Preface to First Edition - lib

ANALYSIS USING R 83set out in the familiar analysis of variance table. The assumptions made inderiving the F-tests are:• The observations are independent of each other,• The observations in each cell arise from a population having a normal distribution,and• The observations in each cell are from populations having the same variance.The multivariate analysis of variance, or MANOVA, is an extension of theunivariate analysis of variance to the situation where a set of variables aremeasured on each individual or object observed. For the data in Table 5.3there is a single factor, epoch, and four measurements taken on each skull; sowe have a one-way MANOVA design. The linear model used in this case isy ijh = µ h + γ jh + ε ijhwhere µ h is the overall mean for variable h, γ jh is the effect of the jth levelof the single factor on the hth variable, and ε ijh is a random error term. Thevector ε ⊤ ij = (ε ij1, ε ij2 , ...,ε ijq ) where q is the number of response variables(four in the skull example) is assumed to have a multivariate normal distributionwith null mean vector and covariance matrix, Σ, assumed to be thesame in each level of the grouping factor. The hypothesis of interest is thatthe population mean vectors for the different levels of the grouping factor arethe same.In the multivariate situation, when there are more than two levels of thegrouping factor, no single test statistic can be derived which is always the mostpowerful, for all types of departures from the null hypothesis of the equalityof mean vector. A number of different test statistics are available which maygive different results when applied to the same data set, although the finalconclusion is often the same. The principal test statistics for the multivariateanalysis of variance are Hotelling-Lawley trace, Wilks’ ratio of determinants,Roy’s greatest root, and the Pillai trace. Details are given in Morrison (2005).5.3 Analysis Using R5.3.1 Weight Gain in RatsBefore applying analysis of variance to the data in Table 5.1 we should try tosummarise the main features of the data by calculating means and standarddeviations and by producing some hopefully informative graphs. The data isavailable in the data.frame weightgain. The following R code produces therequired summary statisticsR> data("weightgain", package = "HSAUR2")R> tapply(weightgain$weightgain,+ list(weightgain$source, weightgain$type), mean)© 2010 by Taylor and Francis Group, LLC