We estimated and reported the adjusted R 2 of each model, which is the portion of the average achievement scores' variance explained by the model.
The standard model explained as much as 77% of the achievement scores' variance in Chinese Taipei (PIRLS), 67% in Korea (TIMSS grade 8) and 66% again in Chinese Taipei (TIMSS grade 4).
For teacher-rated achievement, the total variance of 12-year achievement that is independent of 10-year achievement is.58 (i.e.29+.12+.17 = .58), so the re-standardized heritability of 12-year achievement independent of 10-year achievement is 50% (.29/.58 = .50).50
As a consequence, the variance in achievement between classroom within schools and the between school variance in achievement are confounded, so the proportion of variability between schools is larger than is observed when classroom-to-classroom differences are separated from between school differences.
Rather than removing the g covariance from the variance of achievement scores prior to analysis, multivariate genetic analysis considers all of the variance of achievement scores as well as all of the variance of g, and it decomposes the covariance between them into genetic and environmental components of covariance [7], [40].
Simulation results show its achievement of a low variance close to the Cramer Rao bounds, and that essentially unbiased parameter estimates are obtained.
Table 7 R2 (explained variance of achievement score) by country (PIRLS Countryy Participation rate of schools (before replacement) Standard model Country-specific model R2 (adj).
Likewise, having students decide on their own procedures for solving complex problems did not explain any additional variance in achievement in the U.S., and having students apply facts, concepts and procedures to solve routine problems did not explain any additional variance in achievement in Singapore.
The variance in achievement within schools and between schools (the ICC) varied across the four countries; however, the U.S. is most similar to Singapore with respect to how the variance in mathematics scores is distributed.
In Singapore however, the student and school covariates explained approximately 62% of the total variance in achievement – 22.5% of the 26.1% available within schools, and 75.4% of the 73.9% between schools.
