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NAEP Analysis and Scaling → Estimation of Population and Student Group Distributions → Examining the Population-Structure Models Used in NAEP → Proportion of Variance Accounted For by Principal Components Used in NAEP Population-Structure Models

## Proportion of Variance Accounted For by Principal Components Used in NAEP Population-Structure Models

The population-structure models employed for specific national main, state, and combined national and state assessment samples did not directly use the group variable specifications. As in other statistical analyses where there are a large number of correlated variables, a principal component transformation of the correlation matrix obtained from the variable contrasts derived according to these specifications was performed. The principal components, rather than the original variable contrasts, are used in the analyses so that the estimation procedures are computationally stable. For computational stability and due to computational limitations, a large number, but not all, of the principal components based on this transformation were used as the variables in estimating the population-structure models. For national main assessments, the proportions of variance of the variable contrasts accounted for by the principal components are given for each grade level.

For tables linked to this page starting with the 2002 assessment year, the following information is provided for each type of contrast:

• the number of contrasts for each type of contrast,
• value for the mean proportion of variance explained,
• value for the minimum proportion of variance explained,
• value for the maximum proportion of variance explained, and
• the number of contrasts by proportion of variance explained.

The proportion of variance explained in each table indicates how closely the principal components reflect the variables used to define the groups. If the proportion of variance of a group-defining variable contrast accounted for by the principal components is one, all of the variability of that contrast was taken into account in the population-structure models. If all of the principal components were used in the models, all of the proportions would be one. The number of principal components was selected so that at least 90 percent of the overall variance of the group-defining variable contrasts was accounted for by the principal components included in the population-structure models. This results in proportions that are less than one. The values are provided because results for student groups for which the proportions are high are well described by the population-structure models.

Links to tables of the proportion of variance for each subject area assessment's population-structure models, by year and subject area: Various years, 2000–2012
Arts 2008 R3
Civics 2010 R3 R3 R3
2006 R3 R3 R3
Economics 2012 R3
2006 R3
Geography 2010 R3 R3 R3
2001 R2/R3 R2/R3 R2/R3
Mathematics 2011 R3 R3
2009 R3 R3 R3
2007 R3 R3
2005 R3 R3 R3
2003 R3 R3
2000 R2/R3 R2/R3 R2/R3
2009 R3 R3 R3
2007 R3 R3
2005 R3 R3 R3
2003 R3 R3
2002 R3 R3 R3
2000 R2/R3
2009 R3 R3 R3
Science 2011 R3
2009 R3 R3 R3
2005 R3 R3 R3
2000 R2/R3 R2/R3 R2/R3
U.S. history 2010 R3 R3 R3
2006 R3 R3 R3
2001 R2/R3 R2/R3 R2/R3
Writing 2011 R3 R3
2007 R3 R3
2002 R3 R3 R3
Writing 2002 national main assessment R3 R3 R3
† Not applicable; subject was not assessed at this grade in this year.
NOTE: R2 is the non-accommodated reporting sample; R3 is the accommodated reporting sample. It samples students who are classified students with disabilities (SD) or English language learners (ELL), plus SD/ELL students from sessions in which accommodated were allowed. The R3 sample is more inclusive and excludes a smaller proportion of sampled students. The R3 sample type was the only sample type used in NAEP after 2001.
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, National Assessment of Educational Progress (NAEP), various years, 2000-2012.

Links to tables of the proportion of variance for each subject area assessment's population-structure models, long-term trend assessments, by year and subject area: 2004, 2008, and 2012
Subject area Year Age 9 Age 13 Age 17
Mathematics 2012 R3 R3 R3