Because CPS data are collected from samples of the population, statistical tests are employed to measure differences between estimates to help ensure they are taking into account possible sampling error. 7 The descriptive comparisons in this report were tested using Student’s t statistic. Differences between estimates are tested against the probability of a type I error,8 or significance level. The significance levels were determined by calculating the Student’s t values for the differences between each pair of means or proportions and comparing these with published tables of significance levels for two-tailed hypothesis testing.
Student’s t values may be computed to test the difference between percentages with the following formula:
where P1 and P2 are the estimates to be compared and se1 and se2 are their corresponding standard errors.
Several points should be considered when interpreting t statistics. First, comparisons based on large t statistics may appear to merit special attention. This can be misleading since the magnitude of the t statistic is related not only to the observed differences in means or proportions but also to the number of respondents in the specific categories used for comparison. Hence, a small difference compared across a large number of respondents would produce a large t statistic.
Second, there is a possibility that one can report a “false positive” or type I error. In the case of a t statistic, this false positive would result when a difference measured with a particular sample showed a statistically significant difference when there was no difference in the underlying population. Statistical tests are designed to control this type of error. These tests are set to different levels of tolerance or risk known as alphas. The alpha level of .05 selected for findings in this report indicates that a difference of a certain magnitude or larger would be produced no more than 1 time out of 20 when there was no actual difference in the quantities in the underlying population. When p values are smaller than the .05 level, the null hypothesis that there is no difference between the two quantities is rejected. Finding no difference, however, does not necessarily imply that the values are the same or equivalent.
Third, the probability of a type I error increases with the number of comparisons being made. Bonferroni adjustments are sometimes used to correct for this problem. Bonferroni adjustments do this by reducing the alpha level for each individual test in proportion to the number of tests being done. However, while Bonferroni adjustments help avoid type I errors, they increase the chance of making type II errors. Type II errors occur when there actually is a difference present in a population, but a statistical test applied to estimates from a sample indicates that no difference exists. Prior to the 2001 report in this series, Bonferroni adjustments were employed. Because of changes in NCES reporting standards, Bonferroni adjustments are not employed in this report.
Regression analysis was used to test for trends across age groups and over time. Regression analysis assesses the degree to which one variable (the dependent variable) is related to one or more other variables (the independent variables). The estimation procedure most commonly used in regression analysis is ordinary least squares (OLS). When studying changes in rates over time, the rates were used as dependent measures in the regressions, with a variable representing time and a dummy variable controlling for changes in the educational attainment item in 1992 (= 0 for years 1972 to 1991, = 1 after 1992) used as independent variables. When slope coefficients were positive and significant, rates increased over time. When slope coefficients were negative and significant, rates decreased over time. Because of varying sample sizes over time, some of the observations were less reliable than others (i.e., some years’ standard errors were larger than those for other years). In such cases, OLS estimation procedures do not apply, and it is necessary to modify the regression procedures to obtain unbiased regression parameters. This is accomplished by using weighted least squares regressions.9 Each variable in the analysis was transformed by dividing by the standard error of the relevant year’s rate. The new dependent variable was then regressed on the new time variable, a variable for 1 divided by the standard error for the year’s rate, and the new editing-change dummy variable. All statements about trend changes in this report are statistically significant at the .05 level.
7 The CCD and GEDTS data are universe data collections and therefore do not require statistical testing such as that used for estimates from the CPS sample survey data.
8 A Type I error occurs when one concludes that a difference observed in a sample reflects a true difference in the population from which the sample was drawn, when no such difference is present. It is sometimes referred to as a “false positive.”
9 For a general discussion of weighted least squares analysis, please see Gujarati, D., Basic Econometrics 2nd ed. McGraw Hill, Inc., New York: New York, 1998.