Exactly where is the step where we see that persons and items are independent (the separation of parameters)? Can we see that step, e.g., in Best Test Design, or do you remember if Rasch made this explicit at some point?
The argument is that this shows that the difference between D(i) and D(j) does not depend on B(n). Hence D(i) - D(j) is person independent. But how can we make such conclusion when we know that P(ni) and P(nj) both depend on B(n). How can we know that [P(nj)*(1-P(ni)] / [P(ni)*(1-P(nj)] does not depend on B(n)?
Esk, yes, the particular values of P(nj) etc. depend on the data which depends on B(n). The essential idea behind the Rasch model is that it does not matter which B(n) is chosen, D(i) - D(j) has the same value. The item difficulties do not depend on the choice of n. So there is no interaction between i,j and n. The parameters are separate.
But the data is inferred by B(n). Isn't the argument going in circle? I know that D(i) is person independent (I just cannot see the proof), but if I didn't know, I might complain: If you change n, then you change the data, and then you change D(i). Why isn't this so? Esk
Esk, no, you don't change D(i) - D(j) !!! That was Georg Rasch's revolutionary insight. If we want to measure people in a generalizable way, then we need their relative measurement to be independent of the selection of items we choose, and similarly when we measure items, then we need their relative measurement to be independent of the selection of persons we choose. However, just as when we apply a tape measure to something, we must choose where to locate the zero. The choice of zero does not change D(i) - D(j), though it does change D(i) and D(j). This is the same in physical measurement. We can measure two mountains with zero at sea-level or zero at the center of the earth. The difference in their heights is the same regardless of the choice of zero point.