@Mafoo
That very study recognizes the largest issue when comparing public versus private schooling: parental involvement.
As a public school teacher myself, I, along with many other colleagues, would tell you that the highest correlation in regards to student achievement and external factors is undoubtedly parental involvement.
I don't think anyone here would doubt that there would likely be a significantly larger amount of parental involvement both at school and at home in the private sector. The family types are simply different. Similarly, in Canada, we have two language streams: English and French Immersion (late and early). The French classes nearly in every school outperform the English classes. Is it because of the teachers themselves? Do French teachers have some inexplicably grand teaching technique that they're holding out on us? A much more likely explanation is that of family involvement and background. And studies in pedagogy support this nearly unanimously.
From your very own source:
"NAEP data are collected as part of an observational
study rather than as a randomized experiment. In
particular, families choose to enroll their children in
private schools, and it is possible that there are systematic
differences between those families and the general
population of families that are not captured by the
student characteristics available for analysis. If such
differences are correlated with student achievement,
then the estimated average difference in achievement
between public school students and private school
students (even after adjusting for observed student
characteristics) will be confounded to some degree with
the unobserved differences between the families of the
children in the two school types. This is usually termed
“selection bias.” Although this study employs a powerful
statistical tool, hierarchical linear models, it cannot
fully compensate for the lack of relevant data. The
implication is that the estimated effects obtained should
not be interpreted in terms of causal relationships."
Edit: Kasz, I posted this before your latest post. No longer directed toward you.