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Method

The initial position in parameter space is checked to ensure that it corresponds to a chi-squared minimum. Each parameter in turn is then perturbed from its best fit value and frozen, and the other parameters optimised. From the actual chi-squared values attained, the parameter offset giving the specified chi-sq. min increase (SOFF) is estimated on the assumption that the minimum chi-squared surface is a parabolic function of the parameter whose error is being evaluated. The parabola may have a different shape on each side of the minimum, the curve on each side being defined by the assumption of zero slope at the minimum, and the requirement that it should pass through the single offset point evaluated. e.g. 
     .                             !
      *                            !            .
       .                           !
         .!.      -
            .                      !          *       ^
                 .                 !        .        SOFF
                        .          !     .            ^
                                   M                  ^
where * are the evaluated points, and the resulting confidence limits are shown by .

From Lampton, Margon & Bowyer (Ap.J. 208, p177, 1976) a value SOFF=1 gives a 1 sigma (i.e. 68% confidence) interval and SOFF=2.71 a 90% confidence interval for the case where the model is linear (i.e. the predicted values are linear functions of the parameters). Where this is not the case (i.e. always!) the results may still be approximately right LMB and also Avni Ap.J. 210, p642, 1976). A safe limit in all cases is obtained by projecting the full NPAR-dimensional confidence interval onto the subspace desired (a single parameter axis in this case). This corresponds to taking

        SOFF = chi-squared with NPAR d.o.f.( n% conf)
for a safe upper bound on the n% confidence limit. For large NPAR this will be way above the linear approximation result.


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