          About the chi-square statistic.

The chi-square statistic is

chi^2 = (sum)_i [ [ N(i,S) - B(i,x,pB) - S(i,x,pS) ]^2 / sigma(i)^2 ]

where N(i,S) is the total number of observed counts in bin i of the on-source region; B(i,x,pB) is the number of predicted background model counts in bin i of the on-source region (zero for background-subtracted data), rescaled from bin i of the off-source region, and computed as a function of the model argument x(i) (e.g., energy or time) and set of background model parameter values pB; S(i,x,pS) is the number of predicted source model counts in bin i, as a function of the model argument x(i) and set of source model parameter values pS; and sigma(i) is the error in bin i.

There are several methods for assigning sigma(i):

leastsq
chi2constvar
chi2datavar
chi2gehrels
chi2modvar
chi2xspecvar

In each of the implementations, N(i,B) is the total number of observed counts in bin i of the off-source region; A(B) is the off-source "area", which could be the size of the region from which the background is extracted, or the length of a background time segment, or a product of the two, etc.; and A(S) is the on-source "area".

In the analysis of PHA data, A(B) is the product of the BACKSCAL and EXPTIME FITS header keyword values, provided in the file containing the background data. A(S) is computed similarly, from keyword values in the source data file.

Note that it is currently assumed that there is a one-to-one mapping between a given background region bin and a given source region bin. For instance, in the analysis of PHA data, it is assumed that the input background counts spectrum is binned in exactly the same way as the input source counts spectrum, and any filter applied to the source spectrum automatically applied to the background spectrum. This means that the user cannot, for example, specify arbitrary background and source regions in two dimensions and get correct results.

However, this limitation only applies when analyzing background data that have been entered with the load_bkg command

("ahelp load_bkg"). One can always enter the background as a separate dataset and jointly fit the source and background regions.

See the bugs pages on the Sherpa website for an up-to-date listing of known bugs.

December 2013