Correlation

1. Correlation

2. The Problem • Are two variables related? • Does one increase as the other increases? • e. g. skills and income • Does one decrease as the other increases? • e. g. health problems and nutrition • How can we get a numerical measure of the degree of relationship?

3. Scatterplots • Examples from text • See next three slides • Infant mortality and number of physicians • Life expectancy and health care expenditures • Cancer rate and solar radiation

7. An Example • An actual course with both a lab and an exam component of final grades • Plotting exam component against lab component • Fairly weak relationship • Relationship is positive

9. Exams and Labs • Note relationship is weak, but real. • Note most data cluster on right. • Why do we care about relationship? • What would students conclude if there were no relationship? • What if the relationship were near perfect? • What if the relationship were negative?

10. Heart Disease and Cigarettes • Landwehr & Watkins report data on heart disease and cigarette smoking in 21 developed countries • Data have been rounded for computational convenience. • The results were not affected.

11. The Data Surprisingly, the U.S. is the first country on the list--the country with the highest consumption and highest mortality.

12. Scatterplot of Heart Disease • CHD Mortality goes on ordinate • Why? • Cigarette consumption on abscissa • Why? • What does each dot represent? • Best fitting line included for clarity

13. {X = 6, Y = 11}

14. What Does the Scatterplot Show? • As smoking increases, so does coronary heart disease mortality. • Relationship looks strong • Not all data points on line. • This gives us “residuals” or “errors of prediction” • To be discussed later

15. Correlation Coefficient • A measure of degree of relationship. • Sign refers to direction. • Based on covariance • Measure of degree to which large scores go with large scores, and small scores with small scores

16. Covariance • The formula • How this works, and why • When would covXY be large and positive? • When would covXY be large and negative?

17. Correlation Coefficient • Symbolized by r • Covariance ÷ (product of st. dev.)

18. Calculation • CovXY = 11.13 • sX = 2.33 • sY = 6.69

19. Correlation--cont. • Correlation = .71 • Sign is positive • Why? • If sign were negative • What would it mean? • Would not alter the degree of relationship.

20. Factors Affecting r • Range restrictions • See next slide • Data only for countries with low consumption • Nonlinearity • e.g. age and size of vocabulary • Heterogeneous subsamples • Everyday examples

21. Countries With Low Consumptions Data With Restricted Range Truncated at 5 Cigarettes Per Day 20 18 16 14 12 CHD Mortality per 10,000 10 8 6 4 2 2.5 3.0 3.5 4.0 4.5 5.0 5.5 Cigarette Consumption per Adult per Day

22. Testing r • Population parameter =  • Null hypothesis H0:  = 0 • Test of linear independence • What would a true null mean here? • What would a false null mean here? • Alternative hypothesis (H1)   0 • Two-tailed

23. Tables of Significance • Table in Appendix E.2 • For N - 2 = 19 df, rcrit = .433 • Our correlation > .433 • Reject H0 • Correlation is significant. • Greater cigarette consumption associated with higher CHD mortality.

24. Computer Printout • Printout gives test of significance. • See next slide. • Double asterisks with footnote indicate p < .01.

25. SPSS Printout

26. Intercorrelation Matrix • Matrix of correlations of several variables at once. • Example from Kliewer et al (1998) JCCP • 99 young children • Measured level of • Witness violence, Intrusive thoughts, Social support, and Internalizing symptoms • Define these variables

27. Cont.

28. Intercorrelation Matrix--cont. • Describe the table. • What does this tell us about the effects of witnessing violence? • What role does social support play?