chapter 13: multiple regression
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1
Chapter 13: Multiple Regression
Section 13.1: How Can We Use Several Variables to Predict a Response?
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Learning Objectives
1. Regression Models2. The Number of Explanatory Variables3. Plotting Relationships4. Interpretation of Multiple Regression Coefficients5. Summarizing the Effect While Controlling for a
Variable6. Slopes in Multiple Regression and in Bivariate
Regression7. Importance of Multiple Regression
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Learning Objective 1:Regression Models
The model that contains only two variables, x and y, is called a bivariate model
xy
4
Learning Objective 1:Regression Models
Suppose there are two predictors, denoted by x1 and x2
This is called a multiple regression model
2211xx
y
5
Learning Objective 1: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,…
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Learning Objective 1:Multiple Regression Model
Example: For three explanatory variables, the multiple regression equation is:
332211xxx
y
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Learning Objective 1:Multiple Regression Model
Example: The sample prediction equation with three explanatory variables is:
332211ˆ xbxbxbay
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Learning Objective 1: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
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Output from the analysis:
Learning Objective 1:Example: Predicting Selling Price Using House and Lot Size
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Prediction Equation:
where y = selling price, x1=house size and x2 = lot size
2184.28.53536,10ˆ xxy
Learning Objective 1:Example: Predicting Selling Price Using House and Lot Size
11
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:
276,107
)000,18(84.2)1240(8.53536,10ˆ
y
Learning Objective 1:Example: Predicting Selling Price Using House and Lot Size
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Find its residual:
The residual tells us that the actual selling price was $37,724 higher than predicted
724,37276,107000,145ˆ yy
Learning Objective 1:Example: Predicting Selling Price Using House and Lot Size
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Learning Objective 2: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
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Learning Objective 3: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
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Learning Objective 3:Plotting Relationships
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Learning Objective 4: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)
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Learning Objective 4:Interpretation of Multiple Regression Coefficients Example using the housing data: Suppose we fix x1 = house size at 2000 square
feet The prediction equation becomes:
2
2
2.84x97,022
84.2)2000(8.53536,10ˆ
xy
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Learning Objective 4:Interpretation of Multiple Regression Coefficients Since the slope coefficient of x2 is 2.84,
the predicted selling price increases by $2.84 for every square foot increase in lot size when the house size is 2000 square feet
For a 1000 square-foot increase in lot size, the predicted selling price increases by 1000(2.84) = $2840 when the house size is 2000 square feet
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Learning Objective 4:Interpretation of Multiple Regression CoefficientsExample using the housing data: Suppose we fix x2 = lot size at 30,000 square
feet The prediction equation becomes:
1
1
53.874,676
)000,30(84.28.53536,10ˆ
x
xy
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Learning Objective 4:Interpretation of Multiple Regression Coefficients
Since the slope coefficient of x1 is 53.8, for houses with a lot size of 30,000 square feet, the predicted selling price increases by $53.80 for every square foot increase in house size
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Learning Objective 4: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
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Learning Objective 5: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
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Learning Objective 5: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
2184.28.53536,10ˆ xxy
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Learning Objective 5:Summarizing the Effect While Controlling for a Variable
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Learning Objective 6: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
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Learning Objective 6: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
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Learning Objective 7: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
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Chapter 13:Multiple Regression
Section 13.2 Extending the Correlation and R-Squared for Multiple Regression
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Learning Objectives
1. Multiple Correlation
2. R-squared3. Properties of R2
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Learning Objective 1:Multiple Correlation
To summarize how well a multiple regression model predicts y, we analyze how well the observed y values correlate with the predicted values
The multiple correlation is the correlation between the observed y values and the predicted values It is denoted by R
ˆ y
ˆ y
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Learning Objective 1:Multiple Correlation
For each subject, the regression equation provides a predicted value
Each subject has an observed y-value and a predicted y-value
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Learning Objective 1: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
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Learning Objective 1: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
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Learning Objective 2: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
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Learning Objective 2:R-squared
The error in using the prediction equation to predict y is summarized by the residual sum of squares:
2)ˆ( yy
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Learning Objective 2:R-squared
The error in using to predict y is summarized by the total sum of squares:
2)( yy
y
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Learning Objective 2:R-squared
The proportional reduction in error is:
2
22
2
)(
)ˆ()(
yy
yyyyR
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Learning Objective 2: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
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Learning Objective 2: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
40
The R2 value can be created from the sums of squares in the table
711.090,756
90,756-314,433
)(
)ˆ()(2
22
2
yy
yyyyR
Learning Objective 2:Example: How Well Can We Predict House Selling Prices?
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Using house size and lot size together to predict selling price reduces the prediction error by 71%, relative to using alone to predict selling price
Learning Objective 2:Example: How Well Can We Predict House Selling Prices?
y
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Find and interpret the multiple correlation
There is a strong association between the observed and the predicted selling prices
House size and lot size are very helpful in predicting selling prices
R R2 0.711 0.84
Learning Objective 2:Example: How Well Can We Predict House Selling Prices?
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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
Learning Objective 2:Example: How Well Can We Predict House Selling Prices?
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The multiple regression model has R2 0.71, so it provides better predictions than either bivariate model
Learning Objective 2:Example: How Well Can We Predict House Selling Prices?
45
Learning Objective 2:Example: How Well Can We Predict House Selling Prices?
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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 0.679 to 0.730
As other predictors are added, R2 continues to go up, but not by much
Learning Objective 2:Example: How Well Can We Predict House Selling Prices?
47
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
Learning Objective 2:Example: How Well Can We Predict House Selling Prices?
48
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
Learning Objective 2:Example: How Well Can We Predict House Selling Prices?
49
Learning Objective 3: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
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Learning Objective 3: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
51
Learning Objective 3: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
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Chapter 13:Multiple Regression
Section 13.3: How Can We Use Multiple Regression to Make Inferences?
53
Learning Objectives
1. Inferences about the Population2. Inferences about Individual Regression Parameters3. Significance Test about a Multiple Regression
Parameter4. Confidence Interval for a Multiple Regression
Parameter5. Estimating Variability Around the Regression
Equation6. Do the Explanatory Variables Collectively Have an
Effect?7. Summary of F Test That All Beta Parameters = 0
54
Learning Objective 1: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
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Learning Objective 1: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
56
Learning Objective 2: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
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Learning Objective 3:Significance Test about a Multiple Regression Parameter
1. 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
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Learning Objective 3:Significance Test about a Multiple Regression Parameter
2. Hypotheses: H0: β1= 0
Ha: β1≠ 0
When H0 is true, y is independent of x1, controlling for the other predictors
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Learning Objective 3:Significance Test about a Multiple Regression Parameter3. Test Statistic:
se
bt
01
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Learning Objective 3:Significance Test about a Multiple Regression Parameter
4. 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
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Learning Objective 3:Significance Test about a Multiple Regression Parameter
5. Conclusion: Interpret P-value in context; compare to significance level if decision needed
62
Learning Objective 3: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
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The results of fitting a multiple regression model for predicting weight using the other variables:
Learning Objective 3:Example: What Helps Predict a Female Athlete’s Weight?
64
Interpret the effect of age on weight in the multiple regression equation:
321
32
1
96.036.143.37.97ˆThen
age and fat,body %
height, weight,predicted ˆLet
xxxy
xx
xy
Learning Objective 3:Example: What Helps Predict a Female Athlete’s Weight?
65
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
Learning Objective 3:Example: What Helps Predict a Female Athlete’s Weight?
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Run a hypothesis test to determine whether age helps to predict weight, if you already know height and percent body fat
Learning Objective 3:Example: What Helps Predict a Female Athlete’s Weight?
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1. Assumptions Met?: The 64 female athletes were a
convenience sample, not a random sample
Caution should be taken when making inferences about all female college athletes
Learning Objective 3:Example: What Helps Predict a Female Athlete’s Weight?
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2. Hypotheses:
H0: β3= 0
Ha: β3≠ 0
3. Test statistic:
48.1648.0
960.003 se
bt
Learning Objective 3:Example: What Helps Predict a Female Athlete’s Weight?
69
4. P-value: This value is reported in the output as 0.14
5. 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
Learning Objective 3:Example: What Helps Predict a Female Athlete’s Weight?
70
Learning Objective 4: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) Assumptions are the same as for the t test
)( t slope Estimated.025
se
71
Construct and interpret a 95% CI for β3, the effect of age while controlling for height and % body fat
)3.0,3.2(30.196.0
)648.0(00.296.0)(025.3
setb
Learning Objective 4:Confidence Interval for a Multiple Regression Parameter
72
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
Learning Objective 4:Confidence Interval for a Multiple Regression Parameter
73
Learning Objective 5:Estimating Variability Around the Regression Equation A standard deviation parameter, σ, describes variability
of the observations around the regression equation Its sample estimate is:
eq.) reg.in parameters of (#
)ˆ(
esidual
2
n
yy
df
SSRs
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Learning Objective 5:Example: Estimating Variability of Female Athletes’ Weight Anova Table for the “college athletes” data set:
75
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
2.10260
0.6131 2 df
SSresiduals
Learning Objective 5:Example: Estimating Variability of Female Athletes’ Weight
76
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
1.102.102 s
Learning Objective 5:Example: Estimating Variability of Female Athletes’ Weight
77
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
Learning Objective 5:Example: Estimating Variability of Female Athletes’ Weight
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Learning Objective 5:Do the Explanatory Variables Collectively Have an Effect?
Example: With 3 predictors in a model, we can check this by testing:
0parameter oneleast At :
0:3210
aH
H
79
Learning Objective 5:Do the Explanatory Variables Collectively Have an Effect?
The test statistic for H0 is denoted by F
e errorMean squar
regressionforsquareMeanF
80
Learning Objective 5: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
81
Learning Objective 6:Summary of F Test That All Beta Parameters = 0
1. Assumptions: Multiple regression equation holds, data gathered randomly, normal distribution for y with same standard deviation at each combination of predictors
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Learning Objective 6:Summary of F Test That All Beta Parameters = 0
2.
3. Test statistic:
H0 :1 2 0
Ha : At least one parameter 0
e errorMean squar
regressionforsquareMeanF
83
Learning Objective 6:Summary of F-Test That All Beta Parameters = 0
4. 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)
84
Learning Objective 6:Summary of F-Test That All Beta Parameters = 0
5. 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
85
Learning Objective 6: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 slide
86
Learning Objective 6:Example: The F-Test for Predictors of Athletes’ Weight
87
Use the output in the ANOVA table to test the hypothesis:
0parameter oneleast At :
0:3210
aH
H
Learning Objective 6:Example: The F-Test for Predictors of Athletes’ Weight
88
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
Learning Objective 6:Example: The F-Test for Predictors of Athletes’ Weight
89
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
Learning Objective 6:Example: The F-Test for Predictors of Athletes’ Weight
90
The individual t-tests identify which of the variables are significant (controlling for the other variables)
Learning Objective 6:Example: The F-Test for Predictors of Athletes’ Weight
91
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
Learning Objective 6:Example: The F-Test for Predictors of Athletes’ Weight
92
Chapter 13:Multiple Regression
Section 13.4: Checking a Regression Model Using Residual Plots
93
Learning Objectives
1. Assumptions for Inference with a Multiple Regression Model
2. Checking Shape and Detecting Unusual Observations
3. Plotting Residuals against Each Explanatory Variable
94
Learning Objective 1: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
95
Learning Objective 2: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
96
Learning Objective 2: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 slide
97
Learning Objective 2:Example: Residuals for House Selling Price
98
The residuals are roughly bell shaped about 0
They fall between about -3 and +3 No severe nonnormality is indicated
Learning Objective 2:Example: Residuals for House Selling Price
99
Learning Objective 3: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
100
Learning Objective 3:Plotting Residuals against Each Explanatory Variable
101
Chapter 13:Multiple Regression
Section 13.5: How Can Regression Include Categorical Predictors?
102
Learning Objectives
1. Indicator Variables
2. Is There Interaction?
103
Learning Objective 1:Indicator Variables
Regression models can 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
104
Learning Objective 1: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
105
Learning Objective 1: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:
y (1) , if house is in NW (so x 1)
y (0) , if house is not in NW (so x 0)
y for NW y for other regions
106
Learning Objective 1:Example: Including Region in Regression for House Selling Price Output from the regression model for selling price
of home using house size and region
107
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 not in the NW
Learning Objective 1:Example: Including Region in Regression for House Selling Price
108
The regression equation from the MINITAB output is:
21569,300.78258,15ˆ xxy
Learning Objective 1:Example: Including Region in Regression for House Selling Price
109
For homes not in the NW, x2 = 0
The prediction equation then simplifies to:
1
1
78.015,258-
)0(569,300.78258,15ˆ
x
xy
Learning Objective 1:Example: Including Region in Regression for House Selling Price
110
For homes in the NW, x2 = 1
The prediction equation then simplifies to:
1
1
78.015,311
)1(569,300.78258,15ˆ
x
xy
Learning Objective 1:Example: Including Region in Regression for House Selling Price
111
Learning Objective 1:Example: Including Region in Regression for House Selling Price
112
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)
Learning Objective 1:Example: Including Region in Regression for House Selling Price
113
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
Learning Objective 1:Example: Including Region in Regression for House Selling Price
114
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 0.000 for the test for the coefficient of the indicator variable suggests that this difference is statistically significant
Learning Objective 1:Example: Including Region in Regression for House Selling Price
115
Learning Objective 2: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
116
Learning Objective 2:Example: Interaction in effects on House Selling Price Suppose the actual population relationship
between house size and mean selling price is:
Then the slope for the effect of x1 differs for the two regions - there is interaction between house size and region in their effects on selling price
y 15,000 100x1 for homes in the NW
y 12,000 25x1 for homes in other regions
117
Learning Objective 2:Example: Interaction in effects on House Selling Price
118
Learning Objective 2:Example: Interaction in effects on House Selling Price To allow for interaction with two explanatory
variables, one quantitative and one categorical, you can fit a separate regression line with a different slope between the two quantitative variables for each category of the categorical variable.
119
Chapter 13:Multiple Regression
Section 13.6: How Can We Model A Categorical Response?
120
Learning Objectives
1. Modeling a Categorical Response Variable
2. Examples of Logistic Regression
3. The Logistic Regression Model
121
Learning Objective 1:Modeling a Categorical Response Variable
When y is categorical, a different regression model applies, called logistic regression
122
Learning Objective 2: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
Learning Objective 3: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
Learning Objective 3: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
Learning Objective 3:The Logistic Regression Model
A regression equation for an S-shaped curve for the probability of success p is:
)(
)(
1 x
x
e
ep
126
Learning Objective 3: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
Learning Objective 3:Example: Annual Income and Having a Travel Credit Card
128
Let x = annual income and let y = whether the person possesses a travel credit card (1 = yes, 0 = no)
Learning Objective 3:Example: Annual Income and Having a Travel Credit Card
129
Substituting the α and β estimates into the logistic regression model formula yields:
)105.052.3(
)105.052.3(
1ˆ
x
x
e
ep
Learning Objective 3:Example: Annual Income and Having a Travel Credit Card
130
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
Learning Objective 3:Example: Annual Income and Having a Travel Credit Card
131
For x = 12 thousand euros, the estimated probability of possessing a travel credit card is:
09.01.104
0.104
11ˆ
26.2
26.2
)12(05.152.3
)12(105.052.3
e
e
e
ep
Learning Objective 3:Example: Annual Income and Having a Travel Credit Card
132
For x = 65 thousand euros, the estimated probability of possessing a travel credit card is:
97.028.2485
27.2485
11ˆ
305.3
305.3
)65(05.152.3
)65(105.052.3
e
e
e
ep
Learning Objective 3:Example: Annual Income and Having a Travel Credit Card
133
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
Learning Objective 3:Example: Annual Income and Having a Travel Credit Card
134
Learning Objective 3:Example: Estimating Proportion of Students Who’ve Used Marijuana
A three-variable contingency table from a survey of senior high-school students is shown on the next slide
The students were asked whether they had ever used: alcohol, cigarettes or marijuana
135
Learning Objective 3:Example: Estimating Proportion of Students Who’ve Used Marijuana
136
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)
Learning Objective 3:Example: Estimating Proportion of Students Who’ve Used Marijuana
137
Learning Objective 3:Example: Estimating Proportion of Students Who’ve Used Marijuana
138
The logistic regression prediction equation is:
21
21
85.299.231.5
85.299.231.5
1ˆ
xx
xx
e
ep
Learning Objective 3:Example: Estimating Proportion of Students Who’ve Used Marijuana
139
For those who have not used alcohol or cigarettes, x1= x2 = 0 and:
005.01
ˆ)0(85.2)0(99.231.5
)0(85.2)0(99.231.5
e
ep
Learning Objective 3:Example: Estimating Proportion of Students Who’ve Used Marijuana
140
For those who have used alcohol and cigarettes, x1= x2 = 1 and:
628.01
ˆ)1(85.2)1(99.231.5
)1(85.2)1(99.231.5
e
ep
Learning Objective 3:Example: Estimating Proportion of Students Who’ve Used Marijuana
141
The probability that students have tried marijuana seems to depend greatly on whether they’ve used alcohol and cigarettes
Learning Objective 3:Example: Estimating Proportion of Students Who’ve Used Marijuana
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