# How to judge the best of 2 fits to 1 data set?

Herman Rubin hrubin at b.stat.purdue.edu
Wed Jun 8 09:18:19 EST 1994

```In article <Cr2K5C.Lzp at sci.kun.nl> theos at sci.kun.nl (Theo Schoenmakers) writes:
>
>Dear knowledgeable statisticians,
>
>I am in doubt regarding the analysis of the results of a nonlinear
>regression analysis encountered during my research. I hope you can shed
>some light on the following problem:
>
>-My experiments display an "exponentially" decreasing curve. The analysis
>program offers a one- or a two-exponential decay (plus an offset). I have
>no clue as to which model is correct. So I try both, receiving from the
>program the 3 or 5 parameters plus an R value (4 digits after the comma).
>The curve contains 493 sampled data points, so I guess the degrees of
>freedom are 490 and 488. Okay?
>
>-Usually, the more complicated model fits the data better (not
>surprisingly). The increase in R is from +- 0.9984 to 0.9994. I guess the
>high R is also due to the great number of data points, since I think it is
>calculated as R = sqrt(SS[regression]/(SS[regression]+SS[error])), where
>SS is the sum of squares due to regression, or due to residual error.

You have asked a very important question.  That the regression is nonlinear
may or may not be important; if the sample is large enough, it is not.
To some extent, the problem is discussed in my paper with Sethuraman
(Bayes Risk Efficiency; _Sankhya_ 1965).  But the information needed is
not presented.  BTW, the analysis you have given is the classical one,
comparing the decrease in SS with two df to the error SS with 488 df.

You might consider this a great number of data points; again, without
seeing the likelihood plots, I cannot tell if it a sufficiently large
number (not a great number) or a small one or one which has both aspects.
All of these occur.

The Bayesian approach would compare the integrals of the likelihoods
with respect to the loss x prior measure.  If the situation is local
enough (I could give a better idea if I saw likelihood plots) then
there is a significant sample size effect in the choice of the ratio
of the integrals to use.  The prior probabilities themselves are
unlikely to be of much importance.
--
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hrubin at stat.purdue.edu (Internet, bitnet)
{purdue,pur-ee}!snap.stat!hrubin(UUCP)

```