[Rasch] Sample free

Trevor Bond trevor.bond at jcu.edu.au
Wed Jul 23 17:36:46 EST 2008

Dear Ning,
you asked:
Hi Dr. Bond, 

I have been thinking about the same question. Since person and item difficulty 
parameters are estimated simultaneously as the two facets in a Rasch model 
that the probability of the item endorsement is conditioned on, how does the 
model achieve the "calculation of item difficulty estimates that are independent 
of the distribution of abilities in the particular group of persons for whom the 
items are appropriate?" Further, most empirical data tend to show misfit to the 
Rasch model to some extent, does that mean "item-distribution-free" and 
"person-distribution-free" are not achieved when one or more items' misfit is 
statistically significant? 

Thanks in advance, 

Part 1 (I'll leave the fit questions for others)

Parameter Separation (B&F 2 pp279-280)
Parameter separation implies that one set of parameters (e.g., the items) can be
estimated without knowing the values for the other set (e.g., the persons). This 
is taken advantage of in the estimation procedure known as Conditional 
Maximum Likelihood Estimation. To demonstrate the calculation of the 
relationship between
the abilities of two persons (i.e., Bn and Bm) independent of the actual difficulty
value for an item (Di):
Bn – Di =~ log (Fni/Fin) where Fni is the count of successes by Person n on Item
i; and
Bm =~ Di =~log(Fmi/Fim)
Bn-Bm =~ log(Fni/Fin) − log(Fmi/Fim) (4)
So that the relationship between Bn and Bm can be estimated without 
of the difficulty of Item i.
The ability of the Rasch model to compare persons and items directly means
that we have created person free measures and item free calibrations, as we 
come to expect in the physical sciences, abstract measures that transcend 
persons’ responses to specific items at a specific time. This characteristic, 
unique to
the Rasch model, is called parameter separation. Thus, Rasch measures 
represent a
person’s ability as independent of the specific test items, and item difficulty as
independent of specific samples within standard error estimates. Parameter 
holds for the entire family of Rasch models.

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