I am comparing the results of the NR Lev-Mrq fitter against those of other regression packages (GraFit, DataFit, Excel, others). All fitting applications quote the error on a fit param as the "standard error," but I am confused about the correct definition and derivation of this quantity...

Confusing Observations:

(1) Good News: Given any model function, if I set all the weights sig[i] to one on entry to mrqmin, iterate until converged, then compute sqrt( covar[k][k] ) * sqrt( chisq / (ndata - ma) ), the resulting value exactly matches what other applications quote as the standard error for parameter k.

(2) Bad News: The discussions in NR and in Bevington seem to imply that the factor sqrt( chisq / (ndata - ma) ) is only appropriate for the sig[i]=1 case. Rather, if unequal sig[i] are used throughout, the final sqrt( covar[k][k] ), alone, should be quoted. I must be missing something here because my sqrt( covar[k][k] ) never agrees with the other guy's standard error value...unless I multiply mine by sqrt( chisq / (ndata - ma) ). Again, that "works" but seems entirely contrary to the derivations and advice of NR.

Q1: What is the formal relationship between standard error on a fit parameter a[k] and the standard deviation?

Q2: Is there a reference that further explains whether and why the covariance matrix needs to be multiplied by reduced chisq?

Many Thanks,
billKarsh