Means Of Several Populations Research Articles

Addendum to: An introduction to functional data analysis and a principal component approach for testing the equality of mean curves

We give an introduction to functional data analysis, with examples, and provide a brief review of the literature. We explain how principal component analysis (PCA) can be used to transform curves into finite dimensional data. An application of PCA is developed to test for the equality of the means of several populations (functional analysis of variance). Asymptotics are derived under the null hypothesis that the populations have the same mean curves. The selection of the basis for the projections and the power of the test is discussed for simple random samples and stationary time series samples of curves. We review the part of the literature which is needed to establish the validity of the PCA method. Two data sets, magnetogram records and stock returns, are used to illustrate the applicability of our limit results.

Analysis of variance based on fuzzy observations

Analysis of variance (ANOVA) is an important method in exploratory and confirmatory data analysis. The simplest type of ANOVA is one-way ANOVA for comparison among means of several populations. In this article, we extend one-way ANOVA to a case where observed data are fuzzy observations rather than real numbers. Two real-data examples are given to show the performance of this method.

Analysis of Means Tables Using Mathematical Processors

Introduced by Ellis Ott in the 1960s, the analysis of means (abbreviated ANOM) procedure provides a simple graphical technique for testing for differences between the means of several populations. The ANOM charts are easy to use, similar in appearance and interpretation to control charts, and usually give the same results as the more familiar analysis of variance technique. Unfortunately, the ANOM decision limits depend upon tables of constants that are usually only available in printed form or from specialized software programs. In this article we present new formulas for the ANOM constants and then illustrate how commonly available mathematical processors can be used to calculate the ANOM constants. This approach makes the ANOM constants easily accessible, portable, and unrestricted with regard to the choice of significance level, sample size, and number of populations.

Size performance of some tests in one-way anova

Tsui and Weerahandi (1989) introduced the notion of generalized p-values and since then this idea is used to solve many statistical testing problems. Heteroskedasticity is one of the major practical problems encountered in ANOVA problems. To compare the means of several groups under heteroskedasticity approximate tests are used in the literature. Weerahandi (1995a) introduced a test using the notion of generalized p-values for comparing the means of several populations when the variances are not equal. This test is referred to as a generalized F-test. In this paper we compare the size performance of the Generalized F-test and four other widely used procedures: the Classical F-test for ANOVA, the F-test obtained by the weighted least-squares to adjust for heteroskedasticity, the Brown-Forsythe-test, and the Welch-test. The comparison is based on a simulation study of size performance of tests applied to the balanced one-way model. The intended level of the tests is set at 0.05. While the Generalized F-test was found to have size not exceeding the intended level, as heteroskedasticity becomes severe the other tests were found to have poor size performance. With mild heteroskedasticity the Welch-test and the classical ANOVA F-test have the intended levels, and the Welch-test was found to perform better than the latter. Widely used (due to computational convenience) weighted F-test was found to have very serious size problems. The size advantage of the generalized F-test was also found to be robust even under severe deviations from the assumption of normality.