MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_NextPart_01C80061.B99ED100" This document is a Single File Web Page, also known as a Web Archive file. If you are seeing this message, your browser or editor doesn't support Web Archive files. Please download a browser that supports Web Archive, such as Microsoft Internet Explorer. ------=_NextPart_01C80061.B99ED100 Content-Location: file:///C:/AB542AAC/252regrl.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" ANOVA, REGRESSION, CORRELATION

G. LINEAR REGRESSION-Curve Fitting

1. Exact vs. Inexact Relations

2. The Ordinary Least Squares Formula

We wish to estimate the coefficients in _{. Our ‘prediction’ will be _{and our err=
or
will be _{so that .}}}

(See appendix for derivation)

₌

3. Example

<= sub> _{=
_{=
_₌}}

1= 0 0 0 0 0

2= 2 1 2 1 4

3= 1 2 2 4 1

4= 3 1 3 1 9

5= 1 0 0 0 1

6= 3 3 9 9 9

7= 4 4 16 16 16

8= 2 2 4 4 4

9= 1 2 2 4 1

10 2 1 2 1 4

sum 19 16 40 40 49

First copy _{_{_{_{_=
and}}}}

Then compute means: _{_.}

Use these to compute ‘Spare Parts’: _{<=
!--[if gte vml 1]>

_{(Tota=
l Sum
of Squares) _.}}

Note that _{and _{must be pos=
itive,
while _{can be eith=
er
positive or negative.}}}

We can compute the coefficients:

_₌

So our regression equation is _{or _.}

4. _{, the Coeffi=
cient
of Determination}

_{, _{is Regressi=
on
(Explained) Sum of Squares.}}

_{is the Error
(Unexplained or Residual) Sum of Squares, and is defined as _{, a formula that should never be used for computation.}}

=

An alternate formul= a, if no spare parts have been computed, is

_{. The coefficient of determination is the square of the
correlation. Note that _{, _{and _{all have th=
e same
sign.}}}}

H. LINEAR REGRESSION-Simple Regression

1. Fitting a Line

2. The Gauss Markov Theorem

OLS is BLUE

3. Standard Errors – The standard error is defined as _.

_{_.}

Or, if no spare parts are available,

Note also that if&nbs= p; _{is availabl=
e _.}

Using data from G3, and using our spare parts _{_₌}

4. The Variance of _{and _.}

_and

I. LINEAR REGRESSION-Confidence Intervals and Tests

1. Confidence Intervals for _.

_{=

_{The
interval can be made smaller by increasing either _{or the amou=
nt of
variation in _.}}}

2. Tests for _.

To test _{use _{. Remember _{is most oft=
en
zero – and if the null hypothesis is false in that case we say that <=
sub> is signific=
ant.}}}

To continue the example in G3: _{or _{_{We have already
computed _{, which impl=
ies
that _{and _.}}}}}

The sign= ificance test is now _{_{. Assume that _{, so that for a 2-sided test _{and we reje=
ct the
null hypothesis if _{is below
–2.306 or above 2.306. Since _{is in the
rejection region, we say that =
_{is signific=
ant. A
further test says that _{is not
significantly different from 1.}}}}}}}}

If we want a confidence interval _{. Note that this includes 1,<=
/st1:PersonName>
but not zero.}

Note that since _{, _{. This indicates that both the a large sample size, _{, and a large
variance of _{will tend t=
o make
_{smaller and=
thus
decrease the size of a confidence interval for _{or increase=
the
size (and significance)
of the t-ratio. To put it more
negatively, small amounts of
variation in _{or small sa=
mple
sizes will tend to produce values of _{that are not
significant. The common sense interpretation of this statement is that we n=
eed
a lot of experience with what happens to _{when we var=
y _{to be able =
to put
any confidence in our estimate of the slope of the equation that relates th=
em.}}}}}}}}}}

3. Confidence Intervals and Tests for

We are now testing _{with _.}

_{. So _{<=
!--[if gte vml 1]>

. If we are testing _{_{. Since the
rejection region is the same as in I2,
we accept the null hypothesis and say that _{is not
significant. A confidence interval would be}}}}}

A common way to summarize our results is, _{. The equat=
ion is
written with the standard deviations below the equation. For a Minitab prin=
tout
example of a simple regression problem,
see 252re=
grex1.}

4. Prediction and Confidence Intervals for

The Confidence Interval is _{, where =
_{and the
Prediction Interval is _{, where =
_{. In these two formulas,
for some specific _, _{. For example,
assume that _{so that for=
the
results in G3, _{. Then _{and _{, so that the
confidence interval is _{. This represents a confidence interval for the average v=
alue
that _{will take w=
hen _{. For
the same data _{and _{, so that}}the prediction interval is _{. This is a confidence interval for the value that _{<=
!--[if gte vml 1]>

will take i=
n a
particular instance when _.}}}}}}}}}}}}}}

Ignore the re= mainder of this document unless you have had calculus!

Appendix to G2– Explanation of OLS Formula

Assume that we have three points: _{. We wish to fit a regression line to these points, with the equation _{and the characteristic that the sum of squares, _{is a
minimum. If we imagine that t=
here
is a 'true' regression line _{we can
consider _{to be estim=
ates
of _.}}}}}

&nbs= p; Let us make the definition _{. Note that if we substitute our equation for _{, we find th=
at _{, or _{. This has two consequences: First the sum of squares can=
be
written as _{; and second, that
if we fit the line so that _{or the mean of _{and _{is the same=
we
have _{. Now if we subtract the equation for _{from the equation for _{we find _{. Now=
let
us measure _{and _{as deviations from the mean,<=
/st1:PersonName>
replacing _{with _{and _{with _{. This means that _{or _{. If we substitute this expression in our sum of squares<=
st1:PersonName
w:st=3D"on">,}}}}}} we find that _.}}}}}}}}}}}}}}

&nbs= p; Now write this expression out in terms of our three points and differentiate it= to minimize _{with respect to _{. To do thi=
s, recall that _{is our unkn=
own
and that the _{s and _{s are numbers (constants!),
so that _{and _.}}}}}}

If we now take a derivative of this expression with re= spect to _{and set it equal to zero to find a minimum, we find that:}

&nbs= p; _.

But if _{, then _{=
or _{or _{, so that if=
we
solve for _{, we find . But if we remember that_{and _{, we can wri=
te
this as _or}}}}}}}

&nbs= p; Of course, we still need _{, but rememb=
er
that _{, so that .}}

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252regr 2/26/07 (Open this document in 'Outl= ine' view!) &= nbsp; &nbs= p; Roger Even Bove

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