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Analysis Approaches
Analyses reported in this document involve simple descriptive statistics (e.g., percentages, means), bivariate relationships (i.e., cross-tabulations), and correlations. All statistics are weighted to be representative of a larger population of students (as discussed earlier). These analysis approaches exclude cases with missing values; no imputation of missing values has been conducted.
Statistical tests examining differences between independent subgroups or between responses to different items given by the same group that involve categorical variables with more than two possible response categories were conducted by treating each of the possible response categories as separate dichotomous items. For example, each of the three possible response categories of "very much like me," "a little like me," and "not at all like me" was treated as a separate dichotomous item. The percentages of youth who gave each response were then compared across disability or demographic groups or across different questionnaire/interview items. This approach, rather than using scale scores (e.g., the average response for a disability group on a 3-point scale created by assigning values of 1 through 3 to the three response categories), was adopted for two reasons: the proper scaling for the response categories was not apparent, and it was felt that reporting differences in percentages responding in each of the response categories would be more meaningful and easier to interpret by readers than reporting differences in mean values.
Rather than test for differences between all independent subgroups (e.g., youth in different disability categories) simultaneously (e.g., using a k x 2 chi-square test of homogeneity of distribution, where k is the number of disability groups), the statistical significance of differences between selected pairs of independent subgroups is tested. This approach has been followed because the intent is to identify significant differences between specific groups (e.g., youth with learning disabilities are significantly more likely than those with mental retardation to report that they are cared for "a lot" by parents), rather than to identify a more general "disability effect" (e.g., the observed distribution across disability categories differs significantly from what would be expected from the marginal distributions) for the variable of interest.
The test statistic used to compare Bernoullian-distributed responses (i.e., responses that can be allocated into one of two categories and coded as 0 or 1) for two independent subgroups is analogous to a chi-square test for equality of distribution (Conover 1971) and approximately follows a chi-square distribution with one degree of freedom. However, because the test statistic itself is more similar in form to the square of a two sample t statistic with unequal variances (Satterthwaite 1946), and because a chi-square distribution with one degree of freedom is the same as an F distribution with one degree of freedom in the numerator and infinite degrees of freedom in the denominator (Johnson and Kotz 1970), this statistic can be considered the same as an F value; it also can be considered "x2".18
Tests also were conducted to examine differences in the rates at which youth with disabilities as a whole provided specific kinds of self-representations (for example, the percentage of youth who report relying on parents for support "a lot" compared with the percentage who rely on friends "a lot") using an analogous one-sample statistic based on difference scores.19 The test statistic follows a chi-square distribution with one degree of freedom for sample sizes larger than 30, and for similar reasons to those cited above, is considered roughly equivalent to an F(1, infinity) distribution.
In contrast to the dichotomous approach used in statistical tests examining differences in specific responses given by subgroups or across items by the same group, correlations were calculated by comparing responses on a scale that reflects the number of response category options. For example, a 4-point scale was created for variables with response categories related to youth's perceptions of their strengths of "very good, (4 points)" "pretty good," "not very good," or "not at all good" (1 point).
18 In the case of unweighted data, comparing two percentages is usually accomplished using nonparametric statistics, such as the Fisher exact test. In the case of NLTS2, the data are weighted, and the usual nonparametric tests would yield significance levels that are too small, because the NLTS2 effective sample size is less than the nominal sample size. The p values for the test statistic used as an alternative approach to determine statistical significance are derived from an F(1, infinity) distribution (i.e., a chi-square distribution with one degree of freedom). To test for the equality between the mean values of the responses to a single survey item in two disjoint subpopulations, we begin by computing a ratio where the numerator is the difference of the sample means for those subpopulations. (In the case of Bernoulli variables, each mean is a weighted percentage). The denominator for the ratio is the estimated standard error of the numerator (i.e., the square root of the sum of squares of the estimated standard errors for the two means in the numerator). This test statistic is essentially equivalent to a two-sample t test for independent samples (Welch 1947) using weighted data. Sample sizes (and consequently degrees of freedom) for these student t types of ratios are typically reasonably large (i.e., never fewer than 30 in each group), so the ratio follows by the Central Limit Theorem (Wilks 1962), an approximate normal distribution. For a two-tailed test, the test statistic is the square of the ratio, which then follows an approximate chi-square distribution with one degree of freedom. Because a chi-square distribution with one degree of freedom is the same as an F distribution with one degree of freedom in the numerator and an infinite number of degrees in the denominator, the test statistic approximately follows an F(1, infinity) distribution.
19 Testing for the significance of differences in responses to two survey items for the same individuals involves identifying for each youth the pattern of response to the two items. Responses to each item (e.g., the youth reported relying "a lot" on parents for support—yes or no—and reported relying on friends "a lot" for support—yes or no) are scored as 0 or 1, producing difference values for individual students of +1 (responded affirmatively to the first item but not the second), 0 (responded affirmatively to both or neither item), or -1 (responded affirmatively to the second item but not the first). The test statistic is the square of a ratio, where the numerator of the ratio is the weighted mean change score and the denominator is an estimate of the standard error of that mean. Since the ratio approaches a normal distribution by the Central Limit Theorem, this test statistic approximately follows a chi-square distribution with one degree of freedom, that is, an F(1, infinity) distribution.