Chapter 13: Multiple Regression

Section 13.1: How Can We Use Several Variables to Predict a Response?

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

Learning Objective 1:Regression Models

The model that contains only two variables, x and y, is called a bivariate model

Learning Objective 1:Regression Models

Suppose there are two predictors, denoted by x1 and x2

This is called a multiple regression model

2211xx

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,…

Example: For three explanatory variables, the multiple regression equation is:

332211xxx

Example: The sample prediction equation with three explanatory variables is:

332211ˆ xbxbxbay

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

Output from the analysis:

Learning Objective 1:Example: Predicting Selling Price Using House and Lot Size

Prediction Equation:

where y = selling price, x1=house size and x2 = lot size

2184.28.53536,10ˆ xxy

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

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 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

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

Learning Objective 3:Plotting Relationships

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)

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:

2.84x97,022

84.2)2000(8.53536,10ˆ

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

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:

53.874,676

)000,30(84.28.53536,10ˆ

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

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

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

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

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

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

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

Chapter 13:Multiple Regression

Section 13.2 Extending the Correlation and R-Squared for Multiple Regression

Learning Objectives

1. Multiple Correlation

2. R-squared3. Properties of R2

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

For each subject, the regression equation provides a predicted value

Each subject has an observed y-value and a predicted y-value

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

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

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,

The error in using the prediction equation to predict y is summarized by the residual sum of squares:

2)ˆ( yy

The error in using to predict y is summarized by the total sum of squares:

2)( yy

The proportional reduction in error is:

)ˆ()(

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

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

The R2 value can be created from the sums of squares in the table

711.090,756

90,756-314,433

)ˆ()(2

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

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

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

The multiple regression model has R2 0.71, so it provides better predictions than either bivariate model

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

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

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 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

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

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

Section 13.3: How Can We Use Multiple Regression to Make Inferences?

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

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

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

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

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

2. Hypotheses: H0: β1= 0

Ha: β1≠ 0

When H0 is true, y is independent of x1, controlling for the other predictors

Learning Objective 3:Significance Test about a Multiple Regression Parameter3. Test Statistic:

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

5. Conclusion: Interpret P-value in context; compare to significance level if decision needed

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

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?

Interpret the effect of age on weight in the multiple regression equation:

96.036.143.37.97ˆThen

age and fat,body %

height, weight,predicted ˆLet

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

Run a hypothesis test to determine whether age helps to predict weight, if you already know height and percent body fat

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

2. Hypotheses:

H0: β3= 0

Ha: β3≠ 0

3. Test statistic:

48.1648.0

960.003 se

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 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

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

Learning Objective 4:Confidence Interval for a Multiple Regression Parameter

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

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 (#

esidual

Learning Objective 5:Example: Estimating Variability of Female Athletes’ Weight Anova Table for the “college athletes” data set:

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

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

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: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

The test statistic for H0 is denoted by F

e errorMean squar

regressionforsquareMeanF

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

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

Learning Objective 6:Summary of F Test That All Beta Parameters = 0

3. Test statistic:

H0 :1 2 0

Ha : At least one parameter 0

e errorMean squar

regressionforsquareMeanF

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)

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

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

Use the output in the ANOVA table to test the hypothesis:

0parameter oneleast At :

0:3210

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

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

The individual t-tests identify which of the variables are significant (controlling for the other variables)

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

Section 13.4: Checking a Regression Model Using Residual Plots

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

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

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

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

The residuals are roughly bell shaped about 0

They fall between about -3 and +3 No severe nonnormality is indicated

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

Learning Objective 3:Plotting Residuals against Each Explanatory Variable

Section 13.5: How Can Regression Include Categorical Predictors?

Learning Objectives

1. Indicator Variables

2. Is There Interaction?

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

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

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

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

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

The regression equation from the MINITAB output is:

21569,300.78258,15ˆ xxy

For homes not in the NW, x2 = 0

The prediction equation then simplifies to:

78.015,258-

)0(569,300.78258,15ˆ

For homes in the NW, x2 = 1

The prediction equation then simplifies to:

78.015,311

)1(569,300.78258,15ˆ

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)

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

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 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

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

Learning Objective 2:Example: Interaction in effects on House Selling Price

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.

Section 13.6: How Can We Model A Categorical Response?

Learning Objectives

1. Modeling a Categorical Response Variable

2. Examples of Logistic Regression

3. The Logistic Regression Model

Learning Objective 1:Modeling a Categorical Response Variable

When y is categorical, a different regression model applies, called logistic regression

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

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

The straight-line model is usually inadequate

A more realistic model has a curved S-shape instead of a straight-line trend

A regression equation for an S-shaped curve for the probability of success p is:

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)

Learning Objective 3: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)

Substituting the α and β estimates into the logistic regression model formula yields:

)105.052.3(

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

For x = 12 thousand euros, the estimated probability of possessing a travel credit card is:

09.01.104

)12(05.152.3

)12(105.052.3

For x = 65 thousand euros, the estimated probability of possessing a travel credit card is:

97.028.2485

27.2485

)65(05.152.3

)65(105.052.3

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: 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

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)

The logistic regression prediction equation is:

85.299.231.5

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

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

The probability that students have tried marijuana seems to depend greatly on whether they’ve used alcohol and cigarettes

Chapter 13: Multiple Regression

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