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Correlation and Regression
- 1. Chapter 14 Correlation andRegression PowerPoint Lecture Slides Essentials of Statistics for the Behavioral Sciences Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
- 2. Chapter 14 LearningOutcomes • Understand Pearson r as measure of variables’ relationship1 • Compute Pearson r using definitional or computational formula2 • Use and interpret Pearson r; understand assumptions & limitations3 • Test hypothesis about population correlation (ρ) with sample r4 • Understand the concept of a partial correlation5
- 3. Chapter 14 LearningOutcomes (continued) • Explain/compute Spearman correlation coefficient (ranks)6 • Explain/compute point-biserial correlation coefficient (one dichotomous variable)7 • Explain/compute phi-coefficient for two dichotomous variables8 • Explain/compute linear regression equation to predict Y values9 • Evaluate significance of regression equation10
- 4. Tools You WillNeed • Sum of squares (SS) (Chapter 4) – Computational formula – Definitional formula • z-Scores (Chapter 5) • Hypothesis testing (Chapter 8) • Analysis of Variance (Chapter 12) – MS values and F-ratios
- 5. 14.1 Introduction to Correlation •Measures and describes the relationship between two variables • Characteristics of relationships – Direction (negative or positive; indicated by the sign, + or – of the correlation coefficient) – Form (linear is most common) – Strength or consistency (varies from 0 to 1) • Characteristics are all independent
- 6. Figure 14.1 Scatterplotfor Correlational Data
- 7. Figure 14.2 Positiveand Negative Relationships
- 8. Figure 14.3 DifferentLinear Relationship Values
- 9. 14.2 The PearsonCorrelation • Measures the degree and the direction of the linear relationship between two variables • Perfect linear relationship – Every change in X has a corresponding change in Y – Correlation will be –1.00 or +1.00 yseparatelYandXofvariablity YandXofitycovariabil r
- 10. Sum of Products(SP) • Similar to SS (sum of squared deviations) • Measures the amount of covariability between two variables • SP definitional formula: ))(( YX MYMXSP
- 11. SP – Computationalformula • Definitional formula emphasizes SP as the sum of two difference scores • Computational formula results in easier calculations • SP computational formula: n YX XYSP
- 12. Pearson Correlation Calculation • Ratiocomparing the covariability of X and Y (numerator) with the variability of X and Y separately (denominator) YX SSSS SP r
- 13. Figure 14.4 Example 14.3Scatterplot
- 14. Pearson Correlation and z-Scores •Pearson correlation formula can be expressed as a relationship of z-scores. N zz n zz r YX YX :Population 1 :Sample
- 15. Learning Check • Ascatterplot shows a set of data points that fit very loosely around a line that slopes down to the right. Which of the following values would be closest to the correlation for these data? • 0.75A • 0.35B • -0.75C • -0.35D
- 16. Learning Check -Answer • A scatterplot shows a set of data points that fit very loosely around a line that slopes down to the right. Which of the following values would be closest to the correlation for these data? • 0.75A • 0.35B • -0.75C • -0.35D
- 17. Learning Check • Decideif each of the following statements is True or False • A set of n = 10 pairs of X and Y scores has ΣX = ΣY = ΣXY = 20. For this set of scores, SP = –20 T/F • If the Y variable decreases when the X variable decreases, their correlation is negative T/F
- 18. Learning Check -Answers 204020 10 )20)(20( 20 SP
- 19. 14.3 Using andInterpreting the Pearson Correlation • Correlations used for: – Prediction – Validity – Reliability – Theory verification
- 20. Interpreting Correlations • Correlationdescribes a relationship but does not demonstrate causation • Establishing causation requires an experiment in which one variable is manipulated and others carefully controlled • Example 14.4 (and Figure 14.5) demonstrates the fallacy of attributing causation after observing a correlation
- 21. Figure 14.5 Correlation: Churchesand Serious Crimes
- 22. Correlations and Restricted Rangeof Scores • Correlation coefficient value (size) will be affected by the range of scores in the data • Severely restricted range may provide a very different correlation than would a broader range of scores • To be safe, never generalize a correlation beyond the sample range of data
- 23. Figure 14.6 RestrictedScore Range Influences Correlation
- 24. Correlations and Outliers •An outlier is an extremely deviant individual in the sample • Characterized by a much larger (or smaller) score than all the others in the sample • In a scatter plot, the point is clearly different from all the other points • Outliers produce a disproportionately large impact on the correlation coefficient
- 25. Figure 14.7 OutlierInfluences Size of Correlation
- 26. Correlations and theStrength of the Relationship • A correlation coefficient measures the degree of relationship on a scale from 0 to 1.00 • It is easy to mistakenly interpret this decimal number as a percent or proportion • Correlation is not a proportion • Squared correlation may be interpreted as the proportion of shared variability • Squared correlation is called the coefficient of determination
- 27. Coefficient of Determination •Coefficient of determination measures the proportion of variability in one variable that can be determined from the relationship with the other variable (shared variability) 2 rionDeterminatofoefficientC
- 28. Figure 14.8 ThreeAmounts of Linear Relationship Example
- 29. 14.4 Hypothesis Testswith the Pearson Correlation • Pearson correlation is usually computed for sample data, but used to test hypotheses about the relationship in the population • Population correlation shown by Greek letter rho (ρ) • Non-directional: H0: ρ = 0 and H1: ρ ≠ 0 Directional: H0: ρ ≤ 0 and H1: ρ > 0 or Directional: H0: ρ ≥ 0 and H1: ρ < 0
- 30. Figure 14.9 Correlationin Sample vs. Population
- 31. Correlation Hypothesis Test •Sample correlation r used to test population ρ • Degrees of freedom (df) = n – 2 • Hypothesis test can be computed using either t or F; only t shown in this chapter • Use t table to find critical value with df = n - 2 )2( )1( 2 n r r t
- 32. In the Literature •Report – Whether it is statistically significant • Concise test results – Value of correlation – Sample size – p-value or level – Type of test (one- or two-tailed) • E.g., r = -0.76, n = 48, p < .01, two tails
- 33. Partial Correlation • Apartial correlation measures the relationship between two variables while mathematically controlling the influence of a third variable by holding it constant )1)(1( )( 22 yzxz yzxyxy zxy rr rrr r
- 34. Figure 14.10 Controllingthe Impact of a Third Variable
- 35. 14.5 Alternatives tothe Pearson Correlation • Pearson correlation has been developed – For data having linear relationships – With data from interval or ratio measurement scales • Other correlations have been developed – For data having non-linear relationships – With data from nominal or ordinal measurement scales
- 36. Spearman Correlation • Spearman(rs) correlation formula is used with data from an ordinal scale (ranks) – Used when both variables are measured on an ordinal scale – Also may be used if measurement scales is interval or ratio when relationship is consistently directional but may not be linear
- 37. Figure 14.11 Consistent NonlinearPositive Relationship
- 38. Figure 14.12 Scatterplot ShowingScores and Ranks
- 39. Ranking Tied Scores •Tie scores need ranks for Spearman correlation • Method for assigning rank – List scores in order from smallest to largest – Assign a rank to each position in the list – When two (or more) scores are tied, compute the mean of their ranked position, and assign this mean value as the final rank for each score.
- 40. Special Formula forthe Spearman Correlation • The ranks for the scores are simply integers • Calculations can be simplified – Use D as the difference between the X rank and the Y rank for each individual to compute the rs statistic )1( 6 1 2 2 nn D rs
- 41. Point-Biserial Correlation • Measuresrelationship between two variables – One variable has only two values (called a dichotomous or binomial variable) • Effect size for independent samples t-test in Chapter 10 can be measures by r2 – Point-biserial r2 has same value as the r2 computed from t-statistic – t-statistic tests significance of the mean difference – r statistic measures the correlation size
- 42. Point-Biserial Correlation • Applicablein the same situation as the independent-measures t test in Chapter 10 – Code one group 0 and the other 1 (or any two digits) as the Y score – t-statistic evaluates the significance of mean difference – Point-Biserial r measures correlation magnitude – r2 quantifies effect size
- 43. Phi Coefficient • Bothvariables (X and Y) are dichotomous – Both variables are re-coded to values 0 and 1 (or any two digits) – The regular Pearson formulas is used to calculate r – r2 (coefficient of determination) measures effect size (proportion of variability in one score predicted by the other)
- 44. Learning Check • Participantswere classified as “morning people” or “evening people” then measured on a 50-point conscientiousness scale. Which correlation should be used to measure the relationship? • Pearson correlationA • Spearman correlationB • Point-biserial correlationC • Phi-coefficientD
- 45. Learning Check -Answer • Participants were classified as “morning people” or “evening people” then measured on a 50-point conscientiousness scale. Which correlation should be used to measure the relationship? • Pearson correlationA • Spearman correlationB • Point-biserial correlationC • Phi-coefficientD
- 46. Learning Check • Decideif each of the following statements is True or False • The Spearman correlation is used with dichotomous dataT/F • In a non-directional significance test of a correlation, the null hypothesis states that the population correlation is zero T/F
- 47. Learning Check -Answers • The Spearman correlation uses ordinal (ranked) dataFalse • Null hypothesis assumes no relationship; ρ = zero indicates no relationship in the population True
- 48. 14.6 Introduction toLinear Equations and Regression • The Pearson correlation measures a linear relationship between two variables • Figure 14.13 makes the relationship obvious • The line through the data – Makes the relationship easier to see – Shows the central tendency of the relationship – Can be used for prediction • Regression analysis precisely defines the line
- 49.
- 50. Linear Equations • Generalequation for a line – Equation: Y = bX + a – X and Y are variables – a and b are fixed constant
- 51.
- 52. Regression • Regression isa method of finding an equation describing the best-fitting line for a set of data • How to define a “best fitting” straight line when there are many possible straight lines? • The answer: a line that is the best fit for the actual data that minimizes prediction errors
- 53. Regression • Ŷ isthe value of Y predicted by the regression equation (regression line) for each value of X • (Y- Ŷ) is the distance each data point is from the regression line: the error of prediction • The regression procedure produces a line that minimizes total squared error of prediction • This method is called the least-squared-error solution
- 54. Figure 14.15 Y-ŶDistance: Actual Data Point Minus Predicted Point
- 55. Regression Equations • Regressionline equation: Ŷ = bX + a • The slope of the line, b, can be calculated • The line goes through (MX,MY) therefore X Y X s s rb SS SP b or XY bMMa
- 56. Figure 14.16 DataPoints and Regression Line: Example 14.13
- 57. Standard Error ofEstimate • Regression equation makes a prediction • Precision of the estimate is measured by the standard error of estimate (SEoE) SEoE = 2 )ˆ( 2 n YY df SSresidual
- 58. Figure 14.17 RegressionLines: Perfectly Fit vs. Example 14.13
- 59. Relationship Between Correlation andStandard Error of Estimate • As r goes from 0 to 1, SEoE decreases to 0 • Predicted variability in Y scores: SSregression = r2 SSY • Unpredicted variability in Y scores: SSresidual = (1 - r2) SSY • Standard Error of Estimate based on r: 2 )1( 2 n SSr df SS Yresidual
- 60. Testing Regression Significance •Analysis of Regression – Similar to Analysis of Variance – Uses an F-ratio of two Mean Square values – Each MS is a SS divided by its df • H0: the slope of the regression line (b or beta) is zero
- 61. Mean Squares andF-ratio residual residual residual df SS MS regression regression regression df SS MS residual regression MS MS F
- 62. Figure 14.18 PartitioningSS and df in Regression Analysis
- 63. Learning Check • Alinear regression has b = 3 and a = 4. What is the “predicted Y” (Ŷ) for X = 7? • 14A • 25B • 31C • Cannot be determinedD
- 64. Learning Check -Answer • A linear regression has b = 3 and a = 4. What is the predicted Y for X = 7? • 14A • 25B • 31C • Cannot be determinedD
- 65. Learning Check • Decideif each of the following statements is True or False • It is possible for the regression equation to place none of the actual data points on the regression line T/F • If r = 0.58, the linear regression equation predicts about one third of the variance in the Y scores T/F
- 66. Learning Check -Answers • The line estimates where points should be but there are almost always prediction errors True • When r = .58, r2 = .336 (≈1/3)True
- 67. Figure 14.19 SPSS Outputfor Example 14.13
- 68. Figure 14.20 SPSSOutput for Examples 14.13—14.15
- 69. Figure 14.21 ScatterPlot for Data of Demonstration 14.1
- 70.
Editor's Notes
- #6 Instructors may wish to note that correlation coefficient cannot be smaller than -1 nor larger then +1. If calculation produce a number outside those boundaries, an error was made.
- #7 FIGURE 14.1 Correlational data showing the relationship between family income (X) and student grades (Y) for a sample of n = 14 high school students. The scores are listed in order from lowest to highest family income and are shown in a scatter plot.
- #8 FIGURE 14.2 Examples of positive and negative relationships. (a) Beer sales are positively related to temperature. (b) Coffee sales are negatively related to temperature.
- #9 FIGURE 14.3 Examples of difference values for linear correlations: (a) a perfect negative correlation, -1.00; (b) no linear trend, 0.00; (c) a strong positive relationship, approximately +0.90; (d) a relatively weak negative correlation, approximately -0.40.
- #14 FIGURE 14.4 Scatter plot of the data from Example 14.3.
- #22 FIGURE 14.5 Hypothetical data showing the logical relationship between the number of churches and the number of serious crimes for a sample of U.S. cities.
- #24 FIGURE 14.6 In this example, the full range of X and Y values shows a strong, positive correlation, but the restricted range of scores produces a correlation near zero.
- #26 FIGURE 14.7 A demonstration of how one extreme data point (an outlier) can influence the value of a correlation.
- #29 FIGURE 14.8 Three sets of data showing three different degrees of linear relationship.
- #31 FIGURE 14.9 Scatter plot of a population of X and Y values with near-zero correlation. However, a small sample of n = 3 data points from this population shows a relatively strong, positive correlation. Data points in the sample are circled.
- #35 FIGURE 14.10 Hypothetical data showing the relationship between the number of churches and the number of crimes for three groups of cities. Those with small populations (Z = 1), those with medium populations (Z = 2, and those with large populations (Z = 3).
- #38 FIGURE 14.11 Hypothetical data showing the relationship between practice and performance. Although this relationship is not linear, there is a consistent positive relationship. An increase in performance tends to accompany an increase in practice.
- #39 FIGURE 14.12 Scatter plots showing (a) the scores and (b) the ranks for the data in Example 15.9. Notice that there is a consistent, positive relationship between the X and Y scores, although it is not a linear relationship. Also, notice that the scatter plot of the ranks shows a perfect linear relationship.
- #41 NOTE: This special formula is accurate ONLY when there are no tied ranks; as the number of tied ranks increases, the accuracy of the formula decreases. Also not that the value of the fraction has to be computed first, then subtracted from 1.
- #50 FIGURE 14.13 Hypothetical data showing the relationship between SAT scores and GPA with a regression line drawn through the data points. The regression line defines a precise, one-to-one relationships between each X value (SAT score) and its corresponding Y value (GPA).
- #52 FIGURE 14.14 The relationship between total cost and number of videos rented each month. The video store charges a $5 monthly membership fee and $2 for each video rented. The relationship is described by a linear equation, Y = 2X + 5, where Y is the total cost and X is the number of videos.
- #55 FIGURE 14.15 The distance between the actual data points (Y) and the predicted point on the line (Ŷ) is defined as Y – Ŷ. The goal of regression is to find the equation for the line that minimizes these distances.
- #57 FIGURE 14.16 The X and Y data points and the regression line for the n = 8 pairs of scores in Example 14.13.
- #59 FIGURE 14.17 (a) A scatter plot showing data points that perfectly fit the regression line defined by the equation Ŷ = 2X – 1. Note that the correlation is r = +1.00. (b) A scatter plot for the data in Example 14.15. Notice that there is error between the actual data points and the predicted Y values on the regression line.
- #63 FIGURE 14.18 The partitioning of SS and df for analysis of regression. The variability in the original Y scores (both SSY and dfY) is partitioned into two components: (1) the variability that is explained by the regression equation, and (2) the residual variability.
- #68 FIGURE 14.19 The SPSS output showing the correlation for the data in Example 14.13.
- #69 FIGURE 14.20 Portions of the SPSS output from the analysis of regression for the data in Examples 14.13, 14.14, and 14.15.
- #70 FIGURE 14.21 The scatter plot for the data of Demonstration 14.1. An envelope is drawn around the points to estimate the magnitude of the correlation. A line is drawn through the middle of the envelope.