The continuity principle in psychological research: An introduction to robust statistics.

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AbstractResearch in statistics has demonstrated that the classical estimates of means, variances and correlations are sensitive to small departures from the normal curve. Statisticians have urged caution in the use of classical statistics and have proposed a variety of alternatives which are robust with respect to departures from normality. Robust statistics continue, however, to be little used in psychological research. In this paper we describe common sources of nonnormality in psychological data and examine the distinction between data cleaning and robust estimation. Robust estimation using M - estimators is discussed and recommendations for using these techniques in practice are presented.It is common practice among social and life scientists to adopt an implied continuity principle when interpreting the results of a statistical analysis. It is often assumed, for example, that data which are observed to deviate only slightly in form from that of the familiar normal curve, will only slightly distort the usual estimates of means, standard deviations, correlations and associated hypothesis tests. With increasing departure from an underlying normal model, the greater it is assumed, will be the inaccuracy of the computed statistics.Over the past several decades, research in statistics has demonstrated that a continuity principle of the form described above for normal theory based statistics is invalid. The classical estimates of means, variances and correlations have been shown to be highly sensitive to even small departures from an underlying normal model. A single outlying observation, for example, can strongly bias these statistics and thereby provide misleading or invalid results (see for example, Huber, 1981; Hampel, Ronchetti, Rousseeuw, & Stahel, 1986; Zimmerman, & Zumbo, 1993). For an example where the presence of a single outlier in a sample of 29 observations results in a change of the correlation coefficient from .99 to 0 see Devlin, Gnanadesikan, and Kettenring (1981).The sensitivity of classical statistics to small deviations from normality has important implications for the analysis of research data in psychology. The sensitivity of standard estimates of means and variances to nonnormality can adversely affect analysis of variance (ANOVA) results, while in the case of product moment correlations the lack of robustness will often bias results obtained from principal component analysis, common factor analysis and the analysis of covariance structures (i.e., structural equation modelling). Factor analysis results, for example, which initially appear to provide meaningful factors are often, on a closer examination of the data, simply the result of one or two outliers (Huber, 1981, p. 199).The poor performance of classical statistics in the presence of small departures from normality has led some statisticians (Tukey, 1977, pp. 103 - 106; Hogg, 1977, pp. 1 - 17) to warn that the routine use of classical statistics is unsafe. They recommend that classical estimates of means, variances and correlations only be used in conjunction with alternative methods that are robust with respect to departures from normality. Although there is an increasing amount of statistical software which incorporates robust methods, the use of these methods continues to be, despite some urging by statisticians (Stahel, 1989), little used in applied research. In the behavioural sciences, this is in part likely a result of undergraduate methodology courses that often describe the ANOVA as being robust with respect to type I error and nonnormality (see, for example, Glass & Stanley, 1970, p. 372; Glass, Peckham, & Sanders, 1972). Although ANOVA has some moderate robustness properties with respect to type I error and nonnormality, it is, in relation to type II error, very nonrobust (Hampel, et al., 1986, p. 344; Zimmerman & Zumbo, 1993). This places a researcher in an unusual situation when interpreting ANOVA results. …