cyclic ratio data: mistake in analysis

[From Bill Powers (950729.1550 MDT)]

Bruce Abbott --

Rick and I have both been going slightly nuts over your derivation
showing that the pressing rate is constant. While I've gone ahead to
derive what I think is the correct treatment, I hadn't been able to see

You plotted time per reinforcement versus ratio:

time per reinforcement = 1/r = m/p + c

where p is pressing rate
c is collection time
m is ratio

This looked like a straight line, so you proposed that the equation of
the line was

1/r = k*m + c.

That was your mistake. Without realizing it, you were assuming one of
the unknowns. The correct general form for the straight line would be

1/r = k*m + j, or

m/p + c = k*m + j

What you assumed was j = c. Given that assumption, it follows that
m/p = k*m and

p = 1/k = constant

However, if j is NOT assumed equal to c, the equation becomes

m/p = (j - c) + k*m, or

p = m/(j - c + k*m),

which is not constant. The equation 1/r = j + k*m would still fit the
same data, but now the intercept would only determine j, not c.

The general straight line has two unknown coefficients to determine, j
and k. With only one equation, it is not possible to derive unique
values for j AND k. So your solution was only one of a family of
solutions, the particular solution for j = c. Any number of other
solutions with j <> c are also valid.

···

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Best,

Bill P.