The full study is here. Peterson’s looked at gains in the percentage of students at or above basic & proficient levels in reading and math. He compared scores on fifth and eighth grade PSSA scores for 2 cohorts (those that went from 5th to 8th grade from 2002 to 2005 and from 2003 to 2006). Privately managed schools generally showed greater gains (see back page of the study for complete comparisons).
This differs from a RAND study which showed no significant difference between types of schools, which prompted the Philadelphia Inquire to run the headline “Managers fall short in Phila. schools“. (Though the RAND authors have since praised the Philadelphia school leadership, including the private managers, for overall gains in test scores).