1

3.3 Hypothesis Testing in Multiple Linear Regression

• Questions:– What is the overall adequacy of the model?– Which specific regressors seem important?

• Assume the errors are independent and follow a normal distribution with mean 0 and variance 2

2

3.3.1 Test for Significance of Regression• Determine if there is a linear relationship between

y and xj, j = 1,2,…,k.

• The hypotheses are

H0: β1 = β2 =…= βk = 0

H1: βj 0 for at least one j

• ANOVA

• SST = SSR + SSRes

• SSR/2 ~ 2k, SSRes/2 ~ 2

n-k-1, and SSR and SSRes

are independent1,

ReRe0 ~

)1/(

/

knk

s

R

s

R FMS

MS

knSS

kSSF

3

•

• Under H1, F0 follows F distribution with k and n-

k-1 and a noncentrality parameter of

knkn

kk

c

k

ccR

s

xxxx

xxxx

X

k

XXMSE

MSE

11

1111

1*

2

*'*'2

2Re

)',...,(

)(

)(

2

*'*'

ccXX

4

• ANOVA table

5

6

• Example 3.3 The Delivery Time Data

7

• R2 and Adjusted R2 – R2 always increase when a regressor is added to

the model, regardless of the value of the contribution of that variable.

– An adjusted R2:

– The adjusted R2 will only increase on adding a variable to the model if the addition of the variable reduces the residual mean squares.

)1/(

)/(1 Re2

nSS

pnSSR

T

sadj

8

3.3.2 Tests on Individual Regression Coefficients• For the individual regression coefficient:

– H0: βj = 0 v.s. H1: βj 0

– Let Cjj be the j-th diagonal element of (X’X)-1.

The test statistic:

– This is a partial or marginal test because any estimate of the regression coefficient depends on all of the other regression variables.

– This test is a test of contribution of xj given the

other regressors in the model

120 ~)ˆ(

ˆ

ˆ

ˆ kn

j

j

jj

j tseC

t

9

• Example 3.4 The Delivery Time Data

10

• The subset of regressors:

11

• For the full model, the regression sum of square

• Under the null hypothesis, the regression sum of squares for the reduce model

• The degree of freedom is p-r for the reduce model.

• The regression sum of square due to β2 given β1

• This is called the extra sum of squares due to β2

and the degree of freedom is p - (p - r) = r• The test statistic

yXSSR ''ˆ)(

yXSSR'1

'11

ˆ)(

)()()|( 112 RRR SSSSSS

pnrs

R FMS

rSSF ,

Re

120 ~

/)|(

12

• If β2 0, F0 follows a noncentral F distribution

with

• Multicollinearity: this test actually has no power!

• This test has maximal power when X1 and X2 are

orthogonal to one another!

• Partial F test: Given the regressors in X1, measure

the contribution of the regressors in X2.

22'1

11

'11

'2

'22

])([1

XXXXXIX

13

• Consider y = β0 + β1 x1 + β2 x2 + β3 x3 +

SSR(β1| β0 , β2, β3), SSR(β2| β0 , β1, β3) and SSR(β3|

β0 , β2, β1) are signal-degree-of –freedom sums of

squares.

• SSR(βj| β0 ,…, βj-1, βj, … βk) : the contribution of

xj as if it were the last variable added to the model.

• This F test is equivalent to the t test.

• SST = SSR(β1 ,β2, β3|β0) + SSRes

• SSR(β1 ,β2 , β3|β0) = SSR(β1|β0) + SSR(β2|β1, β0) +

SSR(β3 |β1, β2, β0)

14

• Example 3.5 Delivery Time Data

15

3.3.3 Special Case of Orthogonal Columns in X

• Model: y = Xβ + = X1β1+ X2β2 +

• Orthogonal: X1’X2 = 0

• Since the normal equation (X’X)β= X’y,

•

yX

yX

XX

XX'2

'1

2

1

2'2

1'1

ˆ

ˆ

0

0

yXXXyXXX '2

12

'22

'1

11

'11 )(ˆ and )(ˆ

16

17

3.3.4 Testing the General Linear Hypothesis• Let T be an m p matrix, and rank(T) = r• Full model: y = Xβ +

• Reduced model: y = Z + , Z is an n (p-r) matrix and is a (p-r) 1 vector. Then

• The difference: SSH = SSRes(RM) – SSRes(FM) with r degree of freedom. SSH is called the sum of squares due to the hypothesis H0: Tβ = 0

freedom) of degree p-(n ''ˆ')(Re yXyyFMSS s

freedom) of degreer p-(n ''ˆ')(

')'(ˆ

Re

1

yZyyRMSS

yZZZ

s

18

• The test statistic:

pnrs

H FpnFMSS

rSSF

,Re

~)/()(

/

19

20

• Another form:

• H0: Tβ = c v.s. H1: Tβ c Then

)/()(

/ˆ]')'([''ˆ

Re

11

pnFMSS

rTTXXTTF

s

pnrs

FpnFMSS

rcTTXXTcTF

,Re

11

~)/()(

/)ˆ(]')'([)'ˆ(

21

3.4 Confidence Intervals in Multiple Regression

3.4.1 Confidence Intervals on the Regression Coefficients

• Under the normality assumption,

))'(,(~ˆ 12 XXMN

22

23

3.4.2 Confidence Interval Estimation of the Mean Response

• A confidence interval on the mean response at a particular point.

• x0 = (1,x01,…,x0k)’

• The unbiased estimator of E(y|x0) :

2/10

1'0

2,2/0

01'

02

0

'000

))'((y

responsemean on the C.I. )-100(1 The

)'()ˆ(

)|()ˆ(

xXXxt

xXXxyVar

xxyEyE

pn

24

• Example 3.9 The Delivery Time Data

25

3.4.3 Simultaneous Confidence Intervals on Regression Coefficients

• An elliptically shaped region

26

• Example 3.10 The Rocket Propellant Data

27

28

• Another approach:

is chosen so that a specified probability that all intervals are correct is obtained.

• Bonferroni method: Δ= tα/2p, n-p

• Scheffe S-method: Δ=(2Fα,p, n-p )1/2

• Maximum modulus t procedure: Δ= uα,p, n-2 is the

upper tail point of the distribution of the maximum absolute value of two independent student t r.v.’s each based on n-2 degree of freedom

k ..., 1, 0,j ),ˆ(ˆ jj se

29

• Example 3.11 The Rocket Propellant Data

• Find 90% joint C.I. for β0 and β1 by constructing a 95% C.I. for each parameter.

30

• The confidence ellipse is always a more efficient procedure than the Bonferroni method because the volume of the ellipse is always less than the volume of the space covere3d by the Bonferroni intervals.

• Bonferroni intervals are easier to construct.• The length of C.I.:

Maximum modulus t < Bonferroni method

< Scheffe S-method

31

3.5 Prediction of New Observations

32

3.6 Hidden Extrapolation in Multiple Regression

• Be careful about extrapolating beyond the region containing the original observations!

• Rectangle formed by ranges of regressors NOT data region.

• Regressor variable hull (RVH): the convex hull of the original n data points.

– Interpolation: x0 RVH

– Extrapolation: x0 RVH

33

34

• hii of the hat matrix H = X(XX)-1X’are useful in

detecting hidden extrapolation.

• hmax: the maximum of hii . The point xi that has the

largest value of hii will lie on the boundary of

RVH

• {x | x(XX)-1x h≦ max } is an ellipsoid enclosing all

points inside the RVH.

• Let h00 = x′(X′X)-1x0

– h00 hmax : inside the RVH and the boundary

of RVH

– h00 > hmax : outside the RVH

35

• MCE : minimum covering ellipsoid (Weisberg, 1985).

36

37

3.7 Standardized Regression Coefficients

• Difficult to compare regression coefficients directly.

• Unit Normal Scaling: Standardize a Normal r.v.

38

• New model:

– There is no intercept.– The least-square estimator of b is

nizbzby iikkii ,...,1 ,11*

*1 ')'(ˆ yZZZb

39

• Unit Length Scaling:

40

• New Model:

• The least-square estimator:

niwbwby iikkii ,...,1 ,110

01 ')'(ˆ yWWWb

41

• It does not matter which scaling we use! They both produce the same set of dimensionless regression coefficient.

42

43

44

3.8 Multicollinearity

• A serious problem: Multicollinearity or near-linear dependence among the regression variables.

• The regressors are the columns of X. So an exact linear dependence would result a singular X’X

45

• Unit length scaling

1)ˆ()ˆ(

10

01)'( and

10

01'

22

21

1

bVarbVar

WWWW

46

• Soft drink data:

• Off-diagonal elements are of W’W usually called the simple correlations between regressors.

12.3)ˆ()ˆ(

12.357.2

57.212.3)'( and

1824.0

824.01'

22

21

1

bVarbVar

WWWW

47

• Variance inflation factors (VIFs):– The main diagonal elements of the inverse of

X’X ((W’W)-1 above)

– From above two cases:Soft drink: VIF1 = VIF2 =

3.12 and Figure 3.12: VIF1 = VIF2 = 1

– VIFj = 1/(1-Rj)

– Rj is the coefficient of multiple determination

obtained from regressing xj on the other regressor

variables.

– If xj is nearly linearly dependent on some of the

other regressors, then Rj 1 and VIFj will be

large.– Serious problems: VIFs > 10

48

• Figure 3.13 (a): The plan is unstable and very sensitive to relatively small changes in the data points.

• Figure 3.13 (b): Orthogonal regressors.

49

3.9 Why Do Regression Coefficients Have the Wrong Sign?

• The reasons of the wrong sign:

1. The range of some of the regressors is too small.

2. Important regressors have not been included in the model.

3. Multicollinearity is present.

4. Computational errors have been made.

50

• For reason 1:

n

iixx xxSVar

1

221 ))(/(/)ˆ(

51

• Although it is possible to decrease the variance of the regression coefficients by increase the range of the x’s, it may not be desirable to spread the levels of the regressors out too far:– The true response function may be nonlinear.– Impractical or impossible.

• For reason 2:

52

•

• t.coefficien regression total"" a is ˆ here

, 463.0835.1ˆ 1

xy

given x xofeffect theis ˆ Here

649.3222.1036.1ˆ

211

21

xxy

53

• Fore reason 3: Multicollinearity inflates the variances of the coefficients, and this increases the probability that one or more regression coefficients will have the wrong sign.

• Different computer programs handle round-off or truncation problems in different ways, and some programs are more effective than the others in this regard.