I looked for a ball screw calculator/formula, but I wasn't able to find one... so we can do a little logical assumption to help you come up with an answer. The stanards book I have says that the root diamater of a 16mm ball screw is about 13.4mm. So, if you do a calculation for the deflection of a 16mm steel shaft and for a 13.4mm steel shaft, the deflection of your screw will fall somewhere inbetween these two numbers.
To play on the safe side, you could just skip doing the 16mm shaft and only do the calculations for the 13.4mm shaft.
Sag = L^4 / S
L = length
S = Sag Coefficient
S = E x I x 384 / (5 x sw)
E = Modulus of Elasticity
sw = PI x diam^2 / (4 x density)
If you want to save yourself some math... pacific bearing's catalog has a lot of the math done for you and it's in chart format. So...
Sag = L^4 / S (getting S from a table)
So.. here are the entries for a 14mm shaft:
S = 2.45 x 10^12 for hardened steel
S = 2.31 x 10^12 for stianless
you pick your material
Don't forget to use millimeters for length.