Statistica Sinica

R. C. H. Cheng and O. D. Jones

Abstract:The Cramer-von Mises statistic provides a useful goodness of fit test of whether a random sample has been drawn from some given null distribution. Its use in comparing several samples has also been studied, but not systematically. We show that the statistic is capable of significant generalization. In particular we consider the comparison of the distributions of observations arising from factorial experiments. Provided that observations are replicated, we show that our generalization yields a test statistic capable of decomposition like the sum of squares used in ANOVA. The statistic is calculated using ranked data rather than original observations. We give the asymptotic theory. Unlike ANOVA, the asymptotic distributional properties of the statistic can be obtained without the assumption of normality. Further, the statistic enables differences in distribution other than the mean to be detected. Because it is distribution free, Monte-Carlo sampling can be used to directly generate arbitrarily accurate critical test null values in online analysis irrespective of sample size. The statistic is thus easy to implement in practice. Its use is illustrated with an example based on a man-in-the-loop simulation trial where operators carried out self assessment of the workload that they experienced under different operating conditions.

Key words and phrases:Cramer-von Mises statistic, distribution free, factorial experiment, homoscedasticity, rank based, simulation experiment.