A noted psychoanalyst claims that the causes of mental illness are mainly environmental – extreme poverty, lack of adequate parenting, and sexual abuse –
and chemical imbalances rarely.The phrase included in between the dashes is a modifier phrase that describes the 'causes'. Hence, if you remove this phrase completely, you'll still get a valid sentence. In fact, we need to remove this phrase to visualize what's going on in the passage.
So, after removing this phrase, the sentence effectively becomes:
A noted psychoanalyst claims that the causes of mental illness are mainly environmental and chemical imbalances rarely.
Let's refer to this new version (of the sentence) as
V1. we will now analyze our available options using V1 as our reference.
A)
and chemical imbalances rarely - By analyzing V1, we realize that what follows 'and' is not an independent clause. Hence,
eliminate (A)B)
and chemical imbalance being rare - Same error as in (A). Hence,
eliminate (B)C)
with chemical imbalance as rare - here, chemical imbalances is being equated to 'rare' as though both are analogous to each other. This, however, is incorrect as 'rare' is meant to behave as an adjective that describes the 'chemical imbalance'. Hence,
eliminate (C). D)
chemical imbalances a rarity - here, the preposition 'with' is missing. Without the preposition 'with', we cannot clearly make out what this underlined portion is trying to modify (or whether this underlined portion is a modifier in the first place). There's a lot of room for doubt here. Hence,
eliminate (D)E)
with chemical imbalance as a rarity - here, the preposition phrase 'with... as a rarity' is clearly modifying the noun '' the causes of mental illness''. Hence,
(E) is the right answer choice.