Hi, Stan --
I've inserted a reply at the end of your message. Let me know
how things turn out.
-- Joe
****************************************************************************
****
Joe Ward.........................................Health Careers High School
167 East Arrowhead Dr....................4646 Hamilton Wolfe
San Antonio, TX 78228-2402...........San Antonio, TX 78229
Phone: 210-433-6575.......................Phone: 210-617-5400
Fax: 210-433-2828............................Fax: 210-617-5423
Email: [EMAIL PROTECTED]
http://www.ijoa.org/joeward/wardindex.html
***************************************************************************
----- Original Message -----
From: "Stanley110" <[EMAIL PROTECTED]>
To: <[EMAIL PROTECTED]>
Sent: Sunday, October 08, 2000 1:59 PM
Subject: Q: How to Pool Slopes
> Assume I have three sets of x,y data. I fit each by least-squares to a
straight
> line. I determine that the three fitted lines are homogeneous and
> indistinguishable at a certain significance level. I want to express the
slope
> (of the three) as a single point estimate and as a confidence interval.
What is
> the formula for doing this?
>
> Please reply to this newsgroup and to the writer at <[EMAIL PROTECTED]>.
>
> Thank you for your help.
>
> stan alekman
>
>
> =================================================================
> Instructions for joining and leaving this list and remarks about
> the problem of INAPPROPRIATE MESSAGES are available at
> http://jse.stat.ncsu.edu/
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====== JOE WARD REPLIES ===========
Hi, Stan --
Your Title says
(1)"How to Pool Slopes" and you indicate later that
(2)"I determine that the three fitted lines are homogeneous and
indistinguishable.
For (1) it sounds like you will want THREE DIFFERENT INTERCEPTS, but
for case (2) it sounds like you may want only ONE INTERCEPT.
This is good example of the use of the Regression Option of "NO INT" option
in SAS
or "Y-intercept = zero". The reason that this appears to be a difficult
problem is the use of the frequently-used DEFAULT option in most statistics
packages.
The approach used below for your THREE GROUP DATA is shown for TWO groups
of data
in the Prentice-Hall published book (1973) -- "Introduction to Linear
Models" by
Ward and Jennings. Chapter 8, page 143.
I don't know which Regression Software you are using, but you should be sure
to
FORCE THE Y-intercept THROUGH THE ORIGIN..
First, it is important to put ALL THREE SETS OF DATA in the same model.
Let Y = dependent variable (containing ALL THREE SETS OF DATA)
D1 = 1 if the corresponding element of Y is from DATA SET #1; 0 otherwise
D2 = 1 if the corresponding element of Y is from DATA SET #2; 0 otherwise
D3 = 1 if the corresponding element of Y is from DATA SET #3; 0 otherwise
X1 = Value of x if the corresponding element of Y is from DATA SET #1; 0
otherwise
X2 = Value of x if the corresponding element of Y is from DATA SET #2; 0
otherwise
X3 = Value of x if the corresponding element of Y is from DATA SET #3; 0
otherwise
X = Value of x for ALL corresponding elements of Y.
U = 1 for every element.
Then your ASSUMED MODEL is shown below: (this should give you the same
regression
coefficients that you already have computed -- a check that your new model
is
correct)
Y = a1*D1 + b1*X1 + a2*D2 + b2*X2 + a3*D3 + b3*X3 + E1 (Model #1)
After you have computed this ASSUMED MODEL you may want to TEST THE
HYPOTHESIS
that you imply in CASE (1) above, that the
THREE SLOPES ARE EQUAL, i.e., b1=b2= b3=bc (THE COMMON SLOPE)
Then substituting these restrictions into Model #1 produces the RESTRICTED
MODEL
FOR CASE (1):
Y = a1*D1 + bc*X1 + a2*D2 + bc*X2 + a3*D3 + bc*X3 + E2 (Model #2)
Factoring (or collecting terms) produces:
Y = a1*D1 + a2*D2 + a3*D3 + bc*X + E2 (Model #2)
(Note that the values of a1, a2, and a3 in Model #2 are NOT numerically
equal to
the values in Model #1)
>From Model #2, bc is the least-squares SINGLE POINT estimate of the COMMON
SLOPE.
Your favorite Regression procedure should give what you need to compute a
confidence interval (such as the standard error of bc).
Now for CASE (2) above you may want to test that:
THREE SLOPES ARE EQUAL, i.e., b1=b2= b3=bc ( THE COMMON SLOPE)
and
THREE INTERCEPTS ARE EQUAL, i.e., a1=a2=a3=ac (THE COMMON INTERCEPT)
In which case, the RESTRICTED MODEL becomes:
Y = ac*D1 + bc*X1 + ac*D2 + bc*X2 + ac*D3 + bc*X3 + E3 (Model #3)
Factoring (or collecting terms) produces:
Y = ac*U + bc*X + E3 (Model #3)
(Note that the value of bc in Model #3 is NOT numerically equal to the value
in
Model #2)
And, as before, your favorite Regression procedure should give what you need
to
compute a confidence interval (such as the standard error of bc). Let me
know how
this works out. If you have any problems with this approach you are welcome
to
send me your data and I can run the models.
--- Joe
==================================================================
begin 666 SLOPES-EQUAL-EARL-FINAL.txt
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`
end
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Instructions for joining and leaving this list and remarks about
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