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Calculate Confidence Interval

These functions allow calculating the confidence intervals of s or M of the first component, respectively, of the discrete species model (both functions work very similar).  If the other components are of interest, they should be exchanged and brought into the position of the first species.)

The calculation performed by Sedfit works as follows: Starting from the best fit value, the parameter whose error limit is sought is moved to a non-optimal value and kept fix. Then, all other parameters are floated and allowed to compensate this constraint. The resulting variance of the fit is observed, and the parameter in question is moved a little bit further, again followed by constrained fit, etc. Goal is to bring the variance of the constrained fit close to the value that is predicted by F-statistics for the given confidence limit. The parameter value that corresponds to this variance increase constitutes one error limit. Once one error limit is established, a similar series of constrained fits is performed, moving the parameter in question into the opposite direction. This will give the other error limit. These errors are not necessarily symmetric (!) and take correlation, positivity and other constraints of the parameters into account.

Both functions follow these steps:

1) First, it is necessary for realistic results that before calculating the confidence interval the best-fit has been found. This should be verified by a couple times invoking the Fit function and ensuring that the parameter don’t change anymore.  

2) a confidence level has to be specified (typically 0.68 or 0.95)

3) a test-interval needs to be specified, which should ideally be 1/4 of expected standard deviation of the parameter, or smaller

4) a message box reminds to make sure to start from best-fit situation, and that all unknown parameter should be floated

5) a series of multiple fitting operations will start, one can follow the test parameters in the display (first moving up, then going down).  This step can take a while.  The procedure should not be interrupted, otherwise artificially low error estimates will be produced.  Occasionally, it can be observed that during the series of fits a rms error lower than the initial 'best-fit' is found.  In this case, the parameter values should be noted, the procedure should be aborted, and restarted with the new best-fit values.  This can happen, in particular, with fits that include many floating parameters that can get caught in a combination that represents a local minimum (as opposed to a global minimum).  

6) at the end, a message box appears with the result: best-fit value (lower limit, upper limit). Also displayed is the rmsd of the fit that was calculated to be the statistical cutoff value.

See also: F-statistics, error contour maps