Finding Uncertainty in a coefficient with a Chi Squared Test

In summary, the individual is seeking to find the uncertainty in the parameter B for a function in a chi squared test, and has been told that shifting the parameter until the minimum value of chi squared is altered can help determine this. They are unsure if this method is mathematically sound and are looking for evidence of its validity in other sources.
  • #1
neutrino45
2
0
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?
 
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  • #2
neutrino45 said:
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 [tex]\chi^{2}[/tex] 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:

[tex]\chi^{2}= [n-1]sd^{2}]/\sigma^{2}[/tex] where [tex]\sigma^{2}[/tex] 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 [tex]\chi^{2}=(O-E)^{2}/{E}[/tex] for each degree of freedom with O as the observed value and E as the expected value.
 
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1. What is a Chi Squared Test?

A Chi Squared Test is a statistical test used to determine the relationship between two categorical variables. It helps to determine if there is a significant association between the variables or if the observed results are due to chance.

2. How is a Chi Squared Test used to find uncertainty in a coefficient?

A Chi Squared Test can be used to determine the uncertainty in a coefficient by calculating the p-value. The p-value is a probability value that tells us the likelihood of obtaining the observed results if there is no true relationship between the variables. A low p-value indicates that the results are not due to chance and there is a strong association between the variables, while a high p-value suggests that the results are likely due to chance.

3. What is the significance level in a Chi Squared Test?

The significance level, also known as alpha, is the predetermined threshold used to determine if the results of a Chi Squared Test are statistically significant. The most commonly used significance level is 0.05, which means that there is a 5% chance that the results are due to chance. If the p-value is lower than the significance level, we can reject the null hypothesis and conclude that there is a significant relationship between the variables.

4. How do you interpret the results of a Chi Squared Test?

If the p-value is lower than the significance level, we can reject the null hypothesis and conclude that there is a significant relationship between the variables. In this case, the coefficient can be considered significant and we can have a certain level of confidence in the results. On the other hand, if the p-value is higher than the significance level, we fail to reject the null hypothesis and cannot conclude that there is a significant relationship between the variables.

5. Are there any limitations to using a Chi Squared Test to find uncertainty in a coefficient?

Yes, there are some limitations to using a Chi Squared Test. It can only be used for categorical variables and assumes that the observed values are independent of each other. Additionally, it is sensitive to sample size, so a larger sample size may result in a statistically significant relationship even if the effect size is small. It is important to consider these limitations when interpreting the results of a Chi Squared Test.

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