I actually don't really refer to myself as "independent", partly because most men seem to find the phrase upsetting, and partly because I don't really get the point in announcing that.
In this case I am not referring to independence from God, only to "needing" a husband/provider (though I do value men and I would like to get married). In my early/mid 20's I felt I "needed" a husband, and couldn't be happy without one. Eventually I learned to find my joy and purpose in the Lord. Now I don't feel as if I "need" a husband, but when I mention this (in response to guys stating that women need men), most guys seem to find the notion silly or naive. They refer to studies like the one you mentioned that are supposed to demonstrate to me how incompetent and illogical I must be since I am female. Men who actually know me don't generally say things like that; I'm referring more to casual acquaintances and random men on the internet.
The other time I hear the phrase used is when men in their 30s, 40s, and 50s are explaining that they need to find a wife in her 20s because women over 30 are "too independent". Which is fine, people are entitled to search for a spouse in any demographic they choose, and I know age factors in when you consider childbirth. But the flipside of that is all the women who now HAVE to be independent because most of the men in their age range are still dead-set on finding a cute 20-something.
I realize that plenty of men do not have this mindset; I am merely explaining the context in which I typically hear the phrase used.