  We provide a general preservation theorem for preserving selective
independent families along countable support iterations. The theorem gives a
general framework for a number of results in the literature concerning models
in which the independence number $\mathfrak{i}$ is strictly below
$\mathfrak{c}$, including iterations of Sacks forcing, Miller partition
forcing, $h$-perfect tree forcings, coding with perfect trees. Moreover,
applying the theorem, we show that $\mathfrak{i} = \aleph_1$ in the Miller Lite
model. An important aspect of the preservation theorem is the notion of "Cohen
preservation", which we discuss in detail.
