Category mean-square fit statistics

Computation of Category Mean-Square fit statistics:

 

For all observations in the data:

Xni is the observed value

Eni is the expected value of Xni

Wni is the model variance of the observation around its expectation

Pnik is the probability of observing Xni=k

 

Category Outfit Mean-Square statistic for category k =

[sum ((k-Eni)²/Wni) for all Xni=k] / [sum (Pnik * (k-Eni)²/Wni) for all Xni]

 

Category Infit statistic for category k =

[sum ((k-Eni)²) for all Xni=k] / [sum (Pnik * (k-Eni)²) for all Xni]

 

The expected values of the category mean-squares are 1.0.

 

For dichotomies, with categories 0 and 1:

 

For category 0: k = 0, Eni = Pni = 1 - Pni

 

Category Outfit Mean-Square statistic for category 0

= [sum (Pni/(1-Pni)) for all Xni=0] / [sum (Pni) for all Xni]

 

Category Infit statistic for category 0

= [sum (Pni²) for all Xni=0] / [sum ((1-Pni)*Pni²) for all Xni]

 

For category 1: k = 1

 

Category Outfit Mean-Square statistic for category 0

= [sum ((1-Pni)/Pni) for all Xni=0] / [sum (1-Pni) for all Xni]

 

Category Infit statistic for category 0

= [sum ((1-Pni)²) for all Xni=1] / [sum (Pni*(1-Pni)²) for all Xni]