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Appendix A: Technical Notes

Statistical Procedures

Comparisons in the text based on sample survey data have been tested for statistical significance to ensure that the differences are larger than might be expected due to sampling variation. Findings described in this report with comparative language (e.g., higher, lower, increase, and decrease) are statistically significant at the .05 level. Comparisons based on universe data do not require statistical testing, with the exception of linear trends. Several test procedures were used, depending upon the type of data being analyzed and the nature of the statement being tested. The primary test procedure used in this report was Student's t statistic, which tests the difference between two sample estimates. The t test formula was not adjusted for multiple comparisons. The formula used to compute the t statistic is as follows:

Equation is explained in surrounding text. (1)

where E1 and E2 are the estimates to be compared and se1 and se2 are their corresponding standard errors. Note that this formula is valid only for independent estimates. When the estimates are not independent (for example, when comparing a total percentage with that for a subgroup included in the total), a covariance term (i.e., 2 * r * se1 * se2) must be subtracted from the denominator of the formula: 

Equation is explained in surrounding text. (2)

where r is the correlation coefficient. Once the t value was computed, it was compared to the published tables of values at certain critical levels, called alpha levels. For this report, an alpha value of .05 was used, which has a t value of 1.96. If the t value was larger than 1.96, then the difference between the two estimates is statistically significant at the 95 percent level.

A linear trend test was used when differences among percentages were examined relative to ordered categories of a variable, rather than the differences between two discrete categories. This test allows one to examine whether, for example, the percentage of students using drugs increased (or decreased) over time or whether the percentage of students who reported being physically attacked in school increased (or decreased) with their age. Based on a regression with, for example, student's age as the independent variable and whether a student was physically attacked as the dependent variable, the test involves computing the regression coefficient (b) and its corresponding standard error (se). The ratio of these two (b/se) is the test statistic t. If t is greater than 1.96, the critical value for one comparison at the .05 alpha level, the hypothesis that there is no linear relationship between student's age and being physically attacked is rejected.

Some comparisons among categories of an ordered variable with three or more levels involved a test for a linear trend across all categories, rather than a series of tests between pairs of categories. In this report, when differences among percentages were examined relative to a variable with ordered categories, analysis of variance (ANOVA) was used to test for a linear relationship between the two variables. To do this, ANOVA models included orthogonal linear contrasts corresponding to successive levels of the independent variable. The squares of the Taylorized standard errors (that is, standard errors that were calculated by the Taylor series method), the variance between the means, and the unweighted sample sizes were used to partition the total sum of squares into within- and between-group sums of squares. These were used to create mean squares for the within- and between-group variance components and their corresponding F statistics, which were then compared to published values of F for a significance level of .05. Significant values of both the overall F and the F associated with the linear contrast term were required as evidence of a linear relationship between the two variables.