1. Correlational Problems and Fallacies James H. Steiger

2. Introduction In this module, we discuss some common problems and fallacies regarding correlation coefficients and their interpretation Interpreting a correlation Correlation and causality Perfect correlation and equivalence No Correlation vs. No Relation Combining Populations, and Ignoring Explanatory Variables Restriction of Range

3. Interpreting a Correlation If scores are on roughly similar scales, the shape of the scatterplot can reveal a substantial amount about the correlation.

4. Interpreting a Correlation

7. Anscombe’s Quartet

8. Each of the above 4 data sets has the following summary statistics: Each has a best fitting linear regression line of

9. Anscombe’s Quartet

10. Correlation and Causality Correlation is not causality. This is a standard adage in textbooks on statistics and experimental design, but it is still forgotten on occasion. Example: The correlation between number of fire trucks sent to a fire and the dollar damage done by the fire.

11. Perfect Corrrelation and Equivalence Two variables may correlate highly (or even perfectly), without measuring the same construct. Example: Height and weight on the planet Zorg.

12. Height and Weight on the Planet Zorg

13. Zero Correlation vs. No Relation The Pearson correlation coefficient is a measure of linear relation. Many strong relationships are nonlinear. Always examine the scatterplot!

14. Combining Populations If two groups with different means and/or covariances are combined, the resulting mixture can exhibit spurious correlations. Example. (C.P.) Suppose the correlation between strength and mathematics performance is zero for 6 th grade boys, and zero for 8 th grade boys. Does this mean it will be zero in a combined group of 6 th and 8 th graders?

15. Restriction of Range Often, when linear regression is used to predict performance, the population is restricted. (For example, the GRE is used to predict performance in graduate school, but people with low GRE scores are often refused admission to graduate school. Consequently, the “available data” are a truncated version of the full data set.

16. Restriction of Range

18. The “Third Variable Fallacy” Often people assume, sometimes almost subconsciously, that when two variables correlate highly with a third variable, they correlate highly with each other. Actually, if r XW and r YW are both.7071, r XY can vary anywhere from 0 to 1. Only when r XW and/or r YW become very high does the correlation between X and Y become highly restricted.