1. Chapter 14 Correlation and Regression
PowerPoint Lecture Slides Essentials of Statistics for the Behavioral Sciences Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau

2. Chapter 14 Learning Outcomes
Understand Pearson r as measure of variables’ relationship 2
Compute Pearson r using definitional or computational formula 3
Use and interpret Pearson r; understand assumptions & limitations 4
Test hypothesis about population correlation (ρ) with sample r 5
Understand the concept of a partial correlation

3. Chapter 14 Learning Outcomes (continued)6
Explain/compute Spearman correlation coefficient (ranks) 7
Explain/compute point-biserial correlation coefficient (one dichotomous variable) 8
Explain/compute phi-coefficient for two dichotomous variables 9
Explain/compute linear regression equation to predict Y values 10
Evaluate significance of regression equation

4. Tools You Will Need 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
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.

6. Figure 14.1 Scatterplot for Correlational Data
FIGURE 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.

7. Figure 14.2 Positive and Negative Relationships
FIGURE Examples of positive and negative relationships. (a) Beer sales are positively related to temperature. (b) Coffee sales are negatively related to temperature.

8. Figure 14.3 Different Linear Relationship Values
FIGURE 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

9. 14.2 The Pearson Correlation
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

10. Sum of Products (SP) Similar to SS (sum of squared deviations)
Measures the amount of covariability between two variables
SP definitional formula:

11. SP – Computational formula
Definitional formula emphasizes SP as the sum of two difference scores
Computational formula results in easier calculations
SP computational formula:

12. Pearson Correlation Calculation
Ratio comparing the covariability of X and Y (numerator) with the variability of X and Y separately (denominator)

13. Figure 14.4 Example 14.3 Scatterplot
FIGURE Scatter plot of the data from Example 14.3.

14. Pearson Correlation and z-Scores
Pearson correlation formula can be expressed as a relationship of z-scores.

15. Learning Check
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? A
0.75 B
0.35 C
-0.75 D
-0.35

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? A
0.75 B
0.35 C
-0.75 D
-0.35

17. Learning Check
Decide if each of the following statements is True or False
T/F
A set of n = 10 pairs of X and Y scores has ΣX = ΣY = ΣXY = For this set of scores, SP = –20
If the Y variable decreases when the X variable decreases, their correlation is negative

18. Learning Check - Answers
True
False
The variables change in the same direction, a positive correlation

19. 14.3 Using and Interpreting the Pearson Correlation
Correlations used for:
Prediction
Validity
Reliability
Theory verification

20. Interpreting Correlations
Correlation describes 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: Churches and Serious Crimes
FIGURE Hypothetical data showing the logical relationship between the number of churches and the number of serious crimes for a sample of U.S. cities.

22. Correlations and Restricted Range of 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 Restricted Score Range Influences Correlation
FIGURE 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.

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 Outlier Influences Size of Correlation
FIGURE A demonstration of how one extreme data point (an outlier) can influence the value of a correlation.

26. Correlations and the Strength 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)

28. Figure 14.8 Three Amounts of Linear Relationship Example
FIGURE Three sets of data showing three different degrees of linear relationship.

29. 14.4 Hypothesis Tests with 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 Correlation in Sample vs. Population
FIGURE 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.

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

32. In the Literature Report Concise test results
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
A partial correlation measures the relationship between two variables while mathematically controlling the influence of a third variable by holding it constant

34. Figure 14.10 Controlling the Impact of a Third Variable
FIGURE 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).

35. 14.5 Alternatives to the 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 Nonlinear Positive Relationship
FIGURE 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.

38. Figure 14.12 Scatterplot Showing Scores and Ranks
FIGURE Scatter plots showing (a) the scores and (b) the ranks for the data in Example 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.

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 for the 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
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.

41. Point-Biserial Correlation
Measures relationship 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
Applicable in 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 Both variables (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 Point-biserial correlation Spearman correlation
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? A
Pearson correlation B
Spearman correlation C
Point-biserial correlation D
Phi-coefficient

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? A
Pearson correlation B
Spearman correlation C
Point-biserial correlation D
Phi-coefficient

46. Learning Check
Decide if each of the following statements is True or False
T/F
The Spearman correlation is used with dichotomous data
In a non-directional significance test of a correlation, the null hypothesis states that the population correlation is zero

47. Learning Check - Answers
False
The Spearman correlation uses ordinal (ranked) data
True
Null hypothesis assumes no relationship; ρ = zero indicates no relationship in the population

48. 14.6 Introduction to Linear Equations and Regression
The Pearson correlation measures a linear relationship between two variables
Figure 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. Figure 14.13 Regression line
FIGURE 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).

50. Linear Equations General equation for a line Equation: Y = bX + a
X and Y are variables
a and b are fixed constant

51. Figure 14.14 Linear Equation Graph
FIGURE 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.

52. Regression
Regression is a 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 is the 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-Ŷ Distance: Actual Data Point Minus Predicted Point
FIGURE 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.

55. Regression Equations Regression line equation: Ŷ = bX + a
The slope of the line, b, can be calculated
The line goes through (MX,MY) therefore

56. Figure 14.16 Data Points and Regression Line: Example 14.13
FIGURE The X and Y data points and the regression line for the n = 8 pairs of scores in Example

57. Standard Error of Estimate
Regression equation makes a prediction
Precision of the estimate is measured by the standard error of estimate (SEoE)
SEoE =

58. Figure 14.17 Regression Lines: Perfectly Fit vs. Example 14.13
FIGURE (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 = (b) A scatter plot for the data in Example Notice that there is error between the actual data points and the predicted Y values on the regression line.

59. Relationship Between Correlation and Standard 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:

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 and F-ratio

62. Figure 14.18 Partitioning SS and df in Regression Analysis
FIGURE 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.

63. Learning Check
A linear regression has b = 3 and a = 4. What is the “predicted Y” (Ŷ) for X = 7? A 14 B 25 C 31 D
Cannot be determined

64. Learning Check - Answer
A linear regression has b = 3 and a = 4. What is the predicted Y for X = 7? A 14 B 25 C 31 D
Cannot be determined

65. Learning Check
Decide if each of the following statements is True or False
T/F
It is possible for the regression equation to place none of the actual data points on the regression line
If r = 0.58, the linear regression equation predicts about one third of the variance in the Y scores

66. Learning Check - Answers
True
The line estimates where points should be but there are almost always prediction errors
When r = .58, r2 = .336 (≈1/3)

67. Figure 14.19 SPSS Output for Example 14.13
FIGURE The SPSS output showing the correlation for the data in Example

68. Figure 14.20 SPSS Output for Examples 14.13—14.15
FIGURE Portions of the SPSS output from the analysis of regression for the data in Examples 14.13, 14.14, and

69. Figure 14.21 Scatter Plot for Data of Demonstration 14.1
FIGURE The scatter plot for the data of Demonstration An envelope is drawn around the points to estimate the magnitude of the correlation. A line is drawn through the middle of the envelope.

70. Any Questions?
Concepts?
Equations?