Under broad conditions, two analytic self-maps of the disk fixing 0 commute under composition precisely when they have the same Schroeder map, where the Schroeder map for an analytic $\ph:D\rightarrow D$ with $\ph(0)=0$ is the unique analytic function $\sigma$ on $D$ solving Schroeder's equation $\sigma\circ\ph=\ph'(0)\sigma$ and satisfying $\sigma'(0)=1$. For analytic self-maps of the ball in $C^N$ fixing 0 we may still seek analytic $C^N-$valued solutions $\sigma$ to Schroeder's equation with $\sigma'(0)=I$, but considerable complications for existence and uniqueness of $\sigma$ may ensue. Nevertheless, we show that there are reasonably general hypotheses under which it will still be the case that two analytic self-maps of the ball fixing 0 commute if and only if they share a common Schroeder map $\sigma$ with $\sigma'(0)=I$.