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12.2. The transformed variable has equal variances across the two groups (Levene’s test, p = .119), and the t-test statistic is –1.308 (df = 85, p = .194). Thus, the differences in pollution between watersheds in the East and Midwest are not significant. (The negative sign of the t-test statistic, –1.308, merely reflects the order of the groups for calculating the difference: the testing variable has a larger value in the Midwest than in the East. Reversing the order of the groups results in a positive sign.) Table 12.2 Independent-Samples T-Test: Output For comparison, results for the untransformed variable are shown as well. The untransformed variable has unequal variances across the two groups (Levene’s test, p = .036), and the t-test statistic is –1.801 (df = 80.6, p =.075). Although this result also shows that differences are insignificant, the level of significance is higher; there are instances in which using nonnormal variables could lead to rejecting the null hypothesis. While our finding of insignificant differences is indeed robust, analysts cannot know this in advance. Thus, analysts will need to deal with nonnormality. Variable transformation is one approach to the problem of nonnormality, but transforming variables can be a time-intensive and somewhat artful activity. The search for alternatives has led many analysts to consider nonparametric methods. TWO T-TEST VARIATIONS Paired-Samples T-Test Analysts often use the paired t-test when applying before and after tests to assess student or client progress. Paired t-tests are used when analysts have a dependent rather than an independent sample (see the third t-test assumption, described earlier in this chapter). The paired-samples t-test tests the null hypothesis that the mean difference between the before and after test scores is zero. Consider the following data from Table 12.3. Table 12.3 Paired-Samples Data The mean “before” score is 3.39, and the mean “after” score is 3.87; the mean difference is 0.54. The paired t-test tests the null hypothesis by testing whether the mean of the difference variable (“difference”) is zero. The paired t-test test statistic is calculated as where D is the difference between before and after measurements, and sD is the standard deviation of these differences. Regarding t-test assumptions, the variables are continuous, and the issue of heterogeneity (unequal variances) is moot because this test involves only one variable, D; no Levene’s test statistics are produced. We do test the normality of D and find that it is normally distributed (Shapiro-Wilk = .925, p = .402). Thus, the assumptions are satisfied. We proceed with testing whether the difference between before and after scores is statistically significant. We find that the paired t-test yields a t-test statistic of 2.43, which is significant at the 5 percent level (df = 9, p = .038 < .05).17 Hence, we conclude that the increase between the before and after scores is significant at the 5 percent level.18 One-Sample T-Test Finally, the one-sample t-test tests whether the mean of a single variable is different from a prespecified value (norm). For example, suppose we want to know whether the mean of the before group in Table 12.3 is different from the value of, say, 3.5? Testing against a norm is akin to the purpose of the chi-square goodness-of-fit test described in Chapter 11, but here we are dealing with a continuous variable rather than a categorical one, and we are testing the mean rather than its distribution. The one-sample t-test assumes that the single variable is continuous and normally distributed. As with the paired t-test, the issue of heterogeneity is moot because there is only one variable. The Shapiro-Wilk test shows that the variable “before” is normal (.917, p = .336). The one-sample t-test statistic for testing against the test value of 3.5 is –0.515 (df = 9, p = .619 > .05). Hence, the mean of 3.39 is not significantly
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Evan M. Berman (Essential Statistics for Public Managers and Policy Analysts)