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STAT 224-3 Discussion # 05/07/07 TA: Quoc, Tran
Simple Linear Regression Model
- Simple Linear Regression Model
yi = α + βxi + ǫi, i = 1, 2 , · · · n
where ǫ i
are noise terms which are distributed as N(0, σ
2 ).
Note that yi ∼ N(α + βxi, σ
2
).
- Three important sum squares:
S
XY
(x i
− x¯)(y i
− y¯) =
x i
y i
x i
y i
n
S
XX
(x i
− x¯)
2
=
x
2
i
x i
2
n
SY Y =
(yi − y¯)
2
=
y
2
i
y i
2
n
- Least Square Estimates: b =
β =
S
XY
S
XX
, a = ˆα = ¯y − b¯x
- Fitted(predicted) values: yˆi = a + bxi
- Residuals: ˆǫ i
= y i
− yˆ i
- ANOV A table for simple linear regression
Source df SS MS F
Regression 1 SSR MSR=SSR/1 F = MSR/MSE
Error n-2 SSE MSE=SSE/(n-2)
Total n-1 SST
where
SST = SY Y =
n ∑
i=
(yi − y¯)
2
=
n ∑
i=
y
2
i
n ∑
i=
yi)
2
/n
SSE = S
Y Y
− bS XY
i
(y i
− yˆ i
2
SSR =
S
2
XY
S
XX
2 : s
2 = ˆσ
2 = MSE =
SSE
n − 2
. Here n − 2 is the degree of freedom of residuals.
- Standard error of residuals: s =
MSE
Email: [email protected] 1 Rm B248D, MSC
STAT 224-3 Discussion # 05/07/07 TA: Quoc, Tran
- Coefficient of determination, r
2
= 1 −
SSE
SST
SSR
SST
It is interpreted as the proportion of observed y variation that can be explained by the simple regression
model.
- Inferences on the slope parameter β - The estimated standard deviation of
β: s ˆ β
s
n
i=
(x i
− x¯)
2
- A 100(1 − α)% CI for β
β ± t α/ 2 ,n− 2
s ˆ β
- Hythothesis testing for H 0 : β = β 0
test statistic t =
β − β 0
sˆ β
∼ tn− 2 under Ho.
Alternative Hypo. Reject. Reg. at level α P-value
H
a
: β > β 0
t ≥ t α,n− 2
P (T
n− 2
t)
H
a
: β < β 0
t ≤ −t α,n− 2
P (T
n− 2
< t)
H
a
: β 6 = β 0
Either t ≥ t α/ 2 ,n− 2
or t ≤ −t α/ 2 ,n− 2
2 P (T
n− 2
|t|)
- Model Utility test is the test of H 0
: β = 0 vs. H a
: β 6 = 0, in which case the test statistic value is
t =
β/sˆ β
∗
- Point Estimate of μ Y • x
∗ = a + bx
∗
- A 100(1 − α)% CI for the mean response at x
∗
a + bx
∗
± t α/ 2 ,n− 2
s
n
(x
∗ − ¯x)
2
n
i=
(x i
− x¯)
2
- A 100(1 − α)% Prediction Interval(PI) for a future Y at x = x
∗
a + bx
∗
± t α/ 2 ,n− 2
s
n
(x
∗ − x¯)
2
n
i=
(x i
− ¯x)
2
Email: [email protected] 2 Rm B248D, MSC
