From Homeschooling in Full View: A Reader (Cooper, 2005) p. 4 (internal citations omitted):
Studies done by dozens of researchers during the past 20 years examined the academic achievement of the home-educated. [...] the homeschooled have scored, on average, at the 65th to 80th percentile on standardized academic tests in the United States and Canada, compared to the public school average of the 50th percentile.
This references the work of Brian Ray, which the question already mentions, but also Home Education in Canada: A Summary of the Pan-Canadian Study on Home Education 2003 (Van Pelt/CCHE, 2003).
Over 94% of home-educated students scored
above the Canadian norm for both grade equivalency and
The average home-educated Canadian student in grades
one through eight ranks in the 81st percentile in reading, 76th
percentile in language and 74th percentile in mathematics. The
mean (average) percentile ranks for home-educated students
in grades nine through 12 were, in reading 85th, in language
84th and in mathematics 67th.
These results are correlational, not interventional. (CCHE, 2003), for example, simply distributed packages to 5,800 home-educating families across Canada and invited them to participate in a 16-page survey and self-administer Canadian Achievement Tests.
In The Correlational Relationship Between Homeschooling Demographics and High Test Scores (Burns, 1999), Burns notes that the "vast majority" of homeschooled students have married, biological parents with higher than average incomes and educations, and "watch much less television than their publicly schooled counterparts". These characteristics have each been shown to influence academic achievement.
I could only find one study that attempted to control for any of these other factors. Home schooling: The ameliorator of negative influences on learning (Ray, 2000) found that "children in homeschool families with low income and in which the parents have little education are scoring, on average, above state-school averages" (quote from Homeschoolers on to College (Ray, 2004)). Ray is president of the National Home Education Research Institute.
In my opinion, that there is a difference seems established enough, given the independent results by Ray and Van Pelt, even though they are both associated with home education centres/institutes. The research could be be more convincing and produce a more accurate estimate of the magnitude of the effect if it used more representative sampling methods and removed self-reporting. Why there is a difference has not been well studied.