The purpose of multiple regression analysis is to model the relationship between a dependent variable and multiple independent variables. In Chapters 12 and 13, we will study multiple regression and the general linear model as well as further regression topics. Search for a video, news item, or article (include the link in your discussion post) that gives you a better understanding of multiple regression or is an application in your field of study. Explain in your post why you chose this item and how your linked item corresponds to our Chapters 12 and 13 course objectives (Attatched). Then describe how you could use multiple regression in a life situation.note:: I need 2 different versionsChapter 12

Multiple Regression

and the

General Linear Model

STAT 441/541 Statistical Methods II

1

Sections Covered

• 12.1 Introduction

• 12.2 The General Linear Model

• 12.3 Estimating Multiple Regression Coefficients

• 12.4 Inferences in Multiple Regression

• 12.5 Testing a Subset of Regression Coefficients

• 12.6 Forecasting Using Multiple Regression

• 12.7 Comparing the Slopes of Several Regression Lines

2

Section 12.1 Introduction

• The multiple regression model relates a dependent

variable to a set of independent variables

= 0 + 1 1 + 2 2 + ⋯ + + + ε

• The only restriction is that no independent variable

is a perfect linear function of any other

independent variables

• The parameter 0 is the -intercept and is the

expected value of when each = 0. Only

meaningful if it makes sense to have each = 0

• The other parameters 1 , ⋯ are partial slope

parameters and represent the expected change in

for a unit increase in when all other ′ are held

constant.

Note: Expected value is the same as average value

3

Examples of Multiple Regression

Models

• First-order model

= 0 + 1 1 + 2 2 + 3 3 + ε

• Model with an interaction term

= 0 + 1 1 + 2 2 + 3 1 2 + ε

• Polynomial Model

= 0 + 1 1 + 2 12 + 3 13 + ε

4

Assumptions for Multiple

Regression

• The model has been properly specified

• The variance of the errors is 2 for all observations

• The errors are independent

• The error terms are normally distributed and there

are no outliers

5

Some Limitations of Regression

Analysis

• The existence of a relationship does not imply that

changes in the independent variables cause

changes in the dependent variable (cause and

effect)

• Do not use an estimated regression equation for

extrapolation outside the range of values for all

independent variables

6

Section 12.2 The General Linear

Model

• The general linear model has the form

= 0 + 1 1 + 2 2 + ⋯ + + ε

• The ′ represent

• Quantitative independent variables (this may include

polynomial and cross-product terms)

= 0 + 1 1 + 2 2 + 3 1 2 + 4 12 + 5 22 + ε

• Qualitative independent variables (dummy variables)

• Both quantitative and qualitative independent variables

• The least squares prediction equation is

� = ̂0 + ̂1 1 + ̂2 2 + ⋯ + ̂

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Why is this called a general linear

model?

• The word “linear” in the general linear model refers

to how the ′ are entered in the model

• The word “linear” does not refer to how the

independent variables appear in the model (since

there are polynomial and interaction terms)

• A general linear model is linear in the

• The do not appear as an exponent or as the

argument of a nonlinear function

• Nonlinear example: = 1 1 2 2 +

Note: The general linear model is used in Chapters 12

through 18

8

Section 12.3 Estimating Multiple

Regression Coefficients

• The multiple regression model relates a dependent

variable to a set of quantitative independent variables.

• For a random sample of n measurements, the ith

observation is

= 0 + 1 1 + 2 2 + ⋯ + + + ε

for = 1, 2, … , ; and > ,

where =number of observations and

=number of partial slope parameters in the model for

the ’s

• The method of least-squares is used to estimate all

coefficients in the model 0 , 1 , … ,

• Each coefficient refers to the effect of changing that

variable while other independent variables stay

constant

9

Model Standard Deviation

• It is important to estimate the model standard

deviation

• Residuals, , are used to estimate

• The estimate of the model standard deviation is

the square root of MS(Residual), also called

MS(Error)

Residual: = − �

=

2

− +1

=

10

Section 12.4 Inferences in

Multiple Regression

• Inferences about any of the parameters in the

general linear model are the same as for the simple

linear regression model

• The coefficient of determination, 2 , is the

proportion of the variation in the dependent

variable, , that is explained by the model relating

to 1 , 2 , … ,

• Multicollinearity is present when the independent

variables are themselves highly correlated

11

Overall Model Test

• Hypotheses

• 0 : 1 = 2 = ⋯ = = 0

• : At least one ≠ 0

• Test Statistic: =

( )

( )

• Use p-value from output and compare to

• Check assumptions and draw conclusions

According to the null hypothesis, none of the

variables included in the model has any predictive

value

If the null hypothesis is rejected, there is good

evidence of some degree of predictive value

somewhere among the independent variables

12

Effect of Multicollinearity

• If the independent variable is highly correlated

with one or more other independent variables,

than the parameter estimates are inaccurate and

have large standard errors

• The variance inflation factor (VIF) measures how

much the variance of a coefficient is increased

because of multicollinearity

• If VIF=1, there is no multicollinearity

• If VIF>10, there may be a serious problem

13

Hypothesis Test for = 0

• Hypotheses

• Case 1: 0 : ≤ 0 versus : > 0

• Case 2: 0 : ≥ 0 versus : < 0
• Case 3: 0 : = 0 versus : ≠ 0
• Test Statistic: value from software output
• Compare p-value from output to significance level
• Reject the null hypothesis 0 if p-value ≤
(If p-value is low, 0 must go)
• Fail to reject the null hypothesis 0 if p-value >

(If p-value is high, with 0 we must comply)

• Check assumptions and draw conclusions

14

Hypothesis Test for = 0 (continued)

• The null hypothesis does not assert that the

independent variable has no predictive value by

itself

• It asserts that it has no additional predictive value

over and above that contributed by the other

independent variables in the model

• When two or more independent variables are

highly correlated among themselves, it often

happens that no can be shown to have unique

predictive value, even though the ’s together have

been shown to be useful

15

Section 12.5 Testing a Subset of

Regression Coefficients

• The F test for a subset of regression coefficients

tests simultaneously whether several of the true

coefficients are zero

• If several of the independent variables have no

predictive value then they can be dropped from the

model

16

Test of a Subset of Independent

Variables

• Hypotheses

• 0 : +1 = +2 = ⋯ = = 0

• : The null hypothesis is not true

• Test Statistic: test comparing complete and

reduced models

• Use p-value from output and compare to

• Check assumptions and draw conclusions

17

Section 12.6 Forecasting using

Multiple Regression

• One of the major uses for multiple regression

models is in forecasting a -value given certain

values of the independent variables

• The best forecast is substituting the specified values into the estimated regression equation

• The standard error of a forecast depends on the

interpretation of the forecast

18

Two Interpretations for Forecasts

• The forecast of for given -values can be

interpreted two ways

1. As the estimate for ( ), the long-run average values from averaging many observations of when

the ′ have the specified values

2. The predicted value for one individual case having

the given -values

• We will use software to calculate confidence and

prediction intervals

19

Section 12.7 Comparing the

Slopes of Several Regression Lines

• This topic represents a special case of the general

problem of constructing a multiple regression

equation for both qualitative and quantitative

independent variables

• See Example 12.20 for a comparison of two drug

products (the qualitative variable with levels A & B)

and three doses (the quantitative variable with

levels of 5, 10, and 20 mg)

20

Chapter 13

Further Regression

Topics

STAT 441/541 Statistical Methods II

1

Sections Covered

• 13.1 Introduction

• 13.2 Selecting the Variables (Step 1)

• 13.3 Formulating the Model (Step 2)

• 13.4 Checking Model Assumptions (Step 3)

2

Section 13.1 Introduction

• This chapter is devoted to putting multiple

regression into practice.

• First, decide on the dependent variable and

candidate independent variables for the regression

equation

• Second, consideration is given to selecting the form

of the multiple regression equation

• Third, check for violation of the underlying

assumptions

Note: This is an iterative process

3

Section 13.2 Selecting the

Variables (Step 1)

• Perhaps the most critical decision in constructing a

multiple regression model is the initial selection of

independent variables

• Knowledge of the problem area is critically important in

the initial selection of data

• Identify the dependent variable

• Work with experts to determine what independent variables

affect the dependent variable

• A major consideration in selecting independent

variables is multicollinearity

• Use a scatterplot matrix and (Pearson) correlation matrix to

examine relationships among independent variables

• Use variance inflation factors (VIF) to diagnosis

multicollinearity

4

Selection Procedures for

Independent Variables

• All possible regressions: all one variable models, all two

variable models, etc.

• Best subset regression: best one variable model, best two

variable model

• Backward elimination: start with all candidate independent

variables in the model and remove one variable at a time

until a reasonable regression model is found

• Stepwise regression starts by adding one variable at a time,

checks to see if any variables should be removed, and

continues until a reasonable regression model is found

Note: One procedure is not universally accepted as better

than the others

5

Criterion for selecting the bestfitting model

• Estimated error variance “# = where

smaller is better

• Coefficient of determination # where larger is better

• Adjusted # which provides a penalty for each regression

coefficient included in the model where larger is better

• Mallow’s / statistic where the best-fitting model should

have / ≈ . For explanatory variables, = + 1

• Akaike’s information criterion (AIC) where smallest is best

• Bayesian information criterion (BIC) where smallest is

best

6

Section 13.3 Formulating the

Model (Step 2)

• This step refines the information gleaned from step 1

to develop a useful multiple regression model

• One technique to determine the form of each

independent variable is to examine the scatterplots

of residuals versus each independent variable

• Consider various transformations of the data

• Logarithmic transformations (usual natural log)

• Inverse transformation of the dependent variable (1/ )

7

Section 13.4 Checking Model

Assumptions (Step 3)

• Use diagnostic plots as previously used

• Shapiro-Wilk test for normality of residuals

• Use hat values and Cook’s distance to detect data

points having high leverage and/or high influence

• Hat values greater than 2(k+1) are considered high

leverage (where k=number of independent variables)

• Cook’s distances greater than one identify observations

that have high influence

8

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