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Probability, Statistics and the Logic of Scientific Causality

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Probability, Statistics and the Logic of Scientific Causality

An Introduction to Social Science Data Analysis

- Inter-subjective Standards
- Mathematics
- Logic

- Religion
- Normative philosophy

- To try to use whatever tools we have at our disposal to prove ourselves wrong about our causal theory.
- Tools:
- Logic
- Empirical observation

- Try to disprove our theory as much as possible
- Unfortunately, we cannot prove anything

- Tools:
- Always estimate the level of uncertainty in any claim

- Interpretavism – if we describe the phenomenon, what does it mean
- Behavioralism – the underlying roots of our attitudes and behaviors
- Neo-institutionalism – the relationships among attitudes and behaviors depend on rules and other constraints
- Rational Choice
- Assume preferences
- Deduce outcomes

The Research Question:Why does ‘y’ vary?

- Why do some people vote for Democrats and others for Republicans?
- Why do some ethnic conflicts get resolved and other end in holocausts?
- Why do some democracies remain stable and others fall apart?
- Why are some economies successful and others are not?
- Why do some civil conflicts result in revolution and others do not?
- What causes some people to support the civil liberties of political enemies?
- What causes some people to trust one another and others not to trust?
- What causes some people to participate in their government?
- What causes some people to bring litigation against their government?

Measurement

Measurement

- Inter-subjective Measures
- Inches
- Degrees Fahrenheit
- Dollars

- Continuous v. Discrete

Concepts that are difficult to measure inter-subjectively

- Democracy
- Self-Esteem
- Ideology

Face Validity: Political Tolerance

If your worst political enemy (i.e. Nazi’s, KKK) came to your town, would you support their right to march downtown?

Not support at all

Not really support

Somewhat support

Strongly support

4

1

2

3

Repeat Study: the same people who got a 1 would get about a 1 again and so on.

Multiple questions at the same time.

High Variance

Mean

Low Variance

Mean

Variance

- Total Variance:
- Sum of squared distance of each point to the mean/number of observations.

(4)2=16

(3)2=9

(0)2=0

(2)2=4

(1)2=1

(1)2=1

(1)2=1

Mean

No error

Total Variance = 32/7 = 4.57

Variance: Political Tolerance

Number of People

Variance – Skewed Distribution

Number of People

Low Variance

Number of People

High Variance

Number of People

Standard Deviation

Square root of the average level of variance

We square the deviations from the mean, but then the units of squared numbers do not make sense, so we then take the square root of it.

- The standard deviation is the sum of squared distance of each point to the mean, divided by the number of observations.

(4)2=16

(3)2=9

(0)2=0

(2)2=4

(1)2=1

(1)2=1

(1)2=1

Mean

No error

Total Variance = 4.57;

Standard Deviation = √32/7 = 2.13

Measurement

We will spend a great deal of time on measurement in this class

Probability

- The probability of an outcome …
- Is the frequency of that outcome
- if the process were repeated a large number of times…
- Under similar conditions

- Fire trucks fire damage

- Storks babies

Causal Theory

We will spend a great deal of time on causal theory in this class

- Frequentist statistical theory assumes repeated observations.
- From large sample sizes, we assume that we have repeated observations.
- Large? 60.

Statistical Relationships

40

3

3

3

2

35

2

3

3

1

3

2

30

1

3

Education

3

1

25

1

3

3

3

1

Lowest

Low

Medium

High

Highest

3

3

20

2

2

2

3

2

4

2

3

2

3

2

4

3

2

1

15

4

3

4

2

1

2

3

4

3

4

3

2

2

3

2

4

5

10

1

1

2

3

4

3

4

5

5

4

4

1

1

2

2

3

3

5

5

5

5

1

2

2

2

3

3

4

4

5

5

5

2

1

3

2

3

4

3

5

5

4

5

1

1

2

3

3

3

5

4

5

5

0

1 2 34 5

Political Tolerance

Probability

- The probability of an outcome …
- Is the frequency of that outcome
- if the process were repeated a large number of times…
- Under similar conditions

Statistical Relationships

6

Slope

5

4

Political Tolerance

Mean

3

2

1

0

0

1

2

3

4

5

6

Education

- What is the probable value of tolerance, given condition of education?
- This is what the slope tells us.

6

Slope

5

4

Political Tolerance

Mean

3

2

1

0

0

1

2

3

4

5

6

Education

- The mean is the best guess if all you have is a single variable

Mean

No error

- The purpose of the mean is to minimize error in guessing
- The mean is the most probable expected value

Our job in statistical analysis

- Is to do better than the mean at making the ‘best guess’

Variance: Explained and Unexplained

6

Unexplained: Distance from the points to the slope

5

Slope

Explained: From slope to mean

4

Political Tolerance

Mean

3

2

1

0

0

1

2

3

4

5

6

Education

Remember when we said: Our job in statistical analysis is to do better than the mean?

We use the slope to…

- Minimize Error:
the distance between the points and the slope,

…while, by definition, simultaneously

- Maximizing Explained Variance:
the distance between the mean and the slope.

Decrease Error

6

Slope

5

4

Political Tolerance

Mean

3

2

1

0

0

1

2

3

4

5

6

Variance: Explained and Unexplained

6

5

Slope

4

Political Tolerance

Mean

3

2

1

0

0

1

2

3

4

5

6

Education

Unexplained

- Our job as social scientists is to explain variance.
- Statistically, we do that by separating explained from unexplained.

Explained

Aspects of Political Science Data Analysis

- Causal Theory
- Measurement Theory
- Explain Variation