Splitting families and the Noetherian type of $\beta \omega -\omega$

Extending some results of Malykhin, we prove several independence results about base properties of $\beta \omega -\omega$ and its powers, especially the Noetherian type $Nt(\beta \omega -\omega )$, the least $\kappa$ for which $\beta \omega -\omega$ has a base that is $\kappa$-like with respect to containment. For example, $Nt(\beta \omega -\omega )$ is never less than the splitting number, but can consistently be that ${\omega }_1$, $2^{\omega }$, $(2^{\omega }{)}^{+}$, or strictly between ${\omega }_1$ and $2^{\omega }$. $Nt(\beta \omega -\omega )$ is also consistently less than the additivity of the meager ideal. $Nt(\beta \omega -\omega )$ is closely related to the existence of special kinds of splitting families.