I'm not sure whether or not it was omitted from this article, but I find it interesting to note that the authors of this paper did not acknowledge another potential association that could explain the supposed links between the type of god one believes in and mental health status.

It is fair to assume that every religion has a somewhat unique and distinctive portrayal of the characteristics of its respective higher power.  Also, within every religion, further subdivisions of sects likely have even more unique and distinctive concepts of their religion's god.  It seems that more often than not, individuals of a certain sect are more likely to marry and procreate with other members of that same sect.  Thus, this creates a situation where not only do members of each religion believe in a similar god, but also share similar genetic patterns.

This points to the confounding possibility that perhaps the characteristic beliefs in a certain god does not lead to the mental health of an individual, but rather the genetic pattern of that religious group is instead what leads to potential psychopathology.