Some myths concerning parametric and nonparametric tests.

AbstractFive myths concerning the application of parametric and nonparametric tests are discussed. Well known considerations of power, robustness, and scale of measurement are reviewed briefly. Less well known ideas about the nature of the null hypothesis and generality of application are outlined. It is concluded that in many applications behavioural researchers are using what appear to be parametric tests, but actually are evaluating nonparametric hypotheses and estimating the probability of a Type I error that would be obtained with a nonparametric test.Statisticians have debated the relative merits of parametric and nonparametric inference for over 60 years, and increasingly that literature favours nonparametric inference when applied to data from behavioural research. Nevertheless, even a cursory look at the psychological research literature reveals that the parametric platform has been more convincing to psychologists. Why is there a schism between statisticians and researchers? We hope to answer this question by suggesting that the issue of parametric versus nonparametric inference has been dominated by a collection of interrelated myths and half - truths that have mislead researchers into using, or believing they are using, parametric tests. In debunking these myths we argue that the victory of parametric inference over nonparametric inference is more illusory than real. The myths to be discussed are:1. Parametric tests are more powerful than nonparametric tests.2. Parametric tests are robust.3. Nonparametric tests are tests on non - interval data -- and t - and F - tests are exclusively parametric tests.4. The null hypotheses evaluated by parametric tests are direct and clear, whereas the null hypotheses evaluated by nonparametric tests are indirect and vague.5. Nonparametric tests are restricted in their application.These myths are all tied in one way or another to evaluating the validity of statistical tests performed on real - life data -- that is, evaluating the believability of obtained probabilities of Type I error. Since the validity of all statistical tests relies on satisfying certain assumptions, we begin by briefly reviewing the assumptions underlying parametric and nonparametric tests.AssumptionsThe assumptions underlying the best - known and most frequently usedparametric statistics include:1. All observations are randomly and independently sampled from their parent populations.2. The population distributions from which samples are selected are normal.3. All populations have the same variance.4. The data are measured on at least an interval scale.In contrast, nonparametric statistics make fewer and generally much weaker assumptions. Most importantly, though less well known, nonparametric tests need not assume random sampling. Although all nonparametric tests assume independence of sample observations, that assumption can be tied to random assignment in experiments or exchangeability in observational studies, rather than to random sampling.Given only this information, the behavioural researcher should be convinced to use nonparametric tests. Rarely do we know the extent to which population assumptions are met, and even less often do we randomly sample our subjects. Nevertheless, we persist in using parametric tests primarily because we are persuaded by the myths listed above. The validity of these myths is examined next.I. Parametric Tests are More Powerful than Nonparametric TestsThe power of a test is the probability of correctly rejecting a null hypothesis. Efficiency is a relative term comparing the power of one test to another when both are used to test the same null hypothesis, and the relative efficiency of one test with respect to another is the ratio of the sample sizes needed for both tests to achieve the same power. Thus, another way to phrase the first myth is to say that nonparametric tests require more subjects to achieve the same power as parametric tests. …