Statistical independence of the Boolean function superposition

It is proved here that if a Boolean function $f(x,y)$ is statistically independent on the variables in $x$, then the same is true for any Boolean function $g(f(x,y),z)$, but this may not be so for a superposition $g(f_1(x,y),\dots,f_s(x,y),z)$ where $s\geq2$ and every function $f_1(x,y),\dots,f_s(x,y)$ is statistically independent on $x$.