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Comments on: chi-square distribution [Eqn] http://hea-www.harvard.edu/AstroStat/slog/2008/eotw-chisq-distribution/ Weaving together Astronomy+Statistics+Computer Science+Engineering+Intrumentation, far beyond the growing borders Fri, 01 Jun 2012 18:47:52 +0000 hourly 1 http://wordpress.org/?v=3.4 By: Alex http://hea-www.harvard.edu/AstroStat/slog/2008/eotw-chisq-distribution/comment-page-1/#comment-308 Alex Sun, 27 Jul 2008 01:52:08 +0000 http://hea-www.harvard.edu/AstroStat/slog/?p=342#comment-308 It should also be noted that chi-square error bars and other second-degree statistics (ie, anything relating to squared quantities) are considerably less robust to non-normality than first-order statistics (ie, error bars on means). For example, while normal approximations for the distribution of the mean are usually quite good for n>30 (and certainly for n>100), chi-square and F statistics are often not distributed anywhere close to their nominal distributions for such sample sizes if the data is non-normal. It should also be noted that chi-square error bars and other second-degree statistics (ie, anything relating to squared quantities) are considerably less robust to non-normality than first-order statistics (ie, error bars on means). For example, while normal approximations for the distribution of the mean are usually quite good for n>30 (and certainly for n>100), chi-square and F statistics are often not distributed anywhere close to their nominal distributions for such sample sizes if the data is non-normal.

]]> By: vlk http://hea-www.harvard.edu/AstroStat/slog/2008/eotw-chisq-distribution/comment-page-1/#comment-307 vlk Mon, 21 Jul 2008 03:28:26 +0000 http://hea-www.harvard.edu/AstroStat/slog/?p=342#comment-307 Thanks, Aneta, very useful comment. I would only add that we can of course minimize that statistic Sum{(D-M)^2/var)} to get best-fits (<a href="http://hea-www.harvard.edu/AstroStat/slog/2007/an-example-of-chi2-bias-in-fitting-the-x-ray-spectra/" rel="nofollow">modulo biases</a>) regardless of whether D are normally distributed, but to then follow on and relate the change in the statistic to error bars on the fitted parameters does require that the statistic be chisq distributed, with all the attendant baggage. Thanks, Aneta, very useful comment. I would only add that we can of course minimize that statistic Sum{(D-M)^2/var)} to get best-fits (modulo biases) regardless of whether D are normally distributed, but to then follow on and relate the change in the statistic to error bars on the fitted parameters does require that the statistic be chisq distributed, with all the attendant baggage.

]]> By: aneta http://hea-www.harvard.edu/AstroStat/slog/2008/eotw-chisq-distribution/comment-page-1/#comment-306 aneta Sun, 20 Jul 2008 22:52:40 +0000 http://hea-www.harvard.edu/AstroStat/slog/?p=342#comment-306 of course chi2 equations needs the power of 2, so Sum(D(i)-M(i))^2 of course chi2 equations needs the power of 2, so

Sum(D(i)-M(i))^2

]]> By: aneta http://hea-www.harvard.edu/AstroStat/slog/2008/eotw-chisq-distribution/comment-page-1/#comment-303 aneta Sun, 20 Jul 2008 22:16:23 +0000 http://hea-www.harvard.edu/AstroStat/slog/?p=342#comment-303 I guess in the typical analysis we call chi2 a "random variable" that follows the chi2 distribution: chi2= Sum (D(i)-M(i))/var(i) where D(i) is the observed data, M(i) is the model predicted data and we "silently" assume that D(i) is normally distributed and each i measurement is independent. We minimize this random variable when searching for the best model parameters that fit the data, but we rarely think about probabilities. However, the assumptions are not valid for many X-ray observation, as the number of the observed counts follows the Poisson distribution. Different weighting (choice of var) in this expression is used to overcome the problem when the collected data has a low number of counts. Properties of the chi2 distribution are well understood and this is why we are still using it in our analysis even in case of low counts number. I guess in the typical analysis we call chi2 a “random variable” that follows the chi2 distribution:

chi2= Sum (D(i)-M(i))/var(i)

where D(i) is the observed data, M(i) is the model predicted data and we “silently” assume that D(i) is normally distributed and each i measurement is independent. We minimize this random variable when searching for the best model parameters that fit the data, but we rarely think about probabilities. However, the assumptions are not valid for many X-ray observation, as the number of the observed counts follows the Poisson distribution. Different weighting (choice of var) in this expression is used to overcome the problem when the collected data has a low number of counts. Properties of the chi2 distribution are well understood and this is why we are still using it in our analysis even in case of low counts number.

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