1. Correlations: testing linear relationships between two metric variables
Lecture 18:

2. Agenda
Brief Update on Data for Final
Correlations

3. Some Data Considerations for Final Exam:
Two metric variables
Correlation
One binary categorical variable, one metric variable
T-test
Two categorical variables
Chi-Square (crosstabulations)
Polytomous dependent variable, metric IV
ANOVA
Metric dependent variable, multiple IV’s (categorical or metric)
Linear Regression

4. Checking for simple linear relationships
Pearson’s correlation coefficient
Measures the extent to which two metric or interval-type variables are linearly related
Statistic is Pearson r, or the linear or product-moment correlation
Or, the correlation coefficient is the average of the cross products of the corresponding z-scores.

5. Scatterplots !!! Three ways to summarize this data: point of averages
Horizontal SD
Vertical SD

6. The correlation coefficient
r = average of (x in standard units) x (y in standard units)
The correlation coefficient is a pure number without units.
Thus, the correlation coefficient is not affected by:
Change in scale
Interchanging variables
Adding the same number to one of the variables
Multiplying all values of one variable by a positive number

7. Sidenote: Why N-1?
If we have a randomly distributed variable in a population, extreme cases (i.e., the tails) are less likely to be selected than common cases (i.e., within 1 SD of the mean).
One result of this: sample variance is lower than actual population variance. Dividing by n-1 corrects this bias when calculating sample statistics.

8. Correlations
Ranges from zero to 1, where 1 = perfect linear relationship between the two variables.
Negative relations
Positive relations
The correlation coefficient is a measure of clustering around a single line, relative to the standard deviations.
Remember: correlation ONLY measures linear relationships, not all relationships.
This is why the scatterplot is essential; you can get a coefficient even if there is not a good linear fit.

9. Interpretation
Correlation is a proportional measure; does not depend on specific measurements
Correlation interpretation:
Direction (+/-)
Magnitude of Effect (-1 to 1); shown as r
Statistical Significance (p<.05, p<.01, p<.001)

10. Correlation vs. Causality
Recall that Correlation is a precondition for causality– but by itself it is not sufficient to show causality
Fat in the diet causes cancer?
(but fat and sugar are relatively expensive, and reduce grain consumption as trade off…)
Education and Unemployment during the Great Depression?
(turns out age is a confounding variable: education was going up for the young, and employers seemed to prefer younger job seekers)

11. Factors which limit Correlation coefficient
Homogeneity of sample group
Non-linear relationships
Censored or limited scales
Unreliable measurement instrument
Outliers

12. Homogenous Groups
Limited number of education groups will give you a correlation, but how accurate?

13. Homogenous Groups: Adding Groups

14. Homogenous Groups: Adding More Groups
-The point is NOT that the line changes– b/c it doesn’t– its still a positive correlation every time
-The issue is the overall spread, and whether the homogeneous groups are linear in the same way.

15. Separate Groups: non-homogeneous but similar linear relationship

16. Separate Groups: non-homogeneous and different linear relationship

17. Non-Linear Relationships

18. Censored or Limited Scales…

19. Censored or Limited Scales

20. Unreliable Instrument

21. Unreliable Instrument

22. Unreliable Instrument

23. Outliers

24. Outliers
Outlier

25. Ecological Correlations

26. Ecological Correlations
The ecological correlation overstate the association because they eliminate the variance in the units of analysis (eg., the classroom)
Freedman et al. give Sociology and Poli Sci a hard time for this, which is a valid critique– when the unit of analysis is a summary statistic, you expect to lose variability.
Average by Classroom

27. Correlation: Null and Alt Hypotheses
Null versus Alternative Hypothesis H0
H1, H2, etc
Test Statistics and Significance Level
Test statistic
Calculated from the data
Has a known probability distribution
Significance level
Usually reported as a p-value (probability that a result would occur if the null hypothesis were true).
price mpg
price
mpg
0.0000