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Analysis of variance concepts and computations

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Abstract The analysis of variance (ANOVA) compares means among three or more groups of observations defined by a categorical variable. Any discussion of ANOVA involves intertwined considerations of (1) scientifically informed study design, (2) computational methods, and (3) statistical theory. Our understanding advances at roughly at the same rate in all three aspects. The earliest ANOVA methods were developed in 1925 and required independent Gaussian observations with common variance. The years 1925–1950 saw the flowering of complex variations for evaluating two or more grouping variables (factors) and the possibility of interaction among factors. An interaction occurs whenever the effect of one factor differs across levels of the other. Requiring equal numbers of observations in each group allowed developing sums‐of‐squares formulas practical for electromechanical calculators. The widespread adoption of digital electronic computers during 1950–1975 stimulated a shift to regression‐based calculations allowing unequal group sizes and the use of multivariate methods. During 1975–2000, statisticians generalized ANOVA based on mixed models to allow missing data. The more general methods require iterative solutions of nonlinear systems of equations. The dominance of the mixed‐model approach creates vulnerabilities to computing and modeling errors in practice. Basic numerical analysis concerns (variable scaling, location shift, and error accumulation) can interact badly with some types of coding dummy variables and covariance model selection. Future directions include automation of access to computation and assumption diagnostics. The proliferation of data with large numbers of variables or observations presents challenges at all three interfaces of scientific modeling, computing, and statistical theory. Copyright © 2009 John Wiley & Sons, Inc. This article is categorized under: Statistical Models > Linear Models Statistical Models > Classification Models

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