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Assessing Fit Quality

When performing model-fits to some experimental data it is helpful to be able to gauge how good an individual fit is, how it compares to a fit of the same model to another set of data, or how it compares to a fit of a different model to the same data.

One way is obviously to just inspect the graph of the experimental data and to see how closely (or not!) the 'theory' calculation matches it. But SasView also provides two other measures of the quality of a fit:

  • |chi| 2 (or 'Chi2'; pronounced 'chi-squared')
  • Residuals

Chi2

Chi2 is a statistical parameter that quantifies the differences between an observed data set and an expected dataset (or 'theory').

SasView actually returns this parameter normalized to the number of data points, Npts such that

Chi2/Npts = { SUM[(Y_i - Y_theory_i)^2 / (Y_error_i)^2] } / Npts

This differs slightly from what is sometimes called the 'reduced chi-squared' because it does not take into account the number of fitting parameters (to calculate the number of 'degrees of freedom'), but the 'normalized chi-squared' and the 'reduced chi-squared' are very close to each other when Npts >> number of parameters.

For a good fit, Chi2/Npts tends to 0.

Chi2/Npts is sometimes referred to as the 'goodness-of-fit' parameter.

Residuals

A residual is the difference between an observed value and an estimate of that value, such as a 'theory' calculation (whereas the difference between an observed value and its true value is its error).

SasView calculates 'normalized residuals', R_i, for each data point in the fit:

R_i = (Y_i - Y_theory_i) / (Y_err_i)

For a good fit, R_i ~ 0.

Note

This help document was last changed by Steve King, 08Jun2015

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