This question has come up frequently over the past few years. Although some educators such as Ramsey Musallam and Jerry Overmyer have conducted inspirational studies on related topics, and Jon Bergmann and Aaron Sams released their groundbreaking book, flipped learning research and evidence has been for the most part anecdotal.
I am glad to see that academic studies and further research on flipped learning is in the works. One example is the Flipped Learning Network's recent release of the following three documents:
A Review of Flipped Learning
The Executive Summary of the Literature Review
* Source: http://flippedlearning.org/domain/41
These reports were written by Noora Hamdan and Patrick McKnight (Ph.D. George Mason University), Katherine McKnight (Ph.D. Pearson’s Center for Educator Effectiveness) and Kari M. Arfstrom, (Ph.D. Flipped Learning Network)
I was honored to have been a part of the FLN research committee that provided peer review for these studies. I am also humbled that findings from the research that I conducted at Columbia University ("Flipped Learning & Democratic Education") were included in the literature review (p. 12 & 13) and white paper (p. 10).
I hope you find these studies useful in your journey to improve student learning!