Chapter 9 Correlation

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1 Chapter 9 Correlation PSY 95 - Oswald Outline Why correlation? Scatterplots An example The correlation coefficient Correlations on ranks Factors affecting correlations Chapter 9 Correlation Cont. Outline continued Testing for significance Intercorrelation matrices Other kinds of correlations Review questions Why correlation? 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? Chapter 9 Correlation 3 Chapter 9 Correlation 1

2 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 Infant Mortality 0 - Figure 9.1 Infant Mortaility and Number of Physicians Physicians per 0,000 Population Chapter 9 Correlation 5 Chapter 9 Correlation 7 73 Figure 9. Life Expectancy and Health Care Costs 3 3 Figure 9.3 Cancer Rate and Solar Radiation Life Expectancy (Males) Breast Cancer Rate Solar Radiation Health Care Expenditures Chapter 9 Correlation 7 Chapter 9 Correlation

3 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 Total Points on Exams Rsq = Total Points in Lab Chapter 9 Correlation 9 Chapter 9 Correlation 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? Heart Disease and Cigarettes Landwehr & Watkins report data on heart disease and cigarette smoking in 1 developed countries Data have been rounded for computational convenience. The results were not affected. Chapter 9 Correlation 11 Chapter 9 Correlation 1 3

4 The Data Cigarette Consumption and Coronary Heart Disease Mortality for 1 Countries Cig CHD Cig CHD Cig. = Cigarettes per adult per day CHD = Cornary Heart Disease Mortality per,000 population Scatterplot of Heart Disease CHD Mortality goes on Why? Cigarette consumption on Why? What does each dot represent? Best fitting line included for clarity Surprisingly, the U.S. is the first country on the list--the country with the highest consumption and highest mortality. Chapter 9 Correlation 13 Chapter 9 Correlation 1 CHD Mortality per, {X =, Y = 11} 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 0 1 To be discussed later Cigarette Consumption per Adult per Day Chapter 9 Correlation 15 Chapter 9 Correlation 1

5 Correlation Coefficient A measure of degree of relationship. Sign refers to direction. Based on Measure of degree to which large scores go with large scores, and small scores with small scores Chapter 9 Correlation 17 Covariance The formula Σ( X X)( Y Y) covxy = N 1 How this works, and why When would cov XY be large and positive? When would cov XY be large and negative? Chapter 9 Correlation 1 Correlation Coefficient Symbolized by r Covariance cov XY = s X =.33 s Y =.9 Calculation cov r = s s X XY Y cov r = s s X XY Y = (.33)(.9) = = Chapter 9 Correlation 19 Chapter 9 Correlation 5

6 Correlation--cont. Correlation =.71 Sign is positive Why? If sign were negative What would it mean? Would not alter the degree of relationship. Factors Affecting r Range restriction See next slide Data only for countries with low consumption of cigs Nonlinearity e.g. age and size of vocabulary Heterogeneous subsamples Any groups where relationships/correlations might differ (e.g., body-image perceptions and self-esteem correlations for ) Chapter 9 Correlation 1 Chapter 9 Correlation Countries With Low Consumptions 30 CHD Mortality per, Data With Restricted Range Truncated at 5 Cigarettes Per Day CHD Mortality per,000 0 {X =, Y = 11} Cigarette Consumption per Adult per Day 5.5 Cigarette Consumption per Adult per Day Chapter 9 Correlation 3 Chapter 9 Correlation

7 Testing r Population parameter = ρ Null hypothesis H 0 : ρ = 0 Test of linear independence What would a true null mean here? What would a false null mean here? Alternative hypothesis (H 1 ) ρ 0 Two-tailed Tables of Significance Correlation table For N - = 19 df, r crit =.33 Our correlation >.33 Reject H 0 Correlation is significant. Greater cigarette consumption associated with higher CHD mortality. Chapter 9 Correlation 5 Chapter 9 Correlation Computer Printout Printout gives test of significance. See next slide. Double asterisks with footnote indicate p <.01. Cigarette Consumption per Adult per Day CHD Mortality per,000 SPSS Printout Correlations Cigarette CHD Consumption Mortali per Adult per ty per Day,000 Pearson Correlation Sig. (-tailed) N Pearson.713** Correlation Sig..000 (-tailed) N 1 **. Correlation is significant at the 0.01 level (-tailed). Chapter 9 Correlation 7 Chapter 9 Correlation 7

8 Intercorrelation Matrix Matrix of correlations of several variables at once. Example from Kliewer, et al. (199). The role of social and cognitive processes in children s adjustment to community violence. Journal of Consulting and Clinical Psychology,, young children Measured level of Witness violence, Intrusive thoughts, Social support, and Internalizing symptoms Define these variables Chapter 9 Correlation 9 Witness Violence Intrusive Thoughts Social Support Internalizing Symptoms Witness Violence Intrusive Thoughts Social Support Internalizin Symptoms Chapter 9 Correlation 30 Cont. Intercorrelation Matrix--cont. Describe the table What does this tell us about the effects of witnessing violence? What role does social support play? Review Questions (These are NOT the only things you need to study) What determines what goes on which axis of a scatterplot? What would a correlation of 0 tell us about the relationship between lab grades and exam grades? What factors might affect the relationship between smoking and CHD Mortality? Chapter 9 Correlation 31 Chapter 9 Correlation 3

9 Review Questions--cont. (These are NOT the only things you need to study) Indicate level (high, med, or low) and sign of the correlation for: number of guns in community and number firearm deaths robberies and incidence of drug abuse protected sex and incidence of AIDS community education level and crime rate solar flares and suicide Review Questions--cont. Why would the size of the correlation required for significance decrease with increasing N? Chapter 9 Correlation 33 Cont. Chapter 9 Correlation 3 9

Homework 11. Part 1. Name: Score: / null

Homework 11. Part 1. Name: Score: / null Name: Score: / Homework 11 Part 1 null 1 For which of the following correlations would the data points be clustered most closely around a straight line? A. r = 0.50 B. r = -0.80 C. r = 0.10 D. There is