Meaning of the Fair Average:

The "fair average" transforms the Rasch measure back into an expected average raw response value. This value is in a standardized environment in which all other elements interacting with this element have a zero measure or the mean measure of all elements in their facet. This is "fair" to all elements in the facet, e.g., this adjusts raw ratings for severe and lenient raters. This enables a "fair" comparison to be made in the raw score metric, in the same way that the measure does on the linear latent variable. Fair-M uses the facet element means as the baseline. Fair-Z uses the facet local origins (zero points) as the baseline. These are set by Fair average=.

 

Calculation of the Fair Average Score

The observed average score is the average rating received by the element. The logit measure is the linear measure implied by the observations. This is adjusted for the measures of whatever other elements of other facets participated in producing the observed data. It is often useful to transform these measures back into the original raw score metric to communicate their substantive meaning. Fair Average does this. It is the observed average adjusted for the measures of the other elements encountered. It is the observed average that would have been received if all the measures of the other elements had been located at the average measure of the elements in each of their facets.

 

A basic many-facet Rasch model for observation Xnmij is:

 

       log ( Pnmijk / Pnmij(k-1)) = Bn - Am - Di - Cj - Fk

 

where

       Bn is the ability of person n, e.g., examinee: Mary,

       Am is the challenge of task m, e.g., an essay: "My day at the zoo".

       Di is the difficulty of item i, e.g., punctuation,

       Cj is the severity of judge j, e.g., the grader: Dr. Smith,

       Fk is the barrier to being observed in category k relative to category k-1,

       where k=0 to t, and F0 0.

 

To compute the fair average for person n (or task m, item i, judge j), set all element parameters except Bn (or Am, Di, Cj) to their mean (or zero) values. Thus, the model underlying a fair rating, when Fair=Mean, is:

 

       log ( Pnmijk / Pnmij(k-1)) = Bn - Amean - Dmean - Cmean - Fk

 

or, when Fair=Zero, it becomes:

 

       log ( Pnmijk / Pnmij(k-1)) = Bn - Fk

 

and the Fair average is sum (k Pnmijk) across categories 0 to t.