Lecture 5 Hypothesis Testing in Multiple Linear Regression
[Pages:28]Lecture 5 Hypothesis Testing in Multiple Linear
Regression
BIOST 515
January 20, 2004
1
Types of tests
? Overall test ? Test for addition of a single variable ? Test for addition of a group of variables
2
Overall test
yi = 0 + xi11 + ? ? ? + xipp + i Does the entire set of independent variables contribute significantly to the prediction of y?
3
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
4
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
5
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) .
6
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.
7
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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