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# Types of correlation coefficients

A brief description of different types of correlation coefficients

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### Types of correlation coefficients

1. 1. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Correlation Dr. S. A. Rizwan, M.D., Public Health Specialist (MOH), Lecturer, Saudi Board of Preventive Medicine - Joint Program Riyadh Nov 2019 1
2. 2. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Learning objectives • Describe the conditions in which correlation is used • Describe the steps in calculation of correlation • Describe the various interpretations of correlation coefficient Nov 2019 2
3. 3. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Outline • Questions answered by correlation • Scatterplots • An example • The correlation coefficient • Other kinds of correlations • Factors affecting correlations • Testing for significance Nov 2019 3
4. 4. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course The Question • 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? Nov 2019 4
5. 5. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Scatterplots • AKA scatter diagram or scattergram • Graphically depicts the relationship between two variables in two dimensional space Nov 2019 5
6. 6. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Scatterplot:Video Games and Alcohol Consumption 0 2 4 6 8 10 12 14 16 18 20 0 5 10 15 20 25 Average Hours of Video Games Per Week AverageNumberofAlcoholicDrinks PerWeek Direct Relationship Nov 2019 6
7. 7. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Scatterplot: Video Games and Test Score 0 10 20 30 40 50 60 70 80 90 100 0 5 10 15 20 Average Hours of Video Games Per Week ExamScore Inverse Relationship Nov 2019 7
8. 8. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course An Example • Does smoking cigarettes increase systolic blood pressure? • Plotting number of cigarettes smoked per day against systolic blood pressure – Fairly moderate relationship – Relationship is positive Nov 2019 8
9. 9. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Trend? SMOKING 3020100 SYSTOLIC 170 160 150 140 130 120 110 100 Nov 2019 9
10. 10. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Smoking and BP • Note relationship is moderate, but real. • Why do we care about relationship? – What would conclude if there were no relationship? – What if the relationship were near perfect? – What if the relationship were negative? Nov 2019 10
11. 11. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Heart Disease and Cigarettes • Data on heart disease and cigarette smoking in 21 developed countries (Landwehr and Watkins, 1987) Country Cigarettes CHD 1 11 26 2 9 21 3 9 24 4 9 21 5 8 19 6 8 13 7 8 19 8 6 11 9 6 23 10 5 15 11 5 13 12 5 4 13 5 18 14 5 12 15 5 3 16 4 11 17 4 15 18 4 6 19 3 13 20 3 4 21 3 14 Nov 2019 11
12. 12. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Scatterplot of Heart Disease • CHD Mortality goes on ordinate (Y axis) – Why? • Cigarette consumption on abscissa (X axis) – Why? • What does each dot represent? • Best fitting line included for clarity Nov 2019 12
13. 13. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Cigarette Consumption per Adult per Day 12108642 30 20 10 0 {X = 6, Y = 11} Nov 2019 13
14. 14. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course 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” Nov 2019 14
15. 15. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Correlation • Co-relation • The relationship between two variables • Measured with a correlation coefficient • Most popularly seen correlation coefficient: Pearson Product-Moment Correlation Nov 2019 15
16. 16. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Types of Correlation • Positive correlation – High values of X tend to be associated with high values of Y. – As X increases, Y increases • Negative correlation – High values of X tend to be associated with low values of Y. – As X increases, Y decreases • No correlation • No consistent tendency for values on Y to increase or decrease as X increases Nov 2019 16
17. 17. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Correlation Coefficient • A measure of degree of relationship. • Between 1 and -1 • Sign refers to direction. • Based on covariance – Measure of degree to which large scores on X go with large scores on Y, and small scores on X go with small scores on Y – Think of it as variance, but with 2 variables instead of 1 Nov 2019 17
18. 18. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Nov 2019 18
19. 19. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Covariance • Remember that variance is: • The formula for co-variance is: 2 ( ) ( )( ) 1 1 X X X X X X X Var N N          1 ))((    N YYXX CovXY Nov 2019 19
20. 20. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Example Country X (Cig.) Y (CHD) ( )X X ( )Y Y ( )X X * ( )Y Y 1 11 26 5.05 11.48 57.97 2 9 21 3.05 6.48 19.76 3 9 24 3.05 9.48 28.91 4 9 21 3.05 6.48 19.76 5 8 19 2.05 4.48 9.18 6 8 13 2.05 -1.52 -3.12 7 8 19 2.05 4.48 9.18 8 6 11 0.05 -3.52 -0.18 9 6 23 0.05 8.48 0.42 10 5 15 -0.95 0.48 -0.46 11 5 13 -0.95 -1.52 1.44 12 5 4 -0.95 -10.52 9.99 13 5 18 -0.95 3.48 -3.31 14 5 12 -0.95 -2.52 2.39 15 5 3 -0.95 -11.52 10.94 16 4 11 -1.95 -3.52 6.86 17 4 15 -1.95 0.48 -0.94 18 4 6 -1.95 -8.52 16.61 19 3 13 -2.95 -1.52 4.48 20 3 4 -2.95 -10.52 31.03 21 3 14 -2.95 -0.52 1.53 Mean 5.95 14.52 SD 2.33 6.69 Sum 222.44Nov 2019 20
21. 21. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Example .& ( )( ) 222.44 11.12 1 21 1 cig CHD X X Y Y Cov N         Nov 2019 21
22. 22. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Correlation Coefficient • Pearson’s Product Moment Correlation • Symbolized by r • Covariance ÷ (product of the 2 SDs) • Correlation is a standardized covariance YX XY ss Cov r  Nov 2019 22
23. 23. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Calculation for Example • CovXY = 11.12 • sX = 2.33 • sY = 6.69 cov 11.12 11.12 .713 (2.33)(6.69) 15.59 XY X Y r s s     Nov 2019 23
24. 24. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Example • Correlation = 0.713 • Sign is positive – Why? • If sign were negative – What would it mean? – Would not alter the degree of relationship. Nov 2019 24
25. 25. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Other Kinds of Correlation Attractiveness Symmetry 3 2 4 6 1 1 2 3 5 4 6 5 rsp = 0.77 • Spearman Rank-Order Correlation Coefficient (rsp) – used with 2 ranked/ordinal variables – uses the same Pearson formula Nov 2019 25
26. 26. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Other Kinds of Correlation Attractiveness Date? 3 0 4 0 1 1 2 1 5 1 6 0 rpb = -0.49 • Point biserial correlation coefficient (rpb) – used with one continuous scale and one nominal or ordinal or dichotomous scale. – uses the same Pearson formula Nov 2019 26
27. 27. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Other Kinds of Correlation Attractiveness Date? 0 0 1 0 1 1 1 1 0 0 1 1 F = 0.71 • Phi coefficient (F) – used with two dichotomous scales. – uses the same Pearson formula Nov 2019 27
28. 28. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Factors Affecting r • Range restrictions – Looking at only a small portion of the total scatter plot (looking at a smaller portion of the scores’ variability) decreases r. – Reducing variability reduces r • Nonlinearity – The Pearson r (and its relatives) measure the degree of linear relationship between two variables – If a strong non-linear relationship exists, r will provide a low, or at least inaccurate measure of the true relationship Nov 2019 28
29. 29. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Factors Affecting r • Heterogeneous subsamples – Everyday examples (e.g. height and weight using both men and women) • Outliers – Overestimate Correlation – Underestimate Correlation Nov 2019 29
30. 30. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Countries With Low Consumptions Data With Restricted Range Truncated at 5 Cigarettes Per Day Cigarette Consumption per Adult per Day 5.55.04.54.03.53.02.5 CHDMortalityper10,000 20 18 16 14 12 10 8 6 4 2 Nov 2019 30
31. 31. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Truncation Nov 2019 31
32. 32. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Non-linearity Nov 2019 32
33. 33. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Heterogenous samples Nov 2019 33
34. 34. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Outliers Nov 2019 34
35. 35. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Testing Correlations • So you have a correlation. Now what? • In terms of magnitude, how big is big? – Small correlations in large samples are “big” – Large correlations in small samples aren’t always “big” • Depends upon the magnitude of the correlation coefficient • AND • The size of your sample Nov 2019 35
36. 36. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Testing r • Population parameter =  • Null hypothesis H0:  = 0 – Test of linear independence • Alternative hypothesis (H1)   0 – Two-tailed Nov 2019 36
37. 37. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Tables of Significance • We can convert r to t and test for significance: • Where DF = N - 2 t = r N -2 1- r2 Nov 2019 37
38. 38. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Tables of Significance • In our example r was 0.71 • N - 2 = 21 – 2 = 19 • T-crit (19) = 2.09 • Since 6.90 is larger than 2.09, reject r = 0. 2 2 2 19 19 .71* .71* 6.90 1 1 .71 .4959 N t r r        Nov 2019 38
39. 39. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Take home messages • Correlation coefficient describes the relationship between two variables • There are different types of correlation coefficient • Pearson’s correlation coefficient is used for two continuous variables • It ranges from -1 to +1 Nov 2019 39
40. 40. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course THANK YOU Kindly send your queries to sarizwan1986@gmail.com Nov 2019 40