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
\(\chi^2\) is a statistical parameter that quantifies the differences between an observed data set and an expected dataset (or ‘theory’) calculated as
Fitting typically minimizes the value of \(\chi^2\). For assessing the quality of the model and its “fit” however, SasView displays the traditional reduced \(\chi^2_R\) which normalizes this parameter by dividing it by the number of degrees of freedom (or DOF). The DOF is the number of data points being considered, \(N_\mathrm{pts}\), reduced by the number of free (i.e. fitted) parameters, \(N_\mathrm{par}\). Note that model parameters that are kept fixed do not contribute to the DOF (they are not “free”). This reduced value is then given as
Note that this means the displayed value will vary depending on the number of parameters used in the fit. In particular, when doing a calculation without a fit (e.g. manually changing a parameter) the DOF will now equal \(N_\mathrm{pts}\) and the \(\chi^2_R\) will be the smallest possible for that combination of model, data set, and set of parameter values.
When \(N_\mathrm{pts} \gg N_\mathrm{par}\) as it should for proper fitting, the value of the reduced \(\chi^2_R\) will not change very much.
For a good fit, \(\chi^2_R\) tends to 1.
\(\chi^2_R\) 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:
Think of each normalized residual as the number of standard deviations between the measured value and the theory. For a good fit, 68% of \(R_i\) will be within one standard deviation, which will show up in the Residuals plot as \(R_i\) values between \(-1\) and \(+1\). Almost all the values should be between \(-3\) and \(+3\).
Residuals values larger than \(\pm 3\) indicate that the model is not fit correctly, the wrong model was chosen (e.g., because there is more than one phase in your system), or there are problems in the data reduction. Since the goodness of fit is calculated from the sum-squared residuals, these extreme values will drive the choice of fit parameters. Any uncertainties calculated for the fitting parameters will be meaningless.
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