FWIW, it's likely that no number of studies would change your friends' minds. To understand homeschooling requires a paradigm shift, the ability to imagine childhood without school, and not everyone is in a place to be able to do that.
They may come around in a few years when they observe your dd and how you homeschool, or they may never get it. You don't need their approval (even though we all like approval, when we can get it ).
Funny how fast it can happen though. I had a couple of interesting conversations years ago when visiting Germany and then Switzerland, after telling some acquaintances and an old friend with teens the age of ours at the time that we homeschooled. The first reaction in Germany was shocked disapproval. And then the questions began - "What about the mathematics?," "socialization," etc. It was amazing how soon the expressions of "getting it" came across their faces - within minutes, they were saying things in earnest like "This is a wonderful thing you're doing for your son." Our friend in Switzerland didn't say much till the next day, but first thing in the morning he said he'd been thinking about it a lot during the night, and that it made sense - he said there was no advantage to having to go through bullying and all the rest in a school setting. He commented, "I look at your son and I see that he's a nice young man who carries himself well and has confidence," and that it had obviously served him well. He was clearly moved by the idea that education can work well outside the traditional system. So you never know - when you have confidence in what you're doing, and when it's obvious that your child is thriving, I think that can often carry more weight than any research you can dig up. - Lillian