Is the selection of statistical methods governed by level of measurement?

AbstractThe notion that nonparametric methods are required as a replacement of parametric statistical methods when the scale of measurement in a research study does not achieve a certain level was discussed in light of recent developments in representational measurement theory. A new approach to examining the problem via computer simulation was introduced. Some of the beliefs that have been widely held by psychologists for several decades were examined by means of a computer simulation study that mimicked measurement of an underlying empirical structure and performed two - sample Student t - tests on the resulting sample data. It was concluded that there is no need to replace parametric statistical tests by nonparametric methods when the scale of measurement is ordinal and not interval.Stevens' (1946) classic paper on the theory of scales of measurement triggered one of the longest standing debates in behavioural science methodology. The debate -- referred to as the levels of measurement controversy, or measurement - statistics debate -- is over the use of parametric and nonparametric statistics and its relation to levels of measurement. Stevens (1946; 1951; 1959; 1968), Siegel (1956), and most recently Siegel and Castellan (1988) and Conover (1980) argue that parametric statistics should be restricted to data of interval scale or higher. Furthermore, nonparametric statistics should be used on data of ordinal scale. Of course, since each scale of measurement has all of the properties of the weaker measurement, statistical methods requiring only a weaker scale may be used with the stronger scales. A detailed historical review linking Stevens' work on scales of measurement to the acceptance of psychology as a science, and a pedagogical presentation of fundamental axiomatic (i.e., representational) measurement can be found in Zumbo and Zimmerman (1991).Many modes of argumentation can be seen in the debate about levels of measurement and statistics. This paper focusses almost exclusively on an empirical form of rhetoric using experimental mathematics (Ripley, 1987). The term experimental mathematics comes from mathematical physics. It is loosely defined as the mimicking of the rules of a model of some kind via random processes. In the methodological literature this is often referred to as simulation. However, for the purpose of this paper, the terms experimental mathematics or computer simulation are preferred to because the latter is typically referred to when examining the robustness of a test in relation to particular statistical assumptions. Measurement level is not an assumption of the parametric statistical model (see Zumbo & Zimmerman, 1991 for a discussion of this issue) and to call the method used herein monte carlo would imply otherwise. The term experimental mathematics emphasizes the modelling aspect of the present approach to the debate.The purpose of this paper is to present a new paradigm using experimental mathematics to examine the claims made in the levels of measurement controversy. As Michell (1986) demonstrated, the concern over levels of measurement is inextricably tied to the differing notions of measurement and scaling. Michell further argued that fundamental axiomatic measurement or representational theory (see, for example, Narens & Luce, 1986) is the only measurement theory which implies a relation between measurement scales and statistics. Therefore, the approach advocated in this paper is linked closely to representational theory. The novelty of this approach, to the authors knowledge, is in the use of experimental mathematics to mimic representational measurement. Before describing the methodology used in this paper, we will briefly review its motivation.Admissible TransformationsRepresentational theory began in the late 1950's with Scott and Suppes (1958) and later with Suppes and Zinnes (1963), Pfanzagl (1968), and Krantz, Luce, Suppes & Tversky (1971). …