I'll explain by way of example.
Assume that at a particular school, for a particular course, all of the students are placed in the same performance band (e.g. Band 6).
= raw internal assessment marks
= initial moderated assessment marks
= aligned assessment marks
Their raw assessment marks are moderated via the non-linear transformation:
with
,
and
determined from the distribution of raw examination marks for this moderating group (which is not known).
The initial moderated assessments are then aligned via a linear transformation:
with
and
determined from the raw cut-offs set by the Board.
(The same
and
are also used to align the raw examination marks, for this and all groups.)
Now, substituting for y:
 + e)
The above relationship holds for any individual
)
pair. So, for all
and
:
As
and
are known (by the school), we can create a system of linear equations in three unknowns (
,
and
).
In matrix notation:
where the columns of the first matrix are simply the coefficients of the linear equations above for three particular
.
The fact that
,
and
can be calculated directly but not the individual variables reflects the range of possible distributions of raw examination marks for this group.
In other words, for every distribution of moderated marks produced by applying particular
,
and
to the
, there exist
and
such that the
calculated from the
are the same as the real aligned assessment marks.
This only means that one moderating group isn't enough to determine the cut-offs.
But I think that if you had
pairs from, say, two or three schools, it would be possible to deduce
and
, and thus the cut-offs. In that situation you can hold
and
constant between the schools.
This is where I'm currently up to anyway.
