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Types of correlation coefficients

A brief description of different types of correlation coefficients

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

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