Here are a couple of more recent articles on the issue in the emails below,
usually referred to as the inverse regression or calibration problem. The
second reference is a good review to get you started...
1. Cheng, C. L. and Van Ness, J. W. Robust Calibration. Technometrics.
1997; 39(4):401-411.
2. Mee, R. W. and Eberhardt, K. R. A Comparison of Uncertainty Criteria
for Calibration. Technometrics. 1996; 38(3):221-229.
Regards,
Eric Scharin
-----Original Message-----
From: [EMAIL PROTECTED]
[mailto:[EMAIL PROTECTED]]On Behalf Of Joe Ward
Sent: Friday, October 20, 2000 1:26 PM
To: Teague, Dan; AP Statistics
Cc: EDSTAT-L
Subject: Independent-Dependent Variable Discussion--Inverse Estimation
Hi Dan and all --
I had intended to comment about the independent-dependent variable
discussion
earlier but I got side-tracked. Since Dan reminded us with his comment:
"> This problem statement also brings back the independent-dependent
variable
> discussion. In the real context, the activity level of the crickets
depends
> upon the temperature, so temperature is the independent variable and
number
> of chirps the dependent variable. However, if you want to predict the
> temperature using the number of chirps, you must consider the number of
> chirps as the "independent" variable and temperature as the "dependent"
> variable."
I have inserted some comments below:
=========== Joe Ward writes ======
In the ancient past (1950s), for calibration studies --
Let
Y be a reading from a measuring instrument, SUBJECT TO "ERRORS OF
MEASUREMENT".
and
X be a KNOWN STANDARD, ASSUMED TO BE "WITHOUT ERROR" (FIXED).
Then the least-squares regression model used to PREDICT THE "STANDARD" (X)
from the measurement Y WAS computed as:
Y = b0 + b1*X + Error
Then from this equation to estimate (predict) the KNOWN STANDARD (X) from
the measurement (Y), the past procedure was to solve for X in the above
equation
(leaving off the Error)
Y = b0 + b1*X
or
X = (Y-b0)/b1
is used to PREDICT X from Y.
Dan, you probably are better acquainted with the most recent approach from
the Bureau of Standards since I have not kept up with any changes in the
Standards calibration policy.
Furthermore, in the distant past, it is interesting to note that
simultaneous regression equations were solved to estimate unkown amounts of
chemical compositions in a solution.
An interesting study by Fisher, Hans, R.G. Hansen, and H.W. Norton (1955).
Quantitative determination of glucose and galactose. Anal. Chem. 27,
857-859. is discussed in
E.J Williams' book Regression Analysis, Wiley, 1959, page 163. Williams
refers to this topic as INVERSE ESTIMATION.
Even though the goal is to ESTIMATE (PREDICT) the values of X, the
dependent variables (Y's) are the MEASURES SUBJECT TO ERROR. After the
least-squares solutions are computed then the simultaneous regression
equations are solved, INVERSELY, for unknown X values from
measured(observed) values of Y (which are subject to ERRORS).
It would be interesting to know if this approach is still used. Is the
INVERSE method BETTER? Have there been recent studies comparing the REGULAR
approach with the
INVERSE approach?
Comments from experienced "experts" in this area are welcome.
-- 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
***************************************************************************
======= End of Joe Ward's message =====
----- Original Message -----
From: "Teague, Dan" <[EMAIL PROTECTED]>
To: "AP Statistics" <[EMAIL PROTECTED]>
Sent: Friday, October 20, 2000 10:42 AM
Subject: [ap-stat] RE: effect on LSRL
> Rebecca,
>
> If your student chose values of the independent variable that were very
> large (250-450) and found the y-values that correspond to these x-values
> using y = 56.212 + 0.1356x, then he could increase the slope. For these
> data, the point (249, 55) is below that portion of the regression line on
> the left. The regression line would be pulled towards the point, just as
> you said, but in this situation, it would cause the slope to increase.
>
> The student's argument is flawed to the extent that these values of the
> independent variable do not match the summary statistics (xbar = 167 and s
=
> 31). We expect to find the number of chirps between 70 and 290 and the
> temperature roughly between 50 and 100. For these values of x, the slope
> will be pulled down by the addition of this point.
>
> This problem statement also brings back the independent-dependent variable
> discussion. In the real context, the activity level of the crickets
depends
> upon the temperature, so temperature is the independent variable and
number
> of chirps the dependent variable. However, if you want to predict the
> temperature using the number of chirps, you must consider the number of
> chirps as the "independent" variable and temperature as the "dependent"
> variable.
>
>
> Daniel J. Teague
> NC School of Science and Mathematics
> 1219 Broad Street
> Durham, NC 27705
> [EMAIL PROTECTED]
>
>
> -----Original Message-----
> From: Rebecca Brewer [mailto:[EMAIL PROTECTED]]
> Sent: Friday, October 20, 2000 11:02 AM
> To: AP Statistics
> Subject: [ap-stat] effect on LSRL
>
>
> Help! I am a new AP teacher and need help explaining an answer to one of
my
>
> more challenging students. Here goes:
>
> At summer camp, one of Carla's counselors told her that you can determine
> air temperature from the number of cricket chirps. You are given the
xbar,
> ybar, both standard deviations, and the correlation coefficient r. xbar
> =166.8, ybar = 78.83, standard deviation for x = 31.0, standard deviation
> for y = 9.11, r = 0.461. The LSRL is 56.212 + 0.1356x. Suppose that
Carla
> counted 249 chirps on a day the temperature was 55 degrees. If this point
> were the 13th data point, what effect, if any, would this 13th point have
on
>
> Carla's LSRL? Explain briefly.
>
> I explained that this new data point would pull the LSRL towards it and
the
> slope would decrease. He says it would be higher and to prove this he
> plotted 12 points using the LSRL (nowhere are you given the points so he
> made them up) and then added the 13th point in and recalculated the LSRL.
>
> Is this valid?
>
> Replies asap would be greatly appreciated. Thank you for all of your
help.
>
> Rebecca Brewer
> Brewster High School
> [EMAIL PROTECTED]
>
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