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Fitted Model of Prices, Spring 2001

The following is an attempt to explain the prices of clamps, using data gathered at Brimfield, Spring 2001, on the Saturday. There are 25 clamps in the raw data. Please note the special nature of the sample, and do not over-generalize the findings.

My basic assumption is that the price of a clamp can be estimated from the product of two functions.

Price = Size_Function * Condition_Function

I assume that each size has an individual value, with no given relation to the value of other sizes. (This is in contrast to the Linear Model, where the value depends linearly upon length of jaw.)

Also, I constrain the Condition_Function so that "good" is worth 1, and the function is monotonic, that is, better condition must not imply lower prices. The conditions are defined to be poor, fair, good, and very good. (However, this time there are no "very good" clamps in the data.)

This provides 11 +3 parameters to fit to the data, in contrast to 2 + 3, in the Linear Model. We would expect much better fit. We had an mse = 6.1%.

However, not all combinations of size and condition are in the data set. And, most combinations have only one representative. This means that the fitting algorithm can sometimes establish values that are spurious. This is especially likely when there is only one representative of a particular size, or when all the representatives of a size are of similar condition.

This seems to have happened for the 8.5, 12, and 14 inch sizes, which get very high values in this analysis.

damaged = portions of handles or jaws cut off - not analysed
poor = threads stripped, spindles jammed, jaws covered with glue or paint
fair = threads missing, holes drilled in handles, jaws spattered with glue or paint
good = threads chipped in few places, dings and scratches of normal wear, dust and grime
very good = no damage to spindles or jaws, polished patina

The Excel Solver function was set up, and run.

The Size_function is tabulated below. Note the extremely high values imputed for some sizes.

size	4	6	7	8.5	10	12	14	16	18	19	20
Value	$12.00	$22.01	$10.20	$39.48	$27.40	$48.28	$61.33	$22.67	$24.90	$26.67	$18.28

The Condition_Function is tabluated below:

Cond	p	f	g
Value	+0.48	+0.52	+1.00

The goodness of the estimation is indicated below:

Actual	Est'd	Rel. Error
$ 5.00	$ 4.91	+ 2%
$10.00	$10.20	- 2%
$10.00	$31.71	- 68%
$12.00	$12.00	0%
$12.00	$14.17	- 15%
$12.00	$10.91	+ 10%
$13.00	$18.28	- 29%
$15.00	$27.40	- 45%
$15.00	$31.71	- 53%
$15.00	$22.67	- 34%
$19.00	$18.99	0%
$20.00	$18.28	+ 9%
$20.00	$18.28	+ 9%
$22.00	$22.01	0%
$22.00	$26.67	- 17%
$25.00	$24.97	0%
$25.00	$29.51	- 15%
$25.00	$22.67	+ 10%
$25.00	$24.90	0%
$28.00	$27.40	+ 2%
$30.00	$26.67	+ 13%
$35.00	$27.40	+ 28%
$45.00	$31.71	+ 42%
average	average	average
$20.00	$21.89	+ 7%

last revised and validated