Truthfully, I don't know of any study on this, but from my own personal experience, largely from doing a variety of balancing exercises on one leg in martial arts, the position of the leg has some minor impact on what muscles you have to use to hold yourself erect (a leg drawn straight up with the knee bent will have more of your weight centered compared to doing a front or back scale by lifting the leg with the knee straight), but the primary effect is the muscular effort required to hold that leg in place. Even if you are using your hands to hold the raised leg (common in the bent-knee variation I mentioned before), you still have to engage muscles to maintain height. In more extreme movements, such as straight-leg raises, you will likely have to move your torso to counter-weight your leg.
Personally, I just vary the foot position when I do balances so that I practice it in a variety of positions. The three most common ones are the knee raised and clasped to my chest because it gives me more of a stretch, a straight leg but with the raised leg just slightly off the ground to the front of me because it involves the least muscular effort, and raised with the knee pointing sideways so that I can also work on ankle rotations.