Author Topic: separation reliability chi-squared value  (Read 578 times)

shentong

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separation reliability chi-squared value
« on: October 20, 2016, 01:42:50 AM »
Hi folks!

In the chapter 12 of the Conquest manual, under the Separation Reliability section, it is said that a chi-square test could be used to test the item equality, and the formula for calculating the chi-squared value is  also shown on that page (please see the attachment for the calculation formula). My question is: Is there a proof that the X value follows a chi-square distribution? By definition, the  chi-square distribution is the distribution of the sum of squared standardized random variables. But in this case, the chi-squared value X is the sum of squared item difficulty parameter divided by the standard error, so where is the "standardized"part in the formula?

Thanks in advance for your help!

Tong
Doctoral Research Assistant at Boston College

Eveline Gebhardt

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Re: separation reliability chi-squared value
« Reply #1 on: October 23, 2016, 11:13:53 PM »
Hi Tong

The null hypothesis is that all parameters are equal, i.e. they must be zero.  So est/se is distributed N(0,1)  then (est/se)^2 is chi-square with df=1 and sum of them is chi-square with df=NI-1.

Best wishes
Eveline

shentong

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Re: separation reliability chi-squared value
« Reply #2 on: October 30, 2016, 08:05:05 PM »
Hi Eveline,

Thank you very much for the explanation. However, on the Conquest Version 2.0 manual that I have, the formula for calculating the Chi-square value X is the sum of   (est^2/se) rather than (est/se)^2  (Wu, Adams, Wilson, & Haldane,2007, p160). I guess that is why it confuses me. So is  (est^2/se) a typo?

Second question, could you explain more clearly why (est/se) is a standardized statistic? I understand that (x-mean x)/se = z score and z^2 is chi-square, but what is the proof that (est/se) itself is  standardized?

Thanks!

Tong

Eveline Gebhardt

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Re: separation reliability chi-squared value
« Reply #3 on: November 03, 2016, 12:34:35 AM »
Hi Tong

Yes, sorry, that is a typo indeed.

Regarding the est/SE, Wald statistics are asymptotically normal. It's the standard large sample stats theory.

Cheers
Eveline

shentong

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Re: separation reliability chi-squared value
« Reply #4 on: November 13, 2016, 09:40:17 PM »
Thanks for your clarification, Eveline! Very appreciated.