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NAEP Analysis and Scaling → Summary Statistics for Scale Scores of Groups → Procedures for Estimating Group Scale Score Statistics and Their Standard Errors → Using Plausible Values to Estimate Group Scale Score Statistics and Their Standard Errors

NAEP Technical DocumentationUsing Plausible Values to Estimate Group Scale Score Statistics and Their Standard Errors

Suppose there is a matrix The matrix theta equals the collection of vectors theta 1, theta 2, through theta sub uppercase M containing M plausible values for each respondent. The following steps can be taken to estimate the statistic of interest, t hat of vector theta and vector Y equals f of vector X and vector Y, which can be a mean, a percentage, a correlation, etc.:

1. For each plausible value, f hat sub lowercase m of vector X and vector Ycan be computed, where m equals 1, 2, though uppercase M.

2. The estimate of the statistic of interest is computed as f hat of vector X and vector Y equals 1 over M times the sum over lowercase m of f hat sub lowercase m of vector X and vector Y  

3. The standard error of  f hat of vector X and vector Y contains two components
a. A sampling component, which is computed for each plausible value as U of lowercase m equals the sum over lowercase r of the squared differences between f hat sub lowercase mr of vector X and vector Y and f hat sub lowercase m of vector X and vector Y
 
where f hat sub lowercase mr of vector X and vector Y is the estimate of the statistic based on the mth plausible value and the rth replicate weights. Subsequently, the sampling component is the average Um over plausible values. In practice, U1 is used to approximate this average, substantially reducing the amount of computation required.
b. A measurement component, which is computed as B equals 1 over M minus 1 times the sum over lowercase r of the squared differences between f hat sub lowercase m of vector X and vector Y and f hat of vector X and vector Y
 
4. The final estimate of the standard error is then SE equals square root of V and equals square root of U 1 plus B times 1 plus uppercase M to the -1 power 

From these components, the proportion variance due to the fact that is not directly observed, 1 plus uppercase M raised to the -1 power times B over V

 and the proportion variance due to sampling, U over Vcan be computed. These proportions can be quite different for different subjects, grades, and samples. In general, a larger number of items per student reduced the proportion due to the latency of θ. Also, a more efficient sample, with fewer students per school and many schools reduces the proportion due to sampling.

Note that the proportions are based on the variance of a statistic, which indicates how much confidence can be put into the estimate of this statistic. This variance (squared standard error) should not be confused with the variance of a sample, which is an indication of the distribution of observations, rather than the confidence of a single statistic.

Last updated 27 October 2009 (JL)

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