2. Review

3. . . . . .

4. Statistics Needed
Need to find the best place to draw the regression line on a scatter plot
Need to quantify the cluster of scores around this regression line (i.e., the correlation coefficient)

5. Computational formula

6. Correlation

7. Hypothesis testing of r
Is there a significant relationship between X and Y (or are they independent)?
Are two independent correlations significantly different than each other?

8. Statistics Needed
Need to find the best place to draw the regression line on a scatter plot
Need to quantify the cluster of scores around this regression line (i.e., the correlation coefficient)

9. . . . . .

10. Regression Equation Y = a + bX Where:
Y = value predicted from a particular X value
a = point at which the regression line intersects the Y axis
b = slope of the regression line
X = X value for which you wish to predict a Y value

11. Regression

12. How to draw the regression line. . . . .

13. Hypothesis Testing Have learned
How to calculate r as an estimate of relationship between two variables
How to calculate b as a measure of the rate of change of Y as a function of X
Next determine if these values are significantly different than 0

14. Testing b The significance test for r and b are equivalent
If X and Y are related (r), then it must be true that Y varies with X (b).
Important to learn b significance tests for multiple regression

15. Calculate t-observed
b = Slope
Sb = Standard error of slope

17. Multiple Regression Good news! No Math Bad news!
Too complicated to do by hand
Bad news!
Almost all conceptual

18. Causal Models
X (IV) the cause of Y (DV)

19. Causal Models X (IV) is the cause of Y (DV)
This is an assumption – causation is not demonstrated with statistics! X Y

20. Remember Candy Depression Charlie 5 55 Augustus 7 43 Veruca 4 59 Mike
108
Violet 65

21. Remember Y = 127 + -13.26(X) COV = -30.5 N = 5 r = -.81 Sx = 1.52
Sy = 24.82

22. Causal Models
-13.26
Candy
Depression

23. Example Data collected from 15 people Salary Years since Ph.D.
Publications

24. Example
Predict the salary of a person from the time since their Ph.D. (in years)

26. Example
Predict the salary of a person from the time since their Ph.D. (in years)
Y = 51, (X)
What do these mean?
$51,670 a person tends to earn after graduating (Years = 0)
Each year after that a person’s salary increase $1,218 a year

27. Causal Models
1,218
Years since Ph.D.
Salary

28. Example
Predict the salary of a person from the number of publications they have

30. Causal Models
334
Publications
Salary

31. What if we have two IVs?
It is possible to use two IVs at the same time to predict a DV
Use both publications and years since Ph.D. to predict salary

32. Causal Models
Publications
334
Salary
1,218
Years since Ph.D.

33. Causal Models Publications 334 Salary 1,218 Years since Ph.D.
How to interpret values if IVs are independent

34. Causal Models Publications 334 Salary 1,218 Years since Ph.D.
Problem: Information provided by publications and Years is probably somewhat redundant

36. Causal Models
Publications
Salary
r = .66
Years since Ph.D.

37. Causal Models Publications Salary r = .66 Years since Ph.D.
Must estimate these regression coefficients so this relationship is taken into account (called “partial regression coefficients”)

38. Regression Coefficients
Basic logic is exactly the same as normal regression
Least squares
Has one intercept and each of the IVs has one slope

39. Regression Coefficients
bo the intercept
b1 the slope of the first IV
b2 the slope of the second IV
bp the slope of p IV
Y = bo + b1 (X1) + b2 (X2) bp (Xp)

40. Example
Predict the salary of a person from the number of publications they have and the years since they got their Ph.D.

42. Regression Coefficients
Current Problem
Y = Salary
X1 = Years since Ph.D.; X2 = Publications

43. Causal Models
Publications
122
Salary
r = .66
977
Years since Ph.D.

44. Regression Coefficients
Current Problem
Y = Salary
X1 = Years since Ph.D.; X2 = Publications
What does a person who just graduated (years = 0) with 2 publications likely earn?

45. Regression Coefficients
Current Problem
Y = Salary
X1 = Years since Ph.D.; X2 = Publications
What does a person who just graduated (years = 0) with 2 publications likely earn?

46. Regression Coefficients
Current Problem
Y = Salary
X1 = Years since Ph.D.; X2 = Publications
What does a person who graduated 10 years ago with no publications make?

47. Regression Coefficients
Current Problem
Y = Salary
X1 = Years since Ph.D.; X2 = Publications
What does a person who graduated 10 years ago with no publications make?

48. Question Current Problem Y = Salary
X1 = Years since Ph.D.; X2 = Publications
Which IV has a greater “effect” of salary?

51. Standardized Regression Coefficients
Conceptually the same as standardizing all variables and then doing regression analysis
Why does this work?
Example with Years predicting Salary

52. Standardized Regression Coefficients
With a single predictor – Unstandardized
1,218
Years since Ph.D.
Salary
With a single predictor -- Standardized
.71
Years since Ph.D.
Salary

54. Standardized Regression Coefficients
With single IV
Correlation between years and salary (r = .71) is the SAME as the standardized regression weight!

55. Standardized Regression Coefficients
β1 = Standardized Regression of first IV
β2 = Standardized Regression of second IV
βp = Standardized Regression of p IV
β0 = Intercept (always = 0)

56. Remember
Publications
122
Salary
r = .66
977
Years since Ph.D.

58. Standardized
Publications
.21
Salary
r = .66
.57
Years since Ph.D.

59. Regression Coefficients
Current Problem
Yz = Standardized Salary
Z1 = Years since Ph.D. (Standardized)
Z2 = Publications (Standardized)
Which IV has a greater “effect” of salary?
Can interpret in SD units

60. Regression Coefficients
Current Problem
Yz = Standardized Salary
Z1 = Years since Ph.D. (Standardized)
Z2 = Publications (Standardized)
What would you predict the salary to be if a person’s Years = 1.2 and a persons publications = -.50?
Interpret what these values mean!

61. Testing the full model How well does the model predict?
The fit test for the full model and its significance are equal for both standardized and unstandardized models

62. Person Z1 Z2 ZY 1
-1.26
.35
-.83 2
-.53
-.24 3
-.63
-1.40
-.84 4
.63
1.23
1.56 5
1.26
.36

63. Person Z1 Z2 ZY
Pred 1
-1.26
.35
-.83
-.477 2
-.53
-.24
-.287 3
-.63
-1.40
-.84
-1.097 4
.63
1.23
1.56
1.01 5
1.26
.36
.862

64. Person Z1 Z2 ZY Pred 1 -1.26 .35 -.83 -.477 2 -.53 -.24 -.287 3 -.63
-.53
-.24
-.287 3
-.63
-1.40
-.84
-1.097 4
.63
1.23
1.56
1.01 5
1.26
.36
.862
r = .902

65. Multiple R

66. Multiple R

67. Testing for Significance
Once an equation is created (standardized or unstandardized) typically test for significance.
Two levels
1) Level of each regression coefficient
2) Level of the entire model

68. Testing for Significance
Once an equation is created (standardized or unstandardized) typically test for significance.
Two levels
1) Level of each regression coefficient
2) Level of the entire model

69. Multiple R Commonly used as R2 Can be tested for significance
Pros and Cons
Can be tested for significance
Does the set of variables (taken together) predict Y at better than chance levels?
H1 : R* > 0
Ho : R* = 0

70. Person Z1 Z2 ZY Pred 1 -1.26 .35 -.83 -.477 2 -.53 -.24 -.287 3 -.63
-.53
-.24
-.287 3
-.63
-1.40
-.84
-1.097 4
.63
1.23
1.56
1.01 5
1.26
.36
.862
r = .902

71. Significance testing for Multiple R
p = number of predictors
N = total number of observations

72. Significance testing for Multiple R
p = number of predictors
N = total number of observations

73. Significance testing for Multiple R
p = number of predictors
N = total number of observations

74. Significance testing for Multiple R
Fcrit
Page # 737
Need two df
Numerator df = p
Denominator df = N – p - 1

75. Significance testing for Multiple R
Fcrit
Need two df
Numerator df = p
Denominator df = N – p - 1
F (2, 2) = 19.00

76. Multiple R If F > Fcrit reject Ho and accept H1
If F < or = Fcrit fail to reject Ho
Current problem – fail to reject Ho
These two variables do not predict the outcome

78. Practice
The teaching salary example
Based on 15 people
Two IVs

79. Significance testing for Multiple R
p = number of predictors
N = total number of observations

80. Significance testing for Multiple R
F crit (2,12) = 3.89

81. Multiple R If F > Fcrit reject Ho and accept H1
If F < or = Fcrit fail to reject Ho
Current problem – accept H1
These two variables do predict the outcome

83. Detour
Moving back to issues of correlation
This will help with . . .

84. Testing for Significance
Once an equation is created (standardized or unstandardized) typically test for significance.
Two levels
1) Level of each regression coefficient
2) Level of the entire model

86. How strong is the relationship between publications and salary if we partial out the effect of years?
What this is saying

87. Salary

88. Salary
Publications

89. r2 SP = .35
Salary
Publications

90. r2 SP = .35 Salary Publications
r2 is a ratio = Variance explained / Total Variance
Total variance of Salary = 1 (standardized)

91. r2 SP = .35
.65
Salary
Publications

92. e
Salary a b c
Publications
Years

93. ? e
Salary a b c
Publications
Years

94. ? e
Salary a b c
Publications
Years
How strong is the relationship between publications and salary if we partial out the effect of years?

95. Semipartial correlation of publications and salary
Years

96. Semipartial correlation of publications and salary
Years
Multiple R2 = a + c + b

97. Multiple R

98. R2 = .53 or a + b + c e
Salary a b c
Publications
Years

99. R2 = .53 or a + b + c r2SY = .50 or b + c r2SP = .35 or a + c e Salary
Publications
Years

100. R2 = .53 or a + b + c r2SY = .50 or b + c r2SP = .35 or a + c
So what is just “a”? e
Salary a b c
Publications
Years

101. R2 = .53 or a + b + c r2SY = .50 or b + c r2SP = .35 or a + c
So what is just “a”? e
Salary a b c
Publications
Years
a = (a + b + c) – (b + c)

102. R2 = .53 or a + b + c r2SY = .50 or b + c r2SP = .35 or a + c
So what is just “a”? e
Salary a b c
Publications
Years
a = (a + b + c) – (b + c) or R2 – r2sy

103. R2 = .53 or a + b + c r2SY = .50 or b + c r2SP = .35 or a + c
So what is just “a”? e
Salary a b c
Publications
Years
a = R2 – r2sy = = .03
Thus semipartial correlation = .17

104. R2 = .53 or a + b + c r2SY = .50 or b + c r2SP = .35 or a + c
So what is just “a”? e
Salary a b c
Publications
Years
What is the correlation between years and salary controlling for publications?