Chapter 14 Power Points for Essentials of Statistics for the Behavioral Sciences, Gravetter & Wallnau, 8th ed

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’ 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 Learning Outcomes
(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 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

6. Figure 14.1 Scatterplot for
Correlational Data

7. Figure 14.2 Positive and
Negative Relationships

8. Figure 14.3 Different Linear
Relationship Values

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
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 – Computational formula
• 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
• Ratio comparing 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 14.3 Scatterplot

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
• 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

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
• Decide if 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 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

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

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

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)
2
rionDeterminatofoefficientC 

28. Figure 14.8 Three Amounts of
Linear Relationship Example

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

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
• A partial 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 Controlling the
Impact of a Third Variable

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

38. Figure 14.12 Scatterplot
Showing Scores 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 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
)1(
6
1 2
2



nn
D
rs

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
• 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

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
• Decide if 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 to Linear
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. Figure 14.13 Regression line

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

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

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
X
Y
X s
s
rb
SS
SP
b or 
XY bMMa 

56. Figure 14.16 Data Points and
Regression Line: Example 14.13

57. Standard Error of Estimate
• 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 Regression Lines:
Perfectly Fit vs. Example 14.13

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:
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 and F-ratio
residual
residual
residual
df
SS
MS 
regression
regression
regression
df
SS
MS 
residual
regression
MS
MS
F 

62. Figure 14.18 Partitioning SS
and df in Regression Analysis

63. Learning Check
• A linear 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
• Decide if 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 Output for Example 14.13

68. Figure 14.20 SPSS Output for
Examples 14.13—14.15

69. Figure 14.21 Scatter Plot for
Data of Demonstration 14.1

70. Any
Questions
?
Concepts
?
Equations?