Linear correlation. Contents. 1 Linear correlation. 1.1 Introduction. Anthony Tanbakuchi Department of Mathematics Pima Community College

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1 Introductor Statistics Lectures Linear correlation Testing two variables for a linear relationship Anthon Tanbakuchi Department of Mathematics Pima Communit College Redistribution of this material is prohibited without written permission of the author 2009 (Compile date: Tue Ma 19 14:51: ) Contents 1 Linear correlation Introduction Linear Correlation... 3 Use Computation A complete eample.. 10 Cautions Confidence Interval Belt Graphs Summar Further eamples Linear correlation 1.1 Introduction Motivation Is there a relationship correlation between our height and... (1) our mother s height? (2) our forearm height? (3) our work hours per week? (4) our commute distance? 1

2 2 of Introduction height height height_mother forearm height height work_hours credits Motivation Eample 1. How much of a individual s height is eplained b their mother s height? Use our class data to determine if there is a linear relationship between a mother s height and their child s height (our height) and how much variation in the child s height can be eplained b the mother s height. R: h e i g h t = c l a s s. data $ h e i g h t R: h e i g h t mother = c l a s s. data $ h e i g h t mother The first few data points are: Use a scatter plot to see if a relationship eists R: p l o t ( h e i g h t mother, height, main = Height i n i n c h e s ) Anthon Tanbakuchi

3 Linear correlation 3 of 17 height height mother Height in inches height height_mother Question 1. Does it look like there is a linear relation ship? Draw a best fit line in the data Paired data. Definition 1.1 A set of ( i, i ) data where each pair is related. (Dependent samples.) e: mother height, child height. 1.2 Linear Correlation Correlation. Definition 1.2 eists when two variables have a relationship with one another. Anthon Tanbakuchi

4 4 of Linear Correlation linear correlation non linear correlation Anthon Tanbakuchi

5 Linear correlation 5 of perfect positive strong positive positive no correlation Anthon Tanbakuchi

6 6 of Linear Correlation perfect negative strong negative negative no correlation USE Often used to help answer: 1. Is there a linear relationship between X and Y? 2. Can X be used to predict Y? 3. How much of the variation in X can be predicted with Y? COMPUTATION Definition 1.3 Linear correlation coefficient. The linear correlation coefficient for a population is denoted with ρ. We can estimate ρ via a sample and calculate Pearson s linear correlation coefficient r: (i )( i ȳ) r = (1) (n 1)s s n is the number of pairs of data points (length of or ). Anthon Tanbakuchi

7 Linear correlation 7 of 17 Measures the strength of the linear relationship between and. Larger values of r indicate stronger linear relationship. 1 Positive r indicates positive slope, negative r indicates negative slope. Eamples of r r = 1 r = r = 0.86 r = Larger r does not indicate a steeper slope. We will find the slope later using regression. Anthon Tanbakuchi

8 8 of Linear Correlation r = 1 r = r = 0.88 r = Properties of r (ρ for populations) 1. 1 r r is scale invariant. 3. r is invariant if and are interchanged. 4. r onl measures the strength of linear relationships. Definition 1.4 Coefficient of determination (eplained variation). r 2 is the proportion of linear variation in that is eplained b. 0 r 2 1 The closer r 2 is to 1 the stronger the linear relationship and likewise the more variation in that can be eplained b. Eample of r 2 Anthon Tanbakuchi

9 Linear correlation 9 of 17 r = 0.88 r^2 = Hpothesis test for linear correlation. Definition 1.5 requirements (1) simple paired (, ) random samples, (2) Pairs of (, ) have a bivariate normal distribution 2, (3) correlation is linear. null hpothesis ρ = 0 (no linear correlation) alternative hpothesis ρ 0 ( a linear correlation eists 3 ) Alwas make a scatter plot first to see if the relationship is linear. Linear correlation coefficient r and hpothesis test: cor.test(, ) Calculates r from the sample and conducts the hpothesis test for R Command H 0 ρ = 0. vector of ordered data. vector of ordered data. Test statistic for linear correlation coefficient t = r 0 1 r 2 n 2 where df = n 2. Note: n is number of pairs. Procedure for finding r 1. Define two ordered vectors ( and ) with the data. 2. Make a scatterplot to determine if a linear relationship eists. 3. If a linear relationship eists, run the hpothesis test cor.test() and do all 7 steps. It will give ou a point estimate of ρ (which is r) and allow ou to determine if it is significant (via the p-value). 2 Effectivel, a normal distribution for and. 3 Alternative hpothesis involving < or > also possible and work just as before. (2) Anthon Tanbakuchi

10 10 of Linear Correlation A COMPLETE EXAMPLE Eample of calculating r and testing significance Back to our original eample: Eample 2. How much of a individual s height is eplained b their mother s height? Use our class data to determine if there is a linear relationship between a mother s height and their child s height (our height) and how much variation in the child s height can be eplained b the mother s height. R: h e i g h t [ 1 ] R: h e i g h t mother [ 1 ] First check if relationship looks linear R: p l o t ( h e i g h t mother, height, main = Height i n i n c h e s ) Height in inches height height_mother Question 2. Does is look line a linear relationship? (Strong or weak?) Question 3. What sign do we epect for r? Question 4. What is our null hpothesis? Run hpothesis test... H 0 : ρ = 0, H a : ρ 0, α = Anthon Tanbakuchi

11 Linear correlation 11 of 17 R: r e s = cor. t e s t ( h e i g h t mother, h e i g h t ) R: r e s Pearson ' s product moment c o r r e l a t i o n data : h e i g h t mother and h e i g h t t = , df = 16, p value = a l t e r n a t i v e h p o t h e s i s : t r u e c o r r e l a t i o n i s not equal to 0 95 p e r c e n t c o n f i d e n c e i n t e r v a l : sample e s t i m a t e s : cor Question 5. What is the formal decision? Question 6. What is r? Question 7. What is the percent of variation of an individual s height is eplained b their mother s height? Question 8. What is the formal conclusion? Importance of testing significance of r Is there linear correlation? height work_hours The linear correlation coefficient for the above data is Question 9. Wh is it important to test the hpothesis H 0 : ρ = 0 and H a : ρ 0 since r is not zero? Anthon Tanbakuchi

12 12 of Linear Correlation R: cor. t e s t ( work hours, h e i g h t ) Pearson ' s product moment c o r r e l a t i o n data : work hours and h e i g h t t = , df = 16, p value = a l t e r n a t i v e h p o t h e s i s : t r u e c o r r e l a t i o n i s not equal to 0 95 p e r c e n t c o n f i d e n c e i n t e r v a l : sample e s t i m a t e s : cor Since we fail to reject H 0 : ρ = 0, r is not significant. Sampling error can account for its deviation from the null value of zero. Conclusion: no significant liner correlation. CAUTIONS Cautions when calculating and interpreting correlation If the relationship is nonlinear, r is not meaningful! Correlation does not impl causalit! Calculating correlation based on averages ma falsel inflate r. If no significant linear correlation eists, a relationship ma still eist (namel, a nonlinear one). When relationship is not linear Anthon Tanbakuchi

13 Linear correlation 13 of 17 Anscombe's 4 Regression data sets r = 0.82 r = r = r = Note that r is the same for all plots, but not appropriate for all! 4 CONFIDENCE INTERVAL BELT GRAPHS Anthon Tanbakuchi

14 14 of Linear Correlation % Confidence Belts For Correlation Linear Coefficient 0.8 n = 5 Population Correlation Coefficient, ρ n = 10 n = 20 n = 30 n = 50 n = 100 n = 250 n = 1000 n = infinit n = 1000 n = 250 n = 100 n = 50 n = 30 n = 20 n = n = Sample Correlation Coefficient, r (NOTE: values for n < 25 are onl rough approimations) Anthon Tanbakuchi

15 Linear correlation 15 of 17 95% Confidence Belts For Correlation Linear Coefficient Population Correlation Coefficient, ρ n = 5 n = 10 n = 20 n = 30 n = 50 n = 100 n = 250 n = 1000 n = infinit n = 1000 n = 250 n = 100 n = 50 n = 30 n = 20 n = n = Sample Correlation Coefficient, r (NOTE: values for n < 25 are onl rough approimations) Anthon Tanbakuchi

16 16 of Linear Correlation 99% Confidence Belts For Correlation Linear Coefficient Population Correlation Coefficient, ρ n = 5 n = 10 n = 20 n = 30 n = 50 n = 100 n = 250 n = 1000 n = infinit n = 1000 n = 250 n = 100 n = 50 n = 30 n = 20 n = n = Sample Correlation Coefficient, r (NOTE: values for n < 25 are onl rough approimations) Anthon Tanbakuchi

17 Linear correlation 17 of Summar Linear correlation coefficient r r measures the strength of linear correlation r r is scale invariant. 3. r is invariant if and are interchanged. 4. r onl measures the strength of linear relationships. r 2 proportion of linear variation in that is eplained b. Finding r check for linear relationship, then find r and make sure it is significant. Hpothesis test to find r and test significance: requirements (1) simple paired (, ) random samples, (2) Pairs of (, ) have a bivariate normal distribution, (3) correlation is linear. null hpothesis ρ = 0 (no linear correlation) alternative hpothesis ρ 0 ( a linear correlation eists. test in R : cor.test(, ) vector of ordered data. vector of ordered data. You must make a scatter plot first! 1.4 Further eamples Now test to see if there is a significant linear correlation between a person s height and their forearm length. (Use the class data set, hint: height=class.data$height forearm=class.data$forearm ) Question 10. Does the relationship look linear? Question 11. What is r (Check: r = 0.753) Question 12. Is r significant? (Check: p-value= ) Question 13. What is the 95% confidence interval for r (Check:(0.441, 0.902)) Question 14. What percent of an individual s height is eplained b their forearm length? Question 15. Which is better for predicting an individual s height: their forearm length or mother s height? Anthon Tanbakuchi