.. residuals_help.rst


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.. _Assessing_Fit_Quality:

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*

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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

.. math::

  \chi^2/N_{pts} =  \sum[(Y_i - Y_{theory}_i)^2 / (Y_error_i)^2] / N_{pts}

This differs slightly from what is sometimes called the 'reduced $\chi^2$'
because it does not take into account the number of fitting parameters (to
calculate the number of 'degrees of freedom'), but the 'normalized $\chi^2$
and the 'reduced $\chi^2$ are very close to each other when $N_{pts} \gg
\text{number of parameters}.

For a good fit, $\chi^2/N_{pts}$ tends to 1.

$\chi^2/N_{pts}$ is sometimes referred to as the 'goodness-of-fit' parameter.

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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:

.. math::

  R_i = (Y_i - Y_theory_i) / (Y_err_i)

For a good fit, $R_i \sim 0$.

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