Suppose $\varphi$ is a real analytic plurisubharmonic exhaustion function on a connected noncompact complex manifold $X$. The main result is that if the real analytic set of points at which $\varphi$ is not strongly $q$-convex is of dimension at most $2q+1$, then almost every sufficiently large sublevel of $\varphi$ is strongly $q$-convex as a complex manifold. For $X$ of dimension $2$, this is a special case of a theorem of Diederich and Ohsawa. A version for $\varphi$ real analytic with corners is also obtained.