Degrees of Freedom: A Correction to Chi Square For Physical Hypotheses

Authors: John Michael Williams

In common practice, degrees of freedom (df) may be corrected for the number of theoretical free parameters as though parameters were the same as data categories. However, a free physical parameter generally is not equivalent to a data category in terms of goodness of the fit. Here we use synthetic, nonrandom data to show the effect of choice of categorization and df on goodness of fit. We then explain the origin of the df problem and show how to avoid it in a three-step process: First, the theoretical curve is fit to the data to remove its variance, leaving what, under the null hypothesis, should be structureless residuals. Second, the residuals are fit by a set of orthogonal polynomials up to the degree, should it occur, at which significant variance was removed. Third, the number of nonsignificant polynomial terms in the original + orthogonal set become the df in a standard chi square test. This process reduces a general df problem to one of polynomial df and allows goodness of a fit to be determined by data categorization and significance level alone. An example is given of an evaluation of physical data on neutrino oscillation.

Comments: 47 Pages.

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Submission history

[v1] 16 Aug 2010

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