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NAEP Analysis and Scaling → Estimation of NAEP Score Scales → NAEP Scales → Correlations Among NAEP Subject Area Subscales

Correlations Among NAEP Subject Area Subscales

Two types of correlations are computed to assess the mathematical relationship between the final NAEP subject-area subscales. The first type is designated by NAEP as the conditional correlation coefficient and is directly derived from the population model parameters. The second type is designated by NAEP as the marginal correlation coefficient and follows from the imputation model. The marginal correlation coefficient reflects the correlation between the scales that are actually reported. The conditional correlation coefficient is a maximum likelihood point estimate of the correlation between the scales. Because the reported scales are derived from the imputation model and some regression to the mean will occur, the marginal coefficients are generally somewhat lower than the conditional coefficients. Both coefficients provide a valid measure, but serve different purposes.

The conditional correlation coefficient is equal to the covariance between two subscales divided by the product of the standard deviations of each of the subscales. The covariances and standard deviations (i.e., the square root of the residual variance) are taken from the elements of the ∑, which is one of the population model parameters. For more detailed information about these elements, see the Population Model section.

The marginal correlation coefficient is computed as the Pearson Correlation Coefficient between plausible values. For each of the five plausible values, the correlations are computed and averaged using a Fisher z transformation. Specifically, the correlations are transformed and averaged, and an inverse transformation is applied to the average. Fisher’s z transformation is calculated as follows

 Capital Z sub r bar equals one half times the natural log of the ratio of one plus r bar over one minus r bar, and equals the arctangent of h of r bar 

where r bar is the average marginal correlation and transforms the correlation to an approximately normal distribution. This transformation can also be used to compute standard errors.

Marginal correlations are calculated for the nation and for each jurisdiction in state assessments. The state assessment estimates are summarized as the tenth, fiftieth, and ninetieth percentile correlations across jurisdictions in the assessment. Conditional correlation coefficients are reported for the nation only. Values for correlation coefficients can range from -1 to 1. The further this value is from zero, the stronger the relationship is considered to be, with a correlation coefficient of .90 or above demonstrating a very strong relationship.

Links to tables of the conditional correlations and variances of the subscales, marginal correlations of the subscales for jurisdictions, and average marginal correlations and ranges of scale correlations among the subscales for jurisdictions, by subject assessment type and subject: 2000–2012
Subject Assessment Conditional correlations and variances of the subscales Marginal correlations of the subscales New select percentiles for correlations among the subscales for jurisdictions
Grade 4 Grade 8 Grade 12
Arts 2008 national assessment § § § § §
Civics 2010 national assessment § § § § §
2006 national assessment § § § § §
Economics 2012 national assessment R3 R3
2006 national assessment R3 R3
Geography 2010 national assessment R3 R3
2001 national assessment R3 / R2 R3 / R2
Mathematics 2011 combined national and state assessments R3 R3 R3 R3
2009 combined national and state assessments R3 R3 R3 R3 R3
2007 combined national and state assessments R3 R3 R3 R3
2005 combined national and state assessments R3 R3 R3 R3
2003 combined national and state assessments R3 R3 R3 R3
2000 national assessment R3 / R2 R3 / R2 R3 / R2 R3 / R2
Reading 2011 combined national and state assessments R3 R3 R3 R3
2009 combined national and state assessments R3 R3 R3 R3 R3
2007 combined national and state assessments R3 R3 R3 R3
2005 combined national and state assessments R3 R3 R3 R3
2003 combined national and state assessments R3 R3 R3 R3
2002 combined national and state assessments R3 R3 R3 R3
2000 national assessment R3 / R2 R3 / R2
Reading vocabulary 2011 combined national and state assessments
Science 2011 combined national and state assessments § § § § §
2009 combined national and state assessments § § § § §
2005 combined national and state assessments R3 R3 R3 R3
2000 national assessment R3 / R2 R3 / R2 R3 / R2 R3 / R2
U.S. History 2010 national assessment R3 R3
2006 national assessment R3 R3
2001 national assessment R3 / R2 R3 / R2
Writing 2011 combined national and state assessments § § § § §
2007 combined national and state assessments § § § § §
2002 combined national and state assessments § § § § §
Mathematics 2012 long-term trend assessment § § § § §
2008 long-term trend assessment § § § § §
2004 long-term trend assessment § § § § §
Reading 2012 long-term trend assessment § § § § §
2008 long-term trend assessment § § § § §
2004 long-term trend assessment § § § § §
† Not applicable; conditional and marginal correlation tables are not created for the state assessment. The average marginal correlations and ranges of scales (that is, the average of all states participating in the state assessment) is calculated only for the state assessment.
§ Because each of the civics, writing, and long-term trend assessment subjects has only one scale, tables for these subjects were not produced. Starting in 2009, science was scaled univariately (one scale). As a result, tables for science were not produced.
NOTE: R2 is the non-accommodated reporting sample; R3 is the accommodated reporting sample. It samples students who are classified as students with disabilities (SD) or English language learners (ELL), plus SD/ELL students from sessions in which accommodations were allowed. The R3 sample is more inclusive and excludes a smaller proportion of sampled students.
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, National Assessment of Educational Progress (NAEP), 2000–2012 Assessments.

Last updated 24 February 2016 (GF)