Multiple Regression1 Multiple Linear Regression Multiple Regression Model A regression model that...

Multiple Regression 1

Multiple Linear Regression

Multiple Regression Model A regression model that contains more than one

regressor variable. Multiple Linear Regression Model

A multiple regression model that is a linear function of the unknown parameters 0, 1, 2, and so on.

Examples:

Nonlinear:

iiii

ipipiii

xxY

xxxY

)exp(log 2211010

1,122110

ix

iieY 1

0


Intercept: 0

Partial regression coefficients: 1, 2


Interaction: 12 can be viewed and analyzed as a new

parameter 3

(Replace x12 by a new variable x3)


Interaction: 11 can be viewed and analyzed as a new

parameter 3

(Replace x2 by a new variable x3)


Topics

1.Least Squares Estimation of the Parameters

2.Matrix Approach to Multiple Linear Regression

3.The Covariance Matrix

4.Hypothesis Tests

5.Confidence Intervals

6.Predictions

7.Model Adequacy

8.Polynomial Regression Models

9.Indicator Variables

10. Selection of Variables in Multiple Regression

11. Multicollinearity


A Multiple Regression Analysis A multiple regression analysis involves estimation,

testing, and diagnostic procedures designed to fit the multiple regression model

to a set of data.

The Method of Least SquaresThe prediction equation

is the line that minimizes SSE, the sum of squares of the deviations of the observed values y from the predicted values

kk xxxyE 22110)(

kk xbxbxbby 22110ˆ

.y


2

1 10

1

2

n

i

k

jijji

n

ii xyL

Least Squares Estimation

nix

xxxY

i

k

jijj

ikikiii

,...,1 1

0

,22110

021 1

^

0

^

,...,,0^

1

^

0

^

n

i

k

jijji xy

L

k

kjxxyL

ij

n

i

k

jijji

jk

,...1 021 1

^

0

^

,...,,^

1

^

0

^

The least square function is

The estimates of 0, 1, …, k must satisfy

and


Matrix Approach (I)

y =

11...1

X =

nk

Xy

2

1

1

0

and


Matrix Approach (II)

yXXXL

XyXyLn

ii

'^

'

1

''2

0

Since

Therefore

^

^^

yye

Xy

and


Computer Output for the Example


Estimation of 2

kkk

k

j

CC

CC

XX

where

CVE

yXXX

0

0001'

2^

jj2

^^

'1'^

C

cov and C ,

Covariance matrix


pn

yXyy

pn

SS

therefore

XyXyeeeyySS

and

pn

SSMS

E

n

ii

n

iiiE

EE

'''

,^

2^

^'^'

1

22

1

^

2^


The Analysis of Variance for Multiple Regression

The analysis of variance divides the total variation in the response variable y,

into two portions:- SSR (sum of squares for regression) measures the amount of

variation explained by using the regression equation.- SSE (sum of squares for error) measures the residual variation

in the data that is not explained by the independent variables.

The values must satisfy the equation Total SS SR SSE. There are (n 1) degrees of freedom. There are k regression degrees of freedom. There are (n – p) degrees of freedom for error. MS SS d f

n

yy i

i

2

2TSS


The example ANOVA table:

The conditional or sequential sums of squares each account for one of the k 4 regression degrees of freedom.

Testing the Usefulness of the Regression Model In multiple regression, there is more than one partial

slope—the partial regression coefficients.The t and F tests are no longer equivalent.


The Analysis of Variance F Test

Is the regression equation that uses the information

provided by the predictor variables x1, x2, …, xk substantially

better than the simple predictor that does not rely on any of the x-values?

- This question is answered using an overall F test with the hypotheses At least

one of 1, 2, …, k is not 0.

- The test statistic is found in the ANOVA table as

F = MSR / MSE.

The Coefficient of Determination, R 2

- The regression printout provides a statistical measure of the strength of the model in the coefficient of determination.

- The coefficient of determination is sometimes called multiple R 2

y

: versus0: 1210 HH k


- The F statistic is related to R 2 by the formula

so that when R 2 is large, F is large, and vice versa. Interpreting the Results of a Significant Regression

Testing the Significance of a Partial Regression Coefficients- The individual t test in the first section of the regression printout are designed to test the hypotheses:

for each of the partial regression coefficients, given that the other predictor variables are already in the model.- These tests are based on the Student’s t statistic given by

which has d f (n p) degrees if freedom.

11 2

2

knR

kRF

0: s versu0: 10 ii HH

jj

i

i

i

C

t2^

^

^

^

SE


The Adjusted Value of R 2

- An alternative measure of the strength of the regression model is adjusted for degrees of freedom by using mean squares rather than sums of squares:- An alternative measure if the strength of the regression model is adjusted for degrees of freedom by using mean squares rather than sums of squares:

- For the real estate data in Figure 13.3,

which is provided right next to “R-Sq(adj).”

%1001SS Total

MSE1)adj(2

n

R

%0.96%100142.382,16

9.461)adj(2

R


Tests and Confidence Interval on Individual Regression Coefficients

• Example 11-5 and 11-6, pp. 510~513• Marginal Test Vs. Significance Test


Confidence Interval on the Mean Response


PREDICTION OF NEW OBSERVATIONS


Measures of Model Adequacy

Coefficient of Multiple Determination

Residual Analysis Standardized Residuals Studentized Residuals

Influential Observations Cook Distance Measure

2^

i

E

ii

e

MS

ed


Coefficient of Multiple Determination


Studentized Residuals


Influential Observations


Cook’s Distance


The Analysis Procedure

When you perform multiple regression analysis, use a step-by-step approach:1. Obtain the fitted prediction model.2. Use the analysis of variance F test and R 2 to determine how well the model fits the data.3. Check the t tests for the partial regression coefficients to seewhich ones are contributing significant information in the presence of the others.4. If you choose to compare several different models, use R 2(adj) to compare their effectiveness5. Use-computer generated residual plots to check for violation of the regression assumptions.


The quadratic model is an example of a second-order model because it involves a term whose components sum to 2 (in this case, x2 ).

It is also an example of a polynomial model—a model that takes the form

3111

21110 xxxy

A Polynomial Regression Model

21110 xxy

Example 11-13, pp. 530-531


Using Quantitative and Qualitative Predictor Variables in a Regression

Model

The response variable y must be quantitative. Each independent predictor variable can be either a

quantitative or a qualitative variable, whose levels represent qualities or characteristics and can only be categorized.

We can allow a combination of different variables to be in the model, and we can allow the variables to interact.

A quantitative variable x can be entered as a linear term, x, or to some higher power such as x 2 or x3 .

You could use the first-order model:22110)( xxyE


We can add an interaction term and create a second-order model:

Qualitative predictor variable are entered into a regression model through dummy or indicator variables.

If each employee included in a study belongs to one of three ethnic groups—say, A, B, or C—you can enter the qualitative variable “ethnicity” into your model using two dummy variables:

21322110)( xxxxyE

not if 0

C group if 1

not if 0B group if 1

21

xx


The model allows a different average response for each group.

1 measures the difference in the average

responses between groups B and A, while 2

measures the difference between groups C and A. When a qualitative variable involves k categories, (k 1) dummy variables should be added to the regression model.

Example 11-14, pp. 534~536 <different approach>


Testing Sets of Regression Coefficients

Suppose the demand y may be related to five independent variables, but that the cost of measuring three of them is very high.

If it could be shown that these three contribute little or no information, they can be eliminated.

You want to test the null hypothesis H0 : 3 4 5

0—that is, the independent variables x3, x4, and x5

contribute no infor-mation for the prediction of y—versus the alternative hypothesis: H1 : At least one of the

parameters 3, 4, or 5 differs from 0 —that is, at least

one of the variables x3, x4, or x5 contributes information

for the prediction of y.


To explain how to test a hypothesis concerning a set of model parameters, we define two models:

Model One (reduced model)

Model Two (complete model)

terms in additional terms model 1 in model 2

The test of the null hypothesis

versus the alternative hypothesis

H1 : At least one of the parameters

differs from 0

0: 210 krrH

krr , , , 21

rr xxxyE 221110)(

kkrrrrrr xxxxxxyE 221122110)(


uses the test statistic

where F is based on d f1 (k r ) and d f2 n (k 1).

The rejection region for the test is identical to the rejection forall of the analysis of variance F tests, namely

FF

2

21

MSE

SSESE rkSF


Interpreting Residual Plots The variance of some types of data changes as the

mean changes:

- Poisson data exhibit variation that increases with the mean.

- Binomial data exhibit variation that increases for values of p

from .0 to .5, and then decreases for values of p from .5 to 1.0.

Residual plots for these types of data have a pattern similar to that shown in the next pages.


Plots of residuals against y


If the range of the residuals increases as increases and you know that the data are measurements of Poisson variables, you can stabilize the variance of the response by running the regression analysis on

If the percentages are calculated from binomial data, you can use the arcsin transformation,

If E(y) and a single independent variable x are linearly related, and you fit a straight line to the data, then the observed y values should vary in a random manner about and a plot of the residuals against x will appear as shown in the next page.

If you had incorrectly used a linear model to fit the data, the residual plot would show that the unexplained variation exhibits a curved pattern, which suggests that there is a quadratic effect that has not been included in the model.

y

.* yy

.sin* 1 yy

y


Figure 13.17 Residual plot when the model provides a good

approximation to reality


Stepwise Regression Analysis Try to list all the variables that might affect a college freshman’s

GPA:- Grades in high school courses, high school GPA, SAT score,

ACT score- Major, number of units carried, number of courses taken- Work schedule, marital status, commute or live on campus

A stepwise regression analysis fits a variety of models to the data, adding and deleting variables as their significance in the presence of the other variables is either significant or nonsignificant, respectively.

Once the program has performed a sufficient number of iterations and no more variables are significant when added to the model, and none of the variables are nonsignificant when removed, the procedure stops.

These programs always fit first-order models and are not helpful in detecting curvature or interaction in the data.


Selection of Variables in Multiple Regression

All Possible Regressions R2

p or adj R2p

MSE(p)

Cp

Stepwise Regression Start with the variable with the highest correlation with

Y.

Forward Selection

Backward Selectionpp. 539~549

pn

pSSC

pn

pSSpMS

Rpn

nRor

SS

pSSR

Ep

EE

pp

T

Rp

2

11

1

2^

222


Misinterpreting a Regression Analysis

A second-order model in the variables might provide a very good fit to the data when a first-order model appears to be completely useless in describing the response variable y.

CausalityBe careful not to deduce a causal relationship between a response y and a variable x.

MulticollinearityNeither the size of a regression coefficient nor its t-value indicates the importance of the variable as a contributor of information. This may be because two or more of the predictor variables are highly correlated with one another; this is called multicollinearity.


Multicollinearity can have these effects on the analysis:

- The estimated regression coefficients will have large standarderrors, causing imprecision in confidence and prediction intervals.

- Adding or deleting a predictor variable may cause significant changes in the values of the other regression coefficients.

How can you tell whether a regression analysis exhibits multicollinearity?- The value of R 2 is large, indicating a good fit, but the

individual t-tests are nonsignificant.- The signs of the regression coefficients are contrary to what

you would intuitively expect the contributions of those variables to be.

- A matrix of correlations, generated by the computer, shows you which predictor variables are highly correlated with each other and with the response y.


The last three columns of the matrix show significant correlations between all but one pair of predictor variables:

Multiple Regression1 Multiple Linear Regression Multiple Regression Model A regression model that...

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