User's Manual

Page 18-51
Additional equations for linear regression
The summary statistics such as Σx, Σx
2
, etc., can be used to define the
following quantities:






−=⋅−=−=
∑∑∑
===
n
i
i
n
i
ix
n
i
ixx
x
n
xsnxxS
11
2
2
1
2
1
)1()(
2
11
2
2
1
2
1
)1()(






−=⋅−=−=
∑∑∑
===
n
i
i
n
i
iy
n
i
iy
y
n
ysnyyS












−=⋅−=−−=
∑∑∑∑
====
n
i
i
n
i
i
n
i
iixy
n
i
iixy
yx
n
yxsnyyxxS
1111
2
1
)1())((
From which it follows that the standard deviations of x and y, and the
covariance of x,y are given, respectively, by
1−
=
n
S
s
xx
x
,
1−
=
n
S
s
yy
y
, and
1−
=
n
S
s
yx
xy
Also, the sample correlation coefficient is
.
yyxx
xy
xy
SS
S
r
⋅
=
In terms of x, y, S
xx
, S
yy
, and S
xy
, the solution to the normal equations is:
xbya −= ,
2
x
xy
xx
xy
s
s
S
S
b ==
Prediction error
The regression curve of Y on x is defined as Y = Α + Β⋅x + ε. If we have a set
of n data points (x
i
, y
i
), then we can write Y
i
= Α + Β⋅x
i
+ ε
I
, (i = 1,2,…,n),
where Y
i
= independent, normally distributed random variables with mean
(Α + Β⋅x
i
) and the common variance σ
2
; ε
i
= independent, normally distributed
random variables with mean zero and the common variance σ
2
.