2. Model Diagnostics and OLS Assumptions
Political Analysis II

3. Why should we care about unusual observations?
Because they can drive our results and lead to misleading findings (especially in small samples)
To improve our theory and statistical model
Three types of unusual observations:
Regression outliers
High leverage observations
Influential observations

4. A useful tool: Residuals

5. Regression outliers
Regression outliers = extreme values for Y given their values on X
For example, oil-rich non-democracies
Coding error, peculiarity
Limited effect, but they can increase our standard errors
Detect: large studentized residuals (> |2|)
Fix: Check coding, revise theory
Fox (2008)

6. Example of regression outliers
Lijphart excluded India and Israel from his analysis because they had extreme values on the dependent variable of political stability and absence of violence (i.e. univariate outliers).
But only Israel is a regression outlier.

7. High leverage observations
High leverage = extreme values on one or more independent variables.
They can change the estimate of regression coefficients (if they don’t follow the pattern of the data)
Detect: hat values (measure based on the fitted/Y-hat values)
Fox (2008)

8. Example of high leverage observations
Lijphart described India as an “extreme outlier”, but it is actually a high leverage observation.

9. Example of high leverage observations
Lijphart described India as an “extreme outlier”, but it is actually a high leverage observation.

10. Example of high leverage observations
We can see this clearly when we look at India’s very high hat-values.

11. Influential observations
Influential observations = extreme values for X and Y
Influence = Outlierness and Leverage
Excluding them significantly changes the direction, strength, or significance of the results
Detect: studentized residuals versus leverage, Cook’s Distance
Check coding, “dummying out”, re-run the model without the observation(s) and compare results
Fox (2008)

12. Example of influential observations
No influential observations in Lijphart’s sample…
India: high hat-values, but small residuals
Israel: large residuals, but low hat-values
We find influential observations in the lower-right corner and upper-right corner (not shown here).

13. The infamous butterfly ballot
Wand et al. (2001) show that more than 2,000 Democrats voted for Buchanan in Palm Beach County, a typically Democratic county, due to the butterfly ballot. This type of ballot was only used in this county and only for election-day for president.
As a result, George W. Bush, and not Al Gore, won Florida and the presidency.
Kellstedt and Whitten (2013)

15. Why ordinary least squares (OLS) assumptions?
Describing linear relationships between variables
Interpreting regressions causally
Hypothesis testing and predictions

16. The OLS assumptions Linearity Homoscedasticity Mean independence
No autocorrelation
(Normally distributed errors)
Standard errors

17. The linearity assumption
The relationship between the independent and dependent variables should be linear. A one-unit change in X leads to x-amount of change in Y, regardless of the value of X.

19. Based on the argument of Przeworski and Limongi (1997).
“Modernization: Theories and Facts.” World Politics 49 (02): 155–83.

20. Violations of the linearity assumption
Can you think of other nonlinear relationships?
District magnitude and the number of legislative parties
Age and the likelihood of voting …
Solutions:
Interaction effects
Transform the data (e.g. log, quadratic, exponential)
More on nonlinear relationships next week

21. More articles On influential observations:
Fails and Krieckhaus (2010). Colonialism, Property Rights and the Modern World Income Distribution. British Journal of Political Science, 40(3), Data: property-rights-and-the-modern-world-income-distribution
Wand et al. (2001). The Butterfly Did It: The Aberrant Vote for Buchanan in Palm Beach County, Florida. American Political Science Review, 95(4), Data: 89
On nonlinear relationships:
Przeworski and Limongi (1997). Modernization: Theories and Facts. World Politics 49 (02): 155–83.