Finding Uncertainty in a coefficient with a Chi Squared Test

[neutrino45]

Hello,

I have done a chi squared test on the measurements from a neutron flux experiment to get the best parameters for a function of the form Ysim=Acos(B*X). I used Solver in Excel to find the minimum parameters. The test takes the form

chi^2 / dof = SUM(Ysim-Yi)^2/(sigma i)^2

Where Yi are the measured values of the flux at various heights and (sigma i) is the uncertainty in flux i.

What I want to do is to find the uncertainty in the parameter B. I have been told that if I shift the parameter B until the minimum value of chi^2 is altered to get chi^2 + 1 then the difference between the original value for B and the new value for B can be used to get the uncertainty in B.

Does this make any kind of mathematical sense? I've found hints that this is equivalent to garbageing chi^2 by one standard deviation but I have not found any hard evidence of this. Has anyone seen this method referenced anywhere?

 

[SW VandeCarr]

Does this make any kind of mathematical sense? I've found hints that this is equivalent to garbageing chi^2 by one standard deviation but I have not found any hard evidence of this. Has anyone seen this method referenced anywhere?

The $$\chi^{2}$$ distribution has only one parameter: k, which is the number of degrees of freedom. The mean is simply k, the variance is 2k. The shifting of the parameter by one would correspond to adding or subtracting one degree of freedom.

I can't speak to your application but the standard deviation "sd" (as a measure of uncertainty) is calculated from the sampling distribution and employed in the calculation of the chi square statistic:

$$\chi^{2}= [n-1]sd^{2}]/\sigma^{2}$$ where $$\sigma^{2}$$ is the population variance, n is the sample size. Generally the population variance is not known and the estimate from the sampling distribution is used. So this reduces to $$\chi^{2}=(O-E)^{2}/{E}$$ for each degree of freedom with O as the observed value and E as the expected value.