Lecture 5 Hypothesis Testing in Multiple Linear Regression

[Pages:28]Lecture 5 Hypothesis Testing in Multiple Linear

Regression

BIOST 515

January 20, 2004

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Types of tests

? Overall test ? Test for addition of a single variable ? Test for addition of a group of variables

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Overall test

yi = 0 + xi11 + ? ? ? + xipp + i Does the entire set of independent variables contribute significantly to the prediction of y?

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Test for an addition of a single variable

Does the addition of one particular variable of interest add significantly to the prediction of y acheived by the other independent variables already in the model?

yi = 0 + xi11 + ? ? ? + xipp + i

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Test for addition of a group of variables

Does the addition of some group of independent variables of interest add significantly to the prediction of y obtained through other independent variables already in the model?

yi = 0 + xi11 + ? ? ? + xi,p-1p-1 + xipp + i

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The ANOVA table

Source of variation Regression

Error Total

Sums of squares

SSR = ^ X y - ny?2 SSE = y y - ^ X y SST O = y y - ny?2

Degrees of freedom p

n - (p + 1) n-1

Mean

square

SSR p

SSE n-(p+1)

E[Mean square]

p2 + RXCXCR 2

XC is the matrix of centered predictors:

0 x11 - x?1 x12 - x?2 ? ? ? x1p - x?p 1

XC

=

B B

@

x21

- ...

x?1

x22

- ...

x?2

???

x2p

- ...

x?p

C C A

xn1 - x?1 xn2 - x?2 ? ? ? xnp - x?p

and R = (1, ? ? ? , p) .

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The ANOVA table for

yi = 0 + xi11 + xi22 + ? ? ? + xipp + i

is often provided in the output from statistical software as

Source of Sums of squares

Degrees of F

variation

freedom

Regression x1

1

x2|x1

1

...

Error

xp|xp-1, xp-2, ? ? ? , x1

1

SSE

n - (p + 1)

Total

SST O

n-1

where SSR = SSR(x1) + SSR(x2|x1) + ? ? ? + SSR(xp|xp-1, xp-2, . . . , x1) and has p degrees of freedom.

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Overall test

H0 : 1 = 2 = ? ? ? = p = 0 H1 : j = 0 for at least one j, j = 1, . . . , p

Rejection of H0 implies that at least one of the regressors, x1, x2, . . . , xp, contributes significantly to the model.

We will use a generalization of the F-test in simple linear regression to test this hypothesis.

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