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 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 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 provided show that because results for student groups have high proportions, the groups are well described by the population-structure models.

 

 

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