​​​​​​​​​​​​​​​​​​​​​​​​​​​​​NAEP Technical DocumentationProportion of Variance Accounted for by Principal Components Used in NAEP Population-Structure Models

The population-structure models employed for specific national, 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 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.

To avoid overfitting the model, 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. Prior to 2022, the correlation matrix was used in the principal component transformation. Beginning in 2022, five main reporting variables (school-reported gender, IEP status, LEP status, National School Lunch Program eligibility,​ and school-reported race/ethnicity) were included as direct covariates in the population-structure model. (Please note that the new procedure was implemented for all 2022 assessments other than the ​age 9 long-term trend assessments.) ​The remaining variables were regressed on the five main reporting variables, and a principal component analysis was applied to the residual covariance matrix of this regression. This change was made to ensure that the main reporting variables are fully accounted for in the population-structure models, and to improve parameter estimation with the use of a covariance matrix. For national 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.

For assessments that used the covariance matrix in the principal component transformation, the minimum and maximum proportion of variance explained were also provided for contrasts which met the minimum sample requirements for reporting.​

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 by the population-structure model. If all of the principal components were used in the models, all of the proportions would be one. For the national-level conditioning models, the number of principal components was selected so that at least 95 percent of the overall variance of the group-defining variable contrasts was accounted for by the principal components. For state-level conditioning models, at least 90 percent of the overall variance of the group-defining variable contrasts was accounted for by the principal components to take into account the smaller sample size of state samples​. This results in proportions that are less than one. The values provided show that the groups are well described by the population-structure models, because of the high proportions for student groups. ​Prior to the 2002 and 2003 assessments, for variables with quadratic multi-degree-of-freedom contrasts, the variance explained by the variables was reported.

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