Testing Complex a Priori Contrasts on Means from Independent Samples

Abstract

The robustness of the t statistic on complex contrasts, ty, using MS in the estimate of s. is compared with that of t", a statistic derived by Welch (1947). The t statistic is robust to violations of the homogeneity of variance assumption only if n's are equal and all weights have the same absolute value. The Welch t 11 is robust for all combinations explored with any set of weights and either equal or unequal n f s. Since heterogeneous variances are often encountered in behavioral experiments, the Welch t" is recommended as the benchmark test for complex contrasts.at Yale University Library on July 23, 2015 http://jebs.aera.net Downloaded from 208 (1966, pp. 337-345). K Let lii. be defined as Ew. t u. where w. is a vector of K K weights such that Ew = 0. Then i|i, = Ew,, Y is an unbiased estimate of I|J. that is normally distributed if the Y 's are normally distributed or the n f s are reasonably large. To test the null hypothesis, Ho: ty. = 0^ we may construct the statistic t defined as: KoJw and Games Behavioral examples of usage on a single factor are found in Games (1971) and Tjosvold (1977). Examples using an interaction contrast in a two-factor design are found in Myers

Differential Language Effects on Numerical Skills in Second Grade

“…To unambiguously test an effect of inversion, we used a family of tests to analyze this specific research hypothesis (see Westermann & Hager, 1986). We computed a one-way ANOVA to test a complex a priori contrast on means from independent samples (Kohr & Games, 1977) between all children speaking a language with (Flemish, German, and Austrian children) versus all children speaking a language without inversion (French and Walloon children). Furthermore, we conducted a two-sided independent samples t test between the French and the Walloon as well as a univariate ANOVA between the Flemish, German, and Austrian children to analyze whether unexpected differences were found for groups that share a specific language characteristic (inversion yes or no).…”

Comparisons of Treatments After an Analysis of Variance in Ecology

“…Such tests, however, would have very low power with the small sample sizes commonly used, and it may be best to always use a test robust to unequal variances. Studies by Mehta and Srinivasan ( 1970), Wang (1971), Davenport and Webster (1975), Kohr and Games (1977), and Best and Rayner (1987) all suggest that the Welch (1938) t test, generalized with Satterthwaite's degrees of freedom (see Winer 1971 ), is the most powerful and robust test. Best and Rayner (1987) state that the loss in power is so small for n > 5 that the Welch t test should always be used in place of Student's t test.…”

Theory-Free Statistics and Theory-Based Statistics

“…In several articles, Games has pointed out that considerably greater power (over an omnibus test) is usually available if the experimenter can specify a contrast that corresponds to the experimental prediction (Games, 1971(Games, , 1978a(Games, , 1978bKohr & Games, 1977). This is a form of theorybased statistics in that theory determines the weights used in defining the contrast.…”