random variables are chance processes for generating numbers
  • permit us to use chance process to draw inferences

• to treat data variables as random variables need a model for data generating process

• our models may be good descriptions of chance procedure, or not.

• Expected value of a random variable is a parameter
  • in our examples (e.g. fair die), we have known it
  • usually, we want to estimate it
• statistics (e.g., sample mean) let us draw inferences about parameters from the data.

If we roll a die \(n\) times and add up spots (\(S_i\)) on each roll \(i\):

\[\sum\limits_{i=1}^{n}S_i\]

Each \(S_i\) is a random variable, as is the sum

It turns out that:

\[E\left(\sum\limits_{i=1}^{n}S_i\right) = \sum\limits_{i=1}^{n}E\left(S_i\right) = n \cdot 3.5\]

\[\frac{1}{n}\sum\limits_{i=1}^{n}E\left(S_i\right) = n \cdot 3.5 \cdot \frac{1}{n}\]

If we roll die \(n\) times and take the mean of spots, it is a random variable. The mean of the draws is, in expectation, the mean of the random variable. AND as \(n\) gets large, sample mean will approximate the box/population mean.

We can use repeated observations to estimate parameters of random variable (e.g. expected value).

We did this using independent, identically distributed random draws: good for rolling dice

but reality is complicated.

\[ACE = \bar{\tau} = \frac{1}{N}\sum\limits_{i=1}^N [Y_i^1 - Y_i^0]\]

How do we solve our "missing data" problem?

Random sampling!

Our parameter:

\[\frac{1}{N}\sum\limits_{i=1}^N [Y_i^1 - Y_i^0]\]

Our estimator:

\[\frac{1}{m}\sum\limits_{i=1}^m [Y_i^1 | Z_i = 1] - \frac{1}{N-m}\sum\limits_{i=m + 1}^N [Y_i^0 | Z_i = 0]\]

Where \(m\) units are assigned to treatment \(Z = 1\) and \(N - m\) assigned to control \(Z = 0\).

Because we randomly assign samples to treatment and control

1. Expected values of sample mean are equal to population mean
2. Expected value of mean \(Y_i^1\) for sample in treatment equals population mean of \(Y_i^1\)
3. Expected value of mean \(Y_i^0\) for sample in treatment equals population mean of \(Y_i^0\)
4. Because sample means are random variables, ACE in a sample may differ from the population ACE.
5. But across all possible assignments of subjects to treatment and control, the average sample ACE is the same as the population ACE
6. This estimate is unbiased

\[ACE =\frac{1}{N}\sum\limits_{i=1}^N [Y_i^1 - Y_i^0]\]

\[ACE =E[Y_i^1 - Y_i^0]\]

\[ACE =E[Y_i^1] - E[Y_i^0]\]

And because \(Z_i\) is randomly assigned:

\(E[Y_i^1|Z_i = 1] = E[Y_i^1|Z_i = 0] = E[Y_i^1]\) \(E[Y_i^0|Z_i = 0] = E[Y_i^0|Z_i = 1] = E[Y_i^0]\)

For us to get an unbiased estimate of the average causal effect, we assume:

1. Random assignment: units are sample at random (or as if at random) from the study group and assigned to treatment or control
2. Non-interference/SUTVA: each unit's outcome depends only on its treatment assignment (not on assignment of others)
3. Exclusion restriction: treatment assignment only affects outcomes through receipt of treatment

Can we test or evaluate the validity of these assumptions?