Understanding Confidence Intervals for Fit Parameters

[BillKet]

Hello! Can someone help me understand how are confidence intervals for some parameters of a fit different from the errors on the parameters obtained, for example, from the error matrix. I read Bevington and the whole book he mentions that we can use the error from the error matrix to define the confidence interval (e.g. $68.3\%$ confidence interval for 1 $\sigma$ of a parameter), then in the last chapter he says that, this is not generally correct and we should use confidence intervals which automatically take into account the correlation between parameters. I understand his argument and it makes sense to do that, but now I am not sure I understand what is the error matrix useful for anymore, if the estimates from the error matrix don't take into account the correlations among the parameters? I guess they are useful when the correlations are zero, but does that happen often? Thank you!

 

[tnich]

I assume that you are referring to Data Reduction and Error Analysis for the Physical Sciences, by Bevington and Robinson. It seems to me that equation 3.13 and the discussion around it make clear that covariance terms are important. Also, on page 125 the authors state, "The error matrix can be used to estimate the uncertainty in a calculated result, including the effects of the correlations of the errors." Could you give page numbers and brief quotes of the later contradiction? I am looking at the third edition of the book.

 

[BillKet]

I assume that you are referring to Data Reduction and Error Analysis for the Physical Sciences, by Bevington and Robinson. It seems to me that equation 3.13 and the discussion around it make clear that covariance terms are important. Also, on page 125 the authors state, "The error matrix can be used to estimate the uncertainty in a calculated result, including the effects of the correlations of the errors." Could you give page numbers and brief quotes of the later contradiction? I am looking at the third edition of the book.

Thank you for your reply! Actually I finally understand why using the error matrix is bad. However I am still a bit confused about how Bevington calculates the confidence intervals. He uses a chi squared distribution and for example he uses the fact that for one parameter an increase on 1 for the chi squared is equivalent to a 68% confidence level. However in other online resources (for example: https://lmfit.github.io/lmfit-py/confidence.html) they use an F distribution to define the confidence interval. But the 2 methods don't seem equivalent, as in Bevington the value of number of data points (N in the link I provided) doesn't come in the formula. So which one is the right formula?

 

[tnich]

Why is it bad to use the error matrix?
Can you tell me what page of Bevington you are looking at?