Suppose there is a matrix containing M plausible values for each respondent. The following steps can be taken to estimate the statistic of interest, , which can be a mean, a percentage, a correlation, etc.:
1. For each plausible value, can be computed, where .
2. The estimate of the statistic of interest is computed as
3. The standard error of contains two components
a. A sampling component, which is computed for each plausible value as
where 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
4. The final estimate of the standard error is then
From these components, the proportion variance due to the fact that is not directly observed,
and the proportion variance due to sampling, can 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.