Re: Chi_sqr

["John S. Denker" <jsd@MONMOUTH.COM>]

At 05:48 PM 7/12/00 -0400, Ludwik Kowalski wrote:

> chi_sqr=sum(  (o(i)-e(i))^2 / o(e) )   <-- (i from 1 to n)   [eq 1]

That is not the definition of chisquare.  Overall it appears to be a
misstatement of a special case of the chisquare formula.  The subexpression
o(e) has not been defined and is probably a typo.  Using e(i) would make
more sense;  see below.

> ... errors of measurements of observed quantities are
> NOT specified. In other words we are making a conclusion
> on the basis of an analysis which does not depend on sizes of
> errors. ... It seems to me that such analysis is probably
> not valid in many situations. Any comments?

A chisquare analysis that doesn't depend on the measurement errors is a
contradiction in terms.  Such a thing would be totally invalid, in all
situations.

The correct definition of chisquare is
  chi_sqr=sum( (o(i)-e(i))^2 / (sigma(i))^2 )    [eq 2]
which manifestly depends on the measurement error sigma(i).

> There must be some hidden assumptions in this analysis.
> What are they?

The formula
  chi_sqr=sum(  (o(i)-e(i))^2 / e(i) )     [eq 3]
is a special case of equation [2] for the case of a Poisson process,
further specialized to the subcase where Poisson "root N" counting noise
dominates all other sources of error.