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NAEP Analysis and Scaling → Drawing Inferences from NAEP Results → t Test Comparisons and the Determination of the Appropriate Degrees of Freedom → t Test for Dependent Samples

NAEP Technical Documentationt Test for Dependent Samples

Certain analyses involved the comparison of proportions. One example was the comparison of the proportion of students who reported that a parent graduated from college to the proportion of students who indicated that their parents did not finish high school to determine which proportion was larger. There are other such proportions of interest in this example, such as the proportion of students with at least one parent graduating from high school but neither parent graduating from college. For these types of analyses, NAEP staff determined that the dependencies in the data could not be ignored.

Unlike many other comparisons in NAEP, the correlation between the proportion of students reporting a parent graduated from college and the proportion reporting that their parents did not finish high school is likely to be negative and large. For a particular sample of students, it is likely that the higher the proportion of students reporting "at least one parent graduated from college" is, the lower the proportion of students reporting "neither parent graduated from high school" will be. A negative dependence will result in underestimates of the standard error if the estimation is based on independence assumptions. Such underestimation can result in an unacceptably large number of "nonsignificant" differences being identified as significant.

The procedures for independent groups t tests were modified for analyses that involved comparisons of proportions within a group. The modification involved using a jackknife method for obtaining the standard error of the difference in dependent proportions. The standard error of the difference in proportions was obtained by first obtaining a separate estimate of the difference in question for each jackknife replicate (using the first plausible value only) then taking the standard deviation of the set of replicate estimates as the estimate. The procedures used for proportions within a group differed from the procedures for independent samples t tests only with respect to estimating the standard error of the difference; all other aspects of the procedures were identical.

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