Analisa Regresi Week 7

The Multiple Linear Regression ModelKey ideas from case study Model Description and assumptionThe general linear model and the

least square procedureInference for multiple regression

Key ideas from case study (2)The relationship between a response variable

and an explanatory variable depends on what other explanatory variables are in the model

A variable can be a significant (P<.05) predictor alone and not significant (P>0.5) when other Xs are in the model

Key ideas from case study (1)

First, look at graphical and numerical summaries for one variable at a time

Then, look at relationships between pairs of variables with graphical and numerical summaries.

Use plots and correlations

Key ideas from case study (3)Regression coefficients, standard errors and

the results of significance tests depend on what other explanatory variables are in the model

Key ideas from case study (4)Significance tests (P values) do not tell the

whole storySquared multiple correlations give the

proportion of variation in the response variable explained by the explanatory variables) can give a different view

We often express R2 as a percent

Key ideas from case study (5)You can fully understand the theory in terms of

Y = Xβ + εTo effectively use this methodology in practice

you need to understand how the data were collected, the nature of the variables, and how they relate to each other

Model Description and AssumptionsConsider an experiment in which data are generated of the type

knnnn

k

k

k

xxxy

xxxy

xxxy

xxxy

21

222122

121111

21

Model Description and Assumptions (2)If the experimenter is willing to assume that in the region of the x’s defined by the data, yi is related approximately to the regressor variables, then the model formulation:(1)whereYi is the response variable for the ith caseXi1, Xi2, … , Xik are k explanatory variables for cases i = 1 to n and n ≥ k + 1ε i is a model error β0 is the interceptβ1, β2, … , βk are the regression coefficients for the explanatory variables

ikikiioi XXXY 2211

Model Description and Assumptions (3)What is a Linear Model?A linear model is defined as a model that is

linear in the parameter, i.e., linear in the coefficients, the β’s in the eq (1)

For example : A model quadratic in X

A linear model with interactioniiioi XXY 2

21

iiiiioi XXXXY 21122211

Model Description and Assumptions (4)What is the meaning of Regression Coefficients? βo is the Y intercept of the regression plane. If the scope of the model includes Xi1 = 0, … , Xik = 0, it gives the mean response at Xi1 = 0, … , Xik = 0.The parameter β1 indicates the change in the mean response E(Y) per unit change in X1 when other X’s held constant.The β’s are often caled partial regression coefficients because they reflect the partial effect on one independent variable when the other variables are included in the model and are held constant.

Model Description and Assumptions (5)Assumptionsε i are independent normally distributed

random errors with mean 0 and variance σ2

Xji are not random and are measured with negligible error

The GLM and the Least Square Procedure

In matrix term:

Where:Y is a vector of responsesβ is a vector of parametersX is a matrix of constantsε is a vector of model errors which

ε ~ N(0,σ2I)Consequently, Y ~ N(Xβ,σ2I)

111 nxpxnxpnx

XY

The GLM and the Least Square Procedure (2)

Least Square procedure:

minimize e’e = (Y-Xb)’(Y-Xb)

The least square normal equations:(X’X)b = X’y

Assuming X is of full column rank, b =(X’X)-1X’y

111 nxpxnxpnx

XY

0'

XbYXbYb

The GLM and the Least Square Procedure (3)

The fitted predicted values:

And the residual terms by

Or

The variance-covariance matrix of the residuals is

Which is estimated by s2{e}= MSE(I-H)

HYYX'X)X(X'Xb 1^

Y

XbYYYe ^

YHIe )(

)(}{ 22 HIe

ANOVA TableTo organize arithmeticSources of variation are Model (SAS) or Regression (NKNW)Error (SAS, NKNW) or ResidualTotal

SS and df addSSM + SSE =SSTdfM + dfE = dfT

SS (2)

The sum squares for ANOVA in matrix terms are :

Yn

1''

n

1 - Y'

JIYJYYYSST

Y' Xb)-(YXb)'-(Y e' HIYeSSE

Yn

1''

n

1 - ''

JHYJYYYXbSSM

Source SS df MS Model SSM dfM MSMError SSE dfE MSE Total SST dfT (MST)

F = MSM/MSE

ANOVA F testH0: β1 = β2 = … βp-1 = 0

Ha: βk neq 0, for at least one k=1, … , p-1Another form of the null hypothesis is H0: β1 = 0, and β2 = 0, … , and βp-1 = 0

Under H0, F ~ F(p-1,n-p)

Reject H0 if F is large, use P value

Example NKNW p 249The Zartan company sells a special skin

cream through fashion stores exclusively in 15 districts

Y is salesX1 is target population

X2 is per capita discretionary incomen = 15 districts

Check the data

obs yi x1i x2i obs yi x1i x2i

1 162 274 2450 9 116 195 2137

2 120 180 3254 10 55 53 2560

3 223 375 3802 11 252 430 4020

4 131 205 2838 12 232 372 4427

5 67 86 2347 13 144 236 2660

6 169 265 3782 14 103 157 2088

7 81 98 3008 15 212 370 2605

8 192 330 2450          

•H0: β1 = β2 = … βp-1 = 0 •F = MSM/MSE•Reject H0 if the P value is leq .05

ANOVAb

53844,716 2 26922,358 5679,466 ,000a

56,884 12 4,740

53901,600 14

Regression

Residual

Total

Model1

Sum ofSquares df Mean Square F Sig.

Predictors: (Constant), INCOME, POPa.

Dependent Variable: SALESb.

What do we conclude?

R2

The squared multiple regression correlation (R2) gives the proportion of variation in the response variable explained by the explanatory variables included in the model

It is usually expressed as a percentIt is sometimes called the coefficient of multiple

determination (NKNW p 230)

R2 (2)

R2 = SSM/SST, the proportion of variation explained

R2 = 1 – (SSE/SST), 1 – the proportion of variation not explained

H0: β1 = β2 = … βp-1 = 0 is equivalent to H0: the population R2 is zero

F = [ (R2)/(p-1) ] / [ (1- R2)/(n-p) ]

What and WhyAt this point we have examined the

distribution of the explanatory variables (and the response variable if that is appropriate) and we have taken remedial measures where needed

We have looked at plots and numerical summaries

What and Why (2)The P-value for the F significance test tells us

one of the following:there is no evidence to conclude that any of

our explanatory variables can help us to model the response variable using this kind of model (P gt .05)

one or more of the explanatory variables in our model is potentially useful for predicting the response variable in a linear model (P leq .05)

R-Sq = 0.999Adj R-Sq = 0.999

Coeff Var = 6.0

Inference for individual regression coefficientsb ~ N(β, (s(bi, bk)))

s(bi, bi) = s2(bi)

CI: bi ± t*s(bi)

Significance test for H0i: βi, = 0 uses the test statistic t =bi/s(bi), df=dfE=n-p, and the P-value computed from the t(n-p) distribution

Par StVar Est Err t PInt 3.453 2.431 1.420 0.181Pop 0.496 0.006 81.924 <.0001income 0.009 0.001 9.502 <.0001

Estimation by Doolittle General Format:

X’X | X’Y | I

By using row transformation we’ll have

I | b | (X’X)-1

Example : see worksheet 1

Estimation of E(Yh)

Xh is now a vector

(1, Xh1, Xh2, … , Xh1)’We want an point estimate and a

confidence interval for the subpopulation mean corresponding to Xh

The mean response to be estimated is

The estimated mean response corresponding to Xh, is

This estimated is unbiased and its variance is

bX'h^

hY

hX')( hYE

)E(Y X')( hh

^

hYE

h1-

h2

^2 XX)(X'X')( hY

The estimated variance s2(Ŷh) is given by

The 1 – α CI for E(Yh) are

Example (in the class)

h2

hh1-

h

^2 X)(X')XX)(X'X'()( bsMSEYs h

)();2/1(^^

hh YspntY

It is used when a data set requires repeat observation

Repeat observation in multiple regression are replicate observation on Y coresponding to level of each of the X variables that are constant from trial to trial.

Whit two independent variables repeat observations require that X1 and X2 each remain at given levels from trial to trial.

SSE is decomposed into pure error and lack of fit component

SSE = SSPE + SSLF

The pure error sum of square SSPE is obtained by first calculating for each replicate group the sum of squared deviations of the Y observation around the group mean, where a replicate group has the same values for each of the X variables.

2^

22^

)()()(

ijjjijijij YYYYYY

If the linear regression function is appropriate, then the means Ȳj will be near the fitted values Ŷij calculated from the esyimated linear regression function and SSLF will be small. See the illistration on NWK page 138.

Df(SSPE) = n – c and df(SSLF) = (n-p)-(n-c)=c-p where c is is the number of replicate group

The hypotheses statements are:

The approriate test statistic is

And the appropriate decision rule is conclude Ho if

122111

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}{:

}{:

pko

pko

XXXYEH

XXXYEHo

MSPE

MSLF

cn

SSPE

pc

SSLFFo

*

),;1(* cnpcFFo

Prediction of new observation Yh(new)

Xh is now a vector

(1, Xh1, Xh2, … , Xh1)’

We want a prediction for Yh with an interval that expresses the uncertainty in our prediction

The prediction of new observation Yh(new) corresponding to Xh has 1 – α CI of

Where

Example (in the class)

)XX)(X'X'1()( h1-

h)(2 MSEYs newh

)();2/1( )(

^

newhh YspntY

When m new observation at Xh and their

mean Ȳh(new) is to be predicted, the 1- corresponding to has 1 – α CI of

Where

Example (in the class)

)XX)(X'X'1

()( h1-

h)(

_2

mMSEYs newh

)();2/1( )(

^

newhh YspntY

Last slideReading NKMW 7.1 to 7.8Exercise NKMW page 264 no 7.8-7.11Homework NKMW page 264-267 n0 7.12-

7.19