How does the cost comparison between public and private schools work when costs associated with special needs students and continuation / reform students that public schools have to handle but private schools do not are excluded from the comparison?
Also, private school students do show a self-selection bias in that the student and/or parents care enough to choose the private school, while public schools have some students attending by default where the student and parents care little about education. This could favor the cost and effectiveness metrics of the private schools.
This is not to say that a given private school cannot be more effective at lower cost than nearby public schools. But it does mean that there are confounding factors when comparing cost and effectiveness.