# Compare scores from critics using different scales

Two groups of critics (call them A and B) rate a set of products (labeled $$i=1,\dots,n$$) using different scales.

Assume that product $$i$$ has an innate quality $$Q_i$$ and that critic $$j$$ in group A gives the product the score

$$S^A_{ij} = \alpha^A + \beta^A Q_i + \epsilon^A_{ij}$$

and similarly critic $$j$$ in group B gives the same product the score

$$S^B_{ij} = \alpha^B + \beta^BQ_i + \epsilon^B_{ij}$$

In the general case the number of critics scoring each product can vary, e.g. say that $$n^A_i$$ critics from group A and $$n^B_i$$ critics from group B score product $$i$$.

If we only observe the scores then there is not enough information to determine all of $$\alpha^A, \alpha^B, \beta^A, \beta^B$$ but we can set $$\alpha^A=0$$ and $$\beta^A=1$$ and try to determine values for $$\alpha^B$$ and $$\beta^B$$ which are most consistent with the observed data.

If we make the usual 'nice' assumptions about the errors (e.g. IID normally distributed) then what is the best estimator for $$\alpha^B$$ and $$\beta^B$$?