There's no particular rule of thumb for sizing stair stringers/
carriages/housings that I'm aware of. However you presumably have a
current housed stair to use for comparison purposes - so sizing the
narrowest part of a cut-carriage to equal the depth of the housing
should work. Otherwise you could use the floor joist tables from the
Building Regs to give you a ballpark figure for narrowest point on a
cut-carriage.

Are you planning open risers and is that your motivation for changing
the design? Be aware Building Regulations has quite a bit to say on
stair design.

I'd recommend either one of the general joinery textbooks that cover
stairs (e.g. Porter & Rose) or "Basic Stairbuilding" - Scott Schuttner

http://www.amazon.co.uk/Carpentry-Jo...3562894&sr=1-1

http://www.amazon.co.uk/Carpentry-Jo...3562894&sr=1-2

http://www.amazon.co.uk/Basic-Stairb...3562969&sr=1-3