Statistical overview

If the standard deviation for the population is not constant, as in counting statistics where variance = counts, then each point should be individually weighted when comparing the observed sum of deviations and the expected sum of deviations.

At the conclusion **fit** reports 'stdfit', the standard deviation of the fit,
which is the rms of the residuals, and the variance of the residuals, also
called 'reduced chisquare' when the data points are weighted. The number of
degrees of freedom (the number of data points minus the number of fitted
parameters) is used in these estimates because the parameters used in
calculating the residuals of the datapoints were obtained from the same data.
These values are exported to the variables

FIT_NDF = Number of degrees of freedom FIT_WSSR = Weighted sum-of-squares residual FIT_STDFIT = sqrt(WSSR/NDF)

To estimate confidence levels for the parameters, one can use the minimum chisquare obtained from the fit and chisquare statistics to determine the value of chisquare corresponding to the desired confidence level, but considerably more calculation is required to determine the combinations of parameters which produce such values.

Rather than determine confidence intervals, **fit** reports parameter error
estimates which are readily obtained from the variance-covariance matrix
after the final iteration. By convention, these estimates are called
`"`standard errors`"` or `"`asymptotic standard errors`"`, since they are calculated
in the same way as the standard errors (standard deviation of each parameter)
of a linear least-squares problem, even though the statistical conditions for
designating the quantity calculated to be a standard deviation are not
generally valid for the NLLS problem. The asymptotic standard errors are
generally over-optimistic and should not be used for determining confidence
levels, but are useful for qualitative purposes.

The final solution also produces a correlation matrix, which gives an indication of the correlation of parameters in the region of the solution; if one parameter is changed, increasing chisquare, does changing another compensate? The main diagonal elements, autocorrelation, are all 1; if all parameters were independent, all other elements would be nearly 0. Two variables which completely compensate each other would have an off-diagonal element of unit magnitude, with a sign depending on whether the relation is proportional or inversely proportional. The smaller the magnitudes of the off-diagonal elements, the closer the estimates of the standard deviation of each parameter would be to the asymptotic standard error.