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

Correlations Among NAEP Subject Area Subscales

Two types of correlations are computed to assess the linear 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 subscales. 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. While different, 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' ∑, 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 area and assessment type: Selected years, 2000–2007
Subject area Assessment type 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
Civics 2006 national main assessment § § § §
Economics 2006 national main assessment R3 R3
Geography 2001 national main assessment R3 / R2 R3 / R2
Mathematics 2007 combined national and state main assessment R3 R3 R3 R3
2005 combined national and state main assessment R3 R3 R3 R3
2003 combined national and state main assessment R3 R3 R3 R3
2000 national main assessment R3 / R2 R3 / R2 R3 / R2 R3 / R2
Reading 2007 combined national and state main assessment R3 R3 R3 R3
2005 combined national and state main assessment R3 R3 R3 R3
2003 combined national and state main assessment R3 R3 R3 R3
2002 combined national and state main assessment R3 R3 R3 R3
2000 national main assessment R3 / R2 R3 / R2
Science 2005 combined national and state main assessment R3 R3 R3 R3
2000 national main assessment R3 / R2 R3 / R2 R3 / R2 R3 / R2
U.S. history 2006 national main assessment R3 R3
2001 national main assessment R3 / R2 R3 / R2
Writing 2007 combined national and state main assessment § § § §
2002 combined national and state main assessment § § § §
Mathematics 2004 long-term trend assessment § § § §
Reading 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.
NOTE: R2 refers to accommodations not offered; R3 refers to accommodations offered to students with disabilities (SD) and English language learners (ELL) participating in the NAEP assessment.
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, National Assessment of Educational Progress (NAEP), 2000-2007 Assessments.

Last updated 23 June 2010 (GF)

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