### Author Topic: separation reliability chi-squared value  (Read 867 times)

#### shentong

• Newbie
• • Posts: 14 ##### 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?

Tong
Doctoral Research Assistant at Boston College

#### Eveline Gebhardt

• Full Member
•     • Posts: 103 ##### 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

• Newbie
• • Posts: 14 ##### 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

• Full Member
•     • Posts: 103 ##### 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

• Newbie
• • Posts: 14 ##### Re: separation reliability chi-squared value
« Reply #4 on: November 13, 2016, 09:40:17 PM »
Thanks for your clarification, Eveline! Very appreciated.