Academic journal article Psychology Science

A Plea for More General Tests Than Those for Location Only: Further Considerations on Rasch & Guiard's 'The Robustness of Parametric Statistical Methods'

Academic journal article Psychology Science

A Plea for More General Tests Than Those for Location Only: Further Considerations on Rasch & Guiard's 'The Robustness of Parametric Statistical Methods'

Article excerpt

Abstract

Starting with the discussion between Rasch & Guiard (2004) and von Eye (2004) concerning the use of parametric and nonparametric tests for the comparison of two samples a further approach toward this question is undertaken. Student's t-test requires for its application interval scaled and normally distributed data along with homogeneous variances across groups. In case that at least one of these prerequisites is not fulfilled, common statistical textbooks for social sciences usually refer to the nonparametric Wilcoxon-Mann-Whitney test. Earlier simulation studies revealed the t-test to be rather robust concerning distributional assumptions. The current study extends these findings with respect to the simultaneous violation of distributional and homogeneity assumptions. A simulation study has shown that both tests lead to highly contradicting results, and a more general approach toward the question of whether parametric or nonparametric procedures should be used, is introduced. Results indicate that the U-Test seems to be in general a more proper instrument for psychological research.

Key words: Parametric tests, nonparametric tests, non-normality, heteroscedasticity, power

1. Introduction

The comprehensive article of Rasch & Guiard (2004) provides a detailed overview concerning the robustness of parametric procedures. One aspect of their work leads to the conclusion that there is no further need for the Wilcoxon-Mann-Whitney test, that is why they recommend t-test due to its robustness. In a commentary, von Eye (2004) stresses the point that the exclusive use of the t-test might be inappropriate due to its sensitivity regarding autocorrelation of values. In their reply, Guiard & Rasch (2004) clearly worked out that the same is true for the nonparametric test and therefore conclude: "Summarising we still think there are more disadvantages than advantages in using the Wilcoxon test in place of the t-test." (Guiard & Rasch, 2004, p. 553). This far-reaching conclusion raises the question whether the t-test might preserve its favourable properties even for more extreme distributional aberrations, which it was not designed for. In detail, we investigate the behavior of the t-test in comparison to the U-test in the presence of simultaneous violations of both distributional and homogeneity assumptions.

Rasch & Guiard (2004; referring to Posten, 1978) address the comparison of means (i.e. the "location hypothesis'" H^sub 0^: μ^sub x^ = μ^sub y^). Their results are based on distributions with positive and negative values of skewness over 0 ≤ γ^sub 1^ ≤ 2.0 and kurtosis over 1.4 ≤ γ^sub 2^ ≤ 7.8. We have decided to extend these values by using the lognormal distribution (γ^sub 1^ = 6.19; γ^sub 2^ = 113.93). Our choice reflects (i) the fact that in the context of psychological research (heavily) skewed distributions are likely to occur and (U) depicts such extreme aberrant cases as mentioned above. Examples for lognormally distributed variables are given in e.g. Sachs & Hedderich (2006) and Limpert, Stahel, & Abbt (2001).

2. The two-sample problem

Given two independent, normally distributed samples and homogeneous variances together with interval scaled values, Student's t-test is the most powerful test (Pitman, 1948, as cited in Randies & Wolfe, 1979). Given the same conditions the asymptotic relative efficiency (ARE) of the most powerful nonparametric tests is .955 compared with the t-test, which means that sample size of the nonparametric procedure must be increased by about 4.5% to achieve the same efficiency as the parametric procedure (cf. Randies & Wolfe, 1979; Nikitin, 1995). Numerous studies have dealt with the adequacy of Student's t-test if at least one assumption is violated. In case of unequal variances it has been shown that Student's t-test is only robust if sample sizes are equal (cf. Hsu, 1938; Scheffé, 1970; Posten, Yeh & Owen, 1982; Tuchscherer & Pierer, 1985; Zimmerman & Zumbo, 1993a, 1993b; Bradstreet, 1997; Zimmerman, 2004). …

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