1. Chapter 12 Multiple Regression
Learn….
To use Multiple Regression Analysis to predict a response variable using more than one explanatory variable.

2. How Can We Use Several Variables to Predict a Response?
Section 12.1
How Can We Use Several Variables to Predict a Response?

3. Regression Models
The model that contains only two variables, x and y, is called a bivariate model

4. Regression Models
The regression equation for the bivariate model is:

5. Regression Models
Suppose there are two predictors, denoted by x1 and x2
This is called a multiple regression model

6. Regression Models
The regression equation for this multiple regression model with two predictors is:

7. Multiple Regression Model
The multiple regression model relates the mean µy of a quantitative response variable y to a set of explanatory variables x1, x2,….

8. Multiple Regression Model
Example: For three explanatory variables, the multiple regression equation is:

9. Multiple Regression Model
Example: The sample prediction equation with three explanatory variables is:

10. Example: Predicting Selling Price Using House and Lot Size
The data set “house selling prices” contains observations on 100 home sales in Florida in November 2003
A multiple regression analysis was done with selling price as the response variable and with house size and lot size as the explanatory variables

11. Example: Predicting Selling Price Using House and Lot Size
Output from the analysis:

12. Example: Predicting Selling Price Using House and Lot Size
Prediction Equation:
where y = selling price, x1=house size and x2 = lot size

13. Example: Predicting Selling Price Using House and Lot Size
One house listed in the data set had house size = 1240 square feet, lot size = 18,000 square feet and selling price = $145,000
Find its predicted selling price:

14. Example: Predicting Selling Price Using House and Lot Size
Find its residual:
The residual tells us that the actual selling price was $37,724 higher than predicted

15. The Number of Explanatory Variables
You should not use many explanatory variables in a multiple regression model unless you have lots of data
A rough guideline is that the sample size n should be at least 10 times the number of explanatory variables

16. Plotting Relationships
Always look at the data before doing a multiple regression
Most software has the option of constructing scatterplots on a single graph for each pair of variables
This is called a scatterplot matrix

17. Plotting Relationships

18. Interpretation of Multiple Regression Coefficients
The simplest way to interpret a multiple regression equation looks at it in two dimensions as a function of a single explanatory variable
We can look at it this way by fixing values for the other explanatory variable(s)

19. Interpretation of Multiple Regression Coefficients
Example using the housing data:
Suppose we fix x1 = house size at 2000 square feet
The prediction equation becomes:

20. Interpretation of Multiple Regression Coefficients
Since the slope coefficient of x2 is 2.84, the predicted selling price for 2000 square foot houses increases by $2.84 for every square foot increase in lot size
For a 1000 square-foot increase in lot size, the predicted selling price of 2000 sq. ft. houses increases by 1000(2.84) = $2840

21. Interpretation of Multiple Regression Coefficients
Example using the housing data:
Suppose we fix x2 = lot size at 30,000 square feet
The prediction equation becomes:

22. Interpretation of Multiple Regression Coefficients
Since the slope coefficient of x1 is 53.8, the predicted selling price for houses with a lot size of 30,000 sq. ft. increases by $53.80 for every square foot increase in house size

23. Interpretation of Multiple Regression Coefficients
In summary, an increase of a square foot in house size has a larger impact on the selling price ($53.80) than an increase of a square foot in lot size ($2.84)
We can compare slopes for these explanatory variables because their units of measurement are the same (square feet)
Slopes cannot be compared when the units differ

24. Summarizing the Effect While Controlling for a Variable
The multiple regression model assumes that the slope for a particular explanatory variable is identical for all fixed values of the other explanatory variables

25. Summarizing the Effect While Controlling for a Variable
For example, the coefficient of x1 in the prediction equation:
is 53.8 regardless of whether we plug in x2 = 10,000 or x2 = 30,000 or x2 = 50,000

26. Summarizing the Effect While Controlling for a Variable

27. Slopes in Multiple Regression and in Bivariate Regression
In multiple regression, a slope describes the effect of an explanatory variable while controlling effects of the other explanatory variables in the model

28. Slopes in Multiple Regression and in Bivariate Regression
Bivariate regression has only a single explanatory variable
A slope in bivariate regression describes the effect of that variable while ignoring all other possible explanatory variables

29. Importance of Multiple Regression
One of the main uses of multiple regression is to identify potential lurking variables and control for them by including them as explanatory variables in the model

30. For all students at Walden Univ
For all students at Walden Univ., the prediction equation for y = college GPA and x1= H.S. GPA and x2= study time is:
Find the predicted college GPA of a student who has a H.S. GPA of 3.5 and who studies 3 hrs. per day.
3.67
3.005
3.175
3.4

31. For all students at Walden Univ
For all students at Walden Univ., the prediction equation for y = college GPA and x1= H.S. GPA and x2= study time is:
For students with fixed study time, what is the change in predicted college GPA when H.S. GPA increases from 3.0 to 4.0?
1.13
0.0078
0.643
1.00

32. Extending the Correlation and R-Squared for Multiple Regression
Section 12.2
Extending the Correlation and R-Squared for Multiple Regression

33. Multiple Correlation
To summarize how well a multiple regression model predicts y, we analyze how well the observed y values correlate with the predicted y values
The multiple correlation is the correlation between the observed y values and the predicted y values
It is denoted by R

34. Multiple Correlation
For each subject, the regression equation provides a predicted value
Each subject has an observed y-value and a predicted y-value

35. Multiple Correlation
The correlation computed between all pairs of observed y-values and predicted y-values is the multiple correlation, R
The larger the multiple correlation, the better are the predictions of y by the set of explanatory variables

36. Multiple Correlation The R-value always falls between 0 and 1
In this way, the multiple correlation ‘R’ differs from the bivariate correlation ‘r’ between y and a single variable x, which falls between -1 and +1

37. R-squared
For predicting y, the square of R describes the relative improvement from using the prediction equation instead of using the sample mean, y

38. R-squared
The error in using the prediction equation to predict y is summarized by the residual sum of squares:

39. R-squared
The error in using y to predict y is summarized by the total sum of squares:

40. R-squared
The proportional reduction in error is:

41. R-squared
The better the predictions are using the regression equation, the larger R2 is
For multiple regression, R2 is the square of the multiple correlation, R

42. Example: How Well Can We Predict House Selling Prices?
For the 100 observations on y = selling price, x1 = house size, and x2 = lot size, a table, called the ANOVA (analysis of variance) table was created
The table displays the sums of squares in the SS column

43. Example: How Well Can We Predict House Selling Prices?
The R2 value can be created from the sums of squares in the table

44. Example: How Well Can We Predict House Selling Prices?
Using house size and lot size together to predict selling price reduces the prediction error by 71%, relative to using y alone to predict selling price

45. Example: How Well Can We Predict House Selling Prices?
Find and interpret the multiple correlation
There is a strong association between the observed and the predicted selling prices
House size and lot size very much help us to predict selling prices

46. Example: How Well Can We Predict House Selling Prices?
If we used a bivariate regression model to predict selling price with house size as the predictor, the r2 value would be 0.58
If we used a bivariate regression model to predict selling price with lot size as the predictor, the r2 value would be 0.51

47. Example: How Well Can We Predict House Selling Prices?
The multiple regression model has R , so it provides better predictions than either bivariate model

48. Properties of R2
The previous example showed that R2 for the multiple regression model was larger than r2 for a bivariate model using only one of the explanatory variables
A key factor of R2 is that it cannot decrease when predictors are added to a model

49. Properties of R2 R2 falls between 0 and 1
The larger the value, the better the explanatory variables collectively predict y
R2 =1 only when all residuals are 0, that is, when all regression predictions are prefect
R2 = 0 when the correlation between y and each explanatory variable equals 0

50. Properties of R2
R2 gets larger, or at worst stays the same, whenever an explanatory variable is added to the multiple regression model
The value of R2 does not depend on the units of measurement

51. R2 Values for Various Multiple Regression Models

52. R2 Values for Various Multiple Regression Models
The single predictor in the data set that is most strongly associated with y is the house’s real estate tax assessment
(r2 = 0.679)
When we add house size as a second predictor, R2 goes up from to 0.730
As other predictors are added, R2 continues to go up, but not by much

53. R2 Values for Various Multiple Regression Models
R2 does not increase much after a few predictors are in the model
When there are many explanatory variables but the correlations among them are strong, once you have included a few of them in the model, R2 usually doesn’t increase much more when you add additional ones

54. R2 Values for Various Multiple Regression Models
This does not mean that the additional variables are uncorrelated with the response variable
It merely means that they don’t add much new power for predicting y, given the values of the predictors already in the model

55. In a data set used to predict body weight (in pounds), three predictors were used: height, percent body fat and age.
Their correlations with total body weight were:
Height: Percent Body fat: Age:
Which explanatory variable gives by itself the best prediction of weight?
Height
Percent body fat
Age

56. With height as the sole predictor, what is r2? .745 .555 .625 .825
In a data set used to predict body weight (in pounds), three predictors were used: height, percent body fat and age.
Their correlations with total body weight were:
Height: Percent Body fat: Age:
With height as the sole predictor, what is r2?
.745
.555
.625
.825

57. In a data set used to predict body weight (in pounds), three predictors were used: height, percent body fat and age.
Their correlations with total body weight were:
Height: Percent Body fat: Age:
If Percent Body Fat is added to the model R2 = If Age is then added to the model R2= Once you know height and % body fat, does age seem to help in predicting weight? No
Yes

58. How Can We Use Multiple Regression to Make Inferences?
Section 12.3
How Can We Use Multiple Regression to Make Inferences?

59. Inferences about the Population
Assumptions required when using a multiple regression model to make inferences about the population:
The regression equation truly holds for the population means
This implies that there is a straight-line relationship between the mean of y and each explanatory variable, with the same slope at each value of the other predictors

60. Inferences about the Population
Assumptions required when using a multiple regression model to make inferences about the population:
The data were gathered using randomization
The response variable y has a normal distribution at each combination of values of the explanatory variables, with the same standard deviation

61. Inferences about Individual Regression Parameters
Consider a particular parameter, β1
If β1= 0, the mean of y is identical for all values of x1, at fixed values of the other explanatory variables
So, H0: β1= 0 states that y and x1 are statistically independent, controlling for the other variables
This means that once the other explanatory variables are in the model, it doesn’t help to have x1 in the model

62. Significance Test about a Multiple Regression Parameter
Assumptions:
Each explanatory variable has a straight-line relation with µy with the same slope for all combinations of values of other predictors in the model
Data gathered with randomization
Normal distribution for y with same standard deviation at each combination of values of other predictors in model

63. Significance Test about a Multiple Regression Parameter
Hypotheses:
H0: β1= 0
Ha: β1≠ 0
When H0 is true, y is independent of x1, controlling for the other predictors

64. Significance Test about a Multiple Regression Parameter
Test Statistic:

65. Significance Test about a Multiple Regression Parameter
P-value: Two-tail probability from t- distribution of values larger than observed t test statistic (in absolute value)
The t-distribution has:
df = n – number of parameters in the regression equation

66. Significance Test about a Multiple Regression Parameter
Conclusion: Interpret P-value; compare to significance level if decision needed

67. Example: What Helps Predict a Female Athlete’s Weight?
The “College Athletes” data set comes from a study of 64 University of Georgia female athletes
The study measured several physical characteristics, including total body weight in pounds (TBW), height in inches (HGT), the percent of body fat (%BF) and age

68. Example: What Helps Predict a Female Athlete’s Weight?
The results of fitting a multiple regression model for predicting weight using the other variables:

69. Example: What Helps Predict a Female Athlete’s Weight?
Interpret the effect of age on weight in the multiple regression equation:

70. Example: What Helps Predict a Female Athlete’s Weight?
The slope coefficient of age is -0.96
For athletes having fixed values for x1 and x2, the predicted weight decreases by 0.96 pounds for a 1-year increase in age, and the ages vary only between 17 and 23

71. Example: What Helps Predict a Female Athlete’s Weight?
Run a hypothesis test to determine whether age helps to predict weight, if you already know height and percent body fat

72. Example: What Helps Predict a Female Athlete’s Weight?
Assumptions:
The 64 female athletes were a convenience sample, not a random sample
Caution should be taken when making inferences about all female college athletes

73. Example: What Helps Predict a Female Athlete’s Weight?
Hypotheses:
H0: β3= 0
Ha: β3≠ 0
Test statistic:

74. Example: What Helps Predict a Female Athlete’s Weight?
P-value: This value is reported in the output as 0.14
Conclusion:
The P-value of 0.14 does not give much evidence against the null hypothesis that β3 = 0
Age does not significantly predict weight if we already know height and % body fat

75. Confidence Interval for a Multiple Regression Parameter
A 95% confidence interval for a β slope parameter in multiple regression equals:
The t-score has:
df = (n - # of parameters in the model)

76. Example: What’s Plausible for the Effect of Age on Weight?
Construct and interpret a 95% CI for β3, the effect of age while controlling for height and % body fat

77. Example: What’s Plausible for the Effect of Age on Weight?
At fixed values of x1 and x2, we infer that the population mean of weight changes very little (and maybe not at all) for a 1-year increase in age
The confidence interval contains 0
Age may have no effect on weight, once we control for height and % body fat

78. Estimating Variability Around the Regression Equation
A standard deviation parameter, σ, describes variability of the observations around the regression equation
Its sample estimate is:

79. Example: Estimating Variability of Female Athletes’ Weight
Anova Table for the “college athletes” data set:

80. Example: Estimating Variability of Female Athletes’ Weight
For female athletes at particular values of height, % of body fat, and age, estimate the standard deviation of their weights
Begin by finding the Mean Square Error:
Notice that this value (102.2) appears in the MS column in the ANOVA table

81. Example: Estimating Variability of Female Athletes’ Weight
The standard deviation is:
This value is also displayed in the ANOVA table
For athletes with certain fixed values of height, % body fat, and age, the weights vary with a standard deviation of about 10 pounds

82. Example: Estimating Variability of Female Athletes’ Weight
If the conditional distributions of weight are approximately bell-shaped, about 95% of the weight values fall within about 2s = 20 pounds of the true regression line

83. Do the Explanatory Variables Collectively Have an Effect?
Example: With 3 predictors in a model, we can check this by testing:

84. Do the Explanatory Variables Collectively Have an Effect?
The test statistic for H0 is denoted by F

85. Do the Explanatory Variables Collectively Have an Effect?
When H0 is true, the expected value of the F test statistic is approximately 1
When H0 is false, F tends to be larger than 1
The larger the F test statistic, the stronger the evidence against H0

86. Summary of F Test That All βeta Parameters = 0
Assumptions: Multiple regression equation holds, data gathered randomly, normal distribution for y with same standard deviation at each combination of predictors

87. Summary of F Test That All βeta Parameters = 0
Test statistic:

88. Summary of F-Test That All βeta Parameters = 0
P-value: Right-tail probability above observed F-test statistic value from F- distribution with:
df1 = number of explanatory variables
df2 = n – (number of parameters in regression equation)

89. Summary of F-Test That All βeta Parameters = 0
Conclusion: The smaller the P-value, the stronger the evidence that at least one explanatory variable has an effect on y
If a decision is needed, reject H0 if P-value ≤ significance level, such as 0.05

90. Example: The F-Test for Predictors of Athletes’ Weight
For the 64 female college athletes, the regression model for predicting y = weight using x1 = height, x2 = % body fat and x3 = age is summarized in the ANOVA table on the next page

91. Example: The F-Test for Predictors of Athletes’ Weight

92. Example: The F-Test for Predictors of Athletes’ Weight
Use the output in the ANOVA table to test the hypothesis:

93. Example: The F-Test for Predictors of Athletes’ Weight
The observed F statistic is 40.48
The corresponding P-value is 0.000
We can reject H0 at the 0.05 significance level
We conclude that at least one predictor has an effect on weight

94. Example: The F-Test for Predictors of Athletes’ Weight
The F-test tells us that at least one explanatory variable has an effect
If the explanatory variables are chosen sensibly, at least one should have some predictive power
The F-test result tells us whether there is sufficient evidence to make it worthwhile to consider the individual effects, using t-tests

95. Example: The F-Test for Predictors of Athletes’ Weight
The individual t-tests identify which of the variables are significant (controlling for the other variables)

96. Example: The F-Test for Predictors of Athletes’ Weight
If a variable turns out not to be significant, it can be removed from the model
In this example, ‘age’ can be removed from the model

97. Checking a Regression Model Using Residual Plots
Section 12.4
Checking a Regression Model Using Residual Plots

98. Assumptions for Inference with a Multiple Regression Model
The regression equation approximates well the true relationship between the predictors and the mean of y
The data were gathered randomly
y has a normal distribution with the same standard deviation at each combination of predictors

99. Checking Shape and Detecting Unusual Observations
To test Assumption 3 (the conditional distribution of y is normal at any fixed values of the explanatory variables):
Construction a histogram of the standardized residuals
The histogram should be approximately bell-shaped
Nearly all the standardized residuals should fall between -3 and +3. Any residual outside these limits is a potential outlier

100. Example: Residuals for House Selling Price
For the house selling price data, a MINITAB histogram of the standardized residuals for the multiple regression model predicting selling price by the house size and the lot size was created and is displayed on the following page

101. Example: Residuals for House Selling Price

102. Example: Residuals for House Selling Price
The residuals are roughly bell shaped about 0
They fall between about -3 and +3
No severe nonnormality is indicated

103. Plotting Residuals against Each Explanatory Variable
Plots of residuals against each explanatory variable help us check for potential problems with the regression model
Ideally, the residuals should fluctuate randomly about 0
There should be no obvious change in trend or change in variation as the values of the explanatory variable increases

104. Plotting Residuals against Each Explanatory Variable

105. How Can Regression Include Categorical Predictors?
Section 12.5
How Can Regression Include Categorical Predictors?

106. Indicator Variables
Regression models specify categories of a categorical explanatory variable using artificial variables, called indicator variables
The indicator variable for a particular category is binary
It equals 1 if the observation falls into that category and it equals 0 otherwise

107. Indicator Variables
In the house selling prices data set, the city region in which a house is located is a categorical variable
The indicator variable x for region is
x = 1 if house is in NW (northwest region)
x = 0 if house is not in NW

108. Indicator Variables
The coefficient β of the indicator variable x is the difference between the mean selling prices for homes in the NW and for homes not in the NW

109. Example: Including Region in Regression for House Selling Price
Output from the regression model for selling price of home using house size and region

110. Example: Including Region in Regression for House Selling Price
Find and plot the lines showing how predicted selling price varies as a function of house size, for homes in the NW and for homes no in the NW

111. Example: Including Region in Regression for House Selling Price
The regression equation from the MINITAB output is:

112. Example: Including Region in Regression for House Selling Price
For homes not in the NW, x2 = 0
The prediction equation then simplifies to:

113. Example: Including Region in Regression for House Selling Price
For homes in the NW, x2 = 1
The prediction equation then simplifies to:

114. Example: Including Region in Regression for House Selling Price

115. Example: Including Region in Regression for House Selling Price
Both lines have the same slope, 78
For homes in the NW and for homes not in the NW, the predicted selling price increases by $78 for each square-foot increase in house size
The figure portrays a separate line for each category of region (NW, not NW)

116. Example: Including Region in Regression for House Selling Price
The coefficient of the indicator variable is 30569
For any fixed value of house size, we predict that the selling price is $30,569 higher for homes in the NW

117. Example: Including Region in Regression for House Selling Price
The line for homes in the NW is above the line for homes not in the NW
The predicted selling price is higher for homes in the NW
The P-value of for the test for the coefficient of the indicator variable suggests that this difference is statistically significant

118. Is There Interaction?
For two explanatory variables, interaction exists between them in their effects on the response variable when the slope of the relationship between µy and one of them changes as the value of the other changes

119. Is There Interaction?

120. How Can We Model a Categorical Response?
Section 12.6
How Can We Model a Categorical Response?

121. Modeling a Categorical Response Variable
When y is categorical, a different regression model applies, called a logistic regression

122. Examples of Logistic Regression
A voter’s choice in an election (Democrat or Republican), with explanatory variables: annual income, political ideology, religious affiliation, and race
Whether a credit card holder pays their bill on time (yes or no), with explanatory variables: family income and the number of months in the past year that the customer paid the bill on time

123. The Logistic Regression Model
Denote the possible outcomes for y as 0 and 1
Use the generic terms failure (for outcome = 0) and success (for outcome =1)
The population mean of the scores equals the population proportion of ‘1’ outcomes (successes)
That is, µy = p
The proportion, p, also represents the probability that a randomly selected subject has a success outcome

124. The Logistic Regression Model
The straight-line model is usually inadequate
A more realistic model has a curved S-shape instead of a straight-line trend

125. The Logistic Regression Model
A regression equation for an S-shaped curve for the probability of success p is:

126. Example: Annual Income and Having a Travel Credit Card
An Italian study with 100 randomly selected Italian adults considered factors that are associated with whether a person possesses at least one travel credit card
The table on the next page shows results for the first 15 people on this response variable and on the person’s annual income (in thousands of euros)

127. Example: Annual Income and Having a Travel Credit Card

128. Example: Annual Income and Having a Travel Credit Card
Let x = annual income and let y = whether the person possesses a travel credit card (1 = yes, 0 = no)

129. Example: Annual Income and Having a Travel Credit Card
Substituting the α and β estimates into the logistic regression model formula yields:

130. Example: Annual Income and Having a Travel Credit Card
Find the estimated probability of possessing a travel credit card at the lowest and highest annual income levels in the sample, which were x = 12 and x = 65

131. Example: Annual Income and Having a Travel Credit Card
For x = 12 thousand euros, the estimated probability of possessing a travel credit card is:

132. Example: Annual Income and Having a Travel Credit Card
For x = 65 thousand euros, the estimated probability of possessing a travel credit card is:

133. Example: Annual Income and Having a Travel Credit Card
Annual income has a strong positive effect on having a credit card
The estimated probability of having a travel credit card changes from 0.09 to 0.97 as annual income changes over its range

134. Example: Estimating Proportion of Students Who’ve Used Marijuana
A three-variable contingency table from a survey of senior high-school students in shown on the next page
The students were asked whether they had ever used: alcohol, cigarettes or marijuana

135. Example: Estimating Proportion of Students Who’ve Used Marijuana

136. Example: Estimating Proportion of Students Who’ve Used Marijuana
Let y indicate marijuana use, coded: (1 = yes, 0 = no)
Let x1 be an indicator variable for alcohol use (1 = yes, 0 = no)
Let x2 be an indicator variable for cigarette use (1 = yes, 0 = no)

137. Example: Estimating Proportion of Students Who’ve Used Marijuana

138. Example:Estimating Proportion of Students Who’ve Used Marijuana
The logistic regression prediction equation is:

139. Example: Estimating Proportion of Students Who’ve Used Marijuana
For those who have not used alcohol or cigarettes, x1= x2 = 0 and:

140. Example: Estimating Proportion of Students Who’ve Used Marijuana
For those who have used alcohol and cigarettes, x1= x2 = 1 and:

141. Example: Estimating Proportion of Students Who’ve Used Marijuana
The probability that students have tried marijuana seems to depend greatly on whether they’ve used alcohol and cigarettes