1. Chapter 15 Correlation and Regression
PowerPoint Lecture Slides Essentials of Statistics for the Behavioral Sciences Seventh Edition by Frederick J. Gravetter and Larry B. Wallnau

2. Chapter 15 Learning Outcomes
Explain & compute Pearson correlation to variables’ relationship 2
Test hypothesis about population correlation with sample r 3
Explain & compute Spearman correlation 4
Compute point-biserial correlation and phi-coefficient 5
Explain & compute linear regression equation to predict Y values 6
Evaluate significance of regression equation

3. Concepts to review Sum of squares (SS) (Chapter 4)
Computational formula
Definitional formula
z-Scores (Chapter 5)
Hypothesis testing (Chapter 8)

4. 15.1 Introduction to Correlation and Regression
Measures and describes a relationship between two variables.
Characteristics of relationships
Direction (negative or positive)
Form (linear is most common)
Strength

5. Figure 15.1 Scatterplot for correlational data

6. Figure 15.2 Examples of positive and negative relationships

7. Figure 15.3 Examples of different values for linear relationships

8. 15.2 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

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

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

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

12. Figure 15.4 Example 15.3 Scatterplot

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

14. Learning Check
A scatterplot shows a set of data points that are clustered 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

15. Learning Check
A scatterplot shows a set of data points that are clustered 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 TF
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

17. Answer TF
True
False
The variables change in the same direction, a positive correlation

18. 15.3 Using and Interpreting the Pearson Correlation
Correlations used for prediction
Validity
Reliability
Theory verification

19. Figure 15.5 Number of churches and number of serious crimes

20. Interpreting correlations
Correlation does not demonstrate causation
Value of correlation is affected by the range of scores in the data
Extreme points – outliers – have an impact
Correlation cannot be interpreted as a proportion.
To show the shared variability, need to square the correlation

21. Figure 15.6 Restricted range and correlation

22. Figure 15.7 Influence of outlier on correlation

23. 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.

24. Figure 15.8 Three degrees of linear relationship

25. 15.4 Hypothesis Testing 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)
Nondirectional: H0: ρ = 0 and H1: ρ ≠ 0
Directional: H0: ρ ≤ 0 and H1: ρ > 0

26. Figure 15.9 Correlation of sample and population

27. Hypothesis Test for Correlations
Sample correlation used to test population ρ
Degrees of freedom (df) = n – 2
Hypothesis test can be computed using either t or F.
Critical Values have been computed
See Table B.6
A sample correlation beyond ± Critical Value is very unlikely
A sample correlation beyond ± Critical Value leads to rejecting the null hypothesis.

28. Partial correlation
A partial correlation measures the relationship between two variables while controlling the influence of a third variable by holding it constant

29. Figure 15.10 Controlling the impact of a third variable

30. 15.5 Alternative to the Pearson Correlation
Pearson correlation has been developed
for linear relationships
for interval or ratio data
Other correlations have been developed for
non-linear data
other types of data

31. Spearman correlation
Pearson correlation formula is used with data from an ordinal scale (ranks)
Used when both variables are measured on an ordinal scale
Used when relationship is consistently directional but may not be linear

32. Figure 15.11 Consistent nonlinear positive relationship

33. Figure 15.12 Scatterplot showing scores and ranks

34. 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.

35. 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

36. Point-Biserial Correlation
Measures relationship between two variables
One variable has only two values (dichotomous variable)
Same situation as the independent samples t-test in Chapter 10
Point-biserial r2 has same value as the r2 computed from t-statistic
t-statistic evaluates the significance
r statistic measures its strength

37. Phi Coefficient Both variables (X and Y) are dichotomous
Both variables are re-coded to values 0 and 1
The regular Pearson formulas is used

38. 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

39. 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

40. 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 test for significance of a correlation, the null hypothesis states that the population correlation is zero.

41. Answer The Spearman correlation uses ordinal (ranked) data
False
The Spearman correlation uses ordinal (ranked) data
True
Zero indicates no relationship

42. 15.6 Introduction to Linear Equations and Regression
The Pearson correlation measures a linear relationship between two variables
The line through the data
Makes the relationship easier to see
Shows the central tendency of the relationship
Can be used for prediction

43. Figure 15.13 Regression line

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

45. Figure 15.14 Graph of a linear equation

46. Regression
Regression is the method for determining the best-fitting line through a set of data
The line is called the regression line
Ŷ is the value of Y predicted by the regression equation for each value of X
(Y- Ŷ) is the distance of each data point from the regression line: the error of prediction
Regression minimizes total squared error

47. Figure 15.15 Distance between data point & the predicted point

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

49. Figure 15.16 X and Y points and regression line

50. Figure 15.17 Perfectly fit regression line and regression line for Example 15.13

51. Correlation and the standard error
Predicted variability in Y scores: SSregression = r2 SSY
Unpredicted variability in Y scores: SSresidual = (1 - r2) SSY

52. Standard error of estimate
Regression equation makes prediction
Precision of the estimate is measured by the standard error of estimate

53. Testing significance of regression
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

54. Mean squares and F-ratio

55. Figure 15.18 Partitioning of the SS and df in Analysis of Regression

56. Figure 15.19 Plot of data in Demonstration 15.1

57. Learning Check
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

58. 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

59. Learning Check
Decide if each of the following statements is True or False.
T/F
It is possible for the regression equation to have 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.

60. Answer
True
The line is an estimator.
When r = .58, r2 = .336

61. Any Questions?
Concepts?
Equations?