IntroductionBig PictureAssumptionsMultiple ComparisonsNewman-KeulsBonferroniModelTwo-way ANOVAAnalysis of VarianceBret LargetDepartments of Botany and of StatisticsUniversity of Wisconsin—MadisonStatistics 37129th November 2005Analysis of VarianceIAnalysis of variance (ANOVA) is a statistical procedure foranalyzing data that may be treated as multiple independentsamples with a single quantitative measurement for eachsampled individual.IANOVA is a generalization of the methods we saw earlier inthe course for two-independent samples.IThe bucket of balls model is that we have I different bucketsof balls, each of which contains numbered balls.IThe populations means and standard deviations of thenumbers in each bucket are µiand σirespectively fori = 1, . . . , I .IIn ANOVA, we often assume that all of the populationstandard deviations are equal.Cuckoo BirdsICuckoo birds have a behavior in which they lay their eggs inother birds nests.IThe other birds then raise and care for the newly hatchedcuckoos.ICuckoos return year after year to the same territory and laytheir eggs in the nests of a particular host species.IFurthermore, cuckoos appear to mate only within theirterritory.ITherefore, geographical sub-species are developed, each witha dominant foster-parent species.IA general question is, are the eggs of the different sub-speciesadapted to a particular foster-parent species?ISpecifically, we can ask, are the mean lengths of the cuckooeggs the same in the different sub-species?Display of Cuckoo Bird Egg LengthsIHere is a plot of egg lengths (mm) of cuckoo bird eggscategorized by the species of the host bird.HedgeSparrow MeadowPipet PiedWagtail Robin TreePipet Wren20 21 22 23 24 25birdSpeciesA Dotplot of the DataHedgeSparrow MeadowPipet PiedWagtail Robin TreePipet Wren20 21 22 23 24 25The Big PictureIANOVA is a statistical procedure where we test the nullhypothesis that all population mean are equal versus thealternative hypothesis that they are not all equal.IThe test statistic is a ratio of the variability among samplemeans over the variability within sample means.IWhen this ratio is large, this indicates evidence against thenull hypothesis.IThe test statistic will have a different form than what we havepreviously seen. The null distribution is an F distribution,named after Ronald Fisher.IAn ANOVA table is an accounting method for computing thetest statistic.IWe introduce a lot of new notation on the way. . . .NotationThis notation is used to describe calculations of variability withinsamples and variability among samples, although for historicalreasons of poor grammar, the termbetween samples is morecommonly used.yij= the jth observation in the ith groupI = the number of groupsni= the ith sample size¯yi·= the mean of the ith samplen∗=IXi=1ni= the total number of observations¯y··=PIi=1Pnjj=1yijn∗= the grand meanSums of Squares within GroupsWe measure variability by sums of squared deviations. The sums ofsquares within groups, or SS(within), is a combined measure of thevariability within all groups.SS(within) =IXi=1njXj=1(yij− ¯yi·)2=IXi=1(ni− 1)s2iNotice that this measure of variability is a weighted sum of thesample variances where the weights are the degrees of freedom foreach respective sample.Degrees of FreedomIThe degrees of freedom within samples is simply the sum ofdegrees of freedom for each sample.IThis is equal to the total number of observations minus thenumber of groups.df(within) =IXi=1(ni− 1)= n∗− IMean Square WithinIIn ANOVA, a mean square will be the ratio of a sum ofsquares over the corresponding degrees of freedom.MS(within) =SS(within)df(within)=(n1− 1)s21+ ··· + (nI− 1)s2In∗− IIIn other words, the mean square within is a weighted averageof the sample variances where the weights are the degrees offreedom within each sample.IThe square root of the mean sqaure within is the estimate ofthe common variance for all the I populations.spooled=pMS(within)Sums of Squares Between (Among) MeansIWe measure variability by sums of squared deviations. Thesums of squares between groups, or SS(between), is a measureof the variability among sample means.SS(between) =IXi=1ni(¯yi·− ¯y··)2INotice that this measure of variability is a weighted sum ofthe deviations of the sample means from the grand mean,weighted by sample size.Degrees of FreedomIThe degrees of freedom between samples is simply the numberof groups minus one.df(between) = I − 1Mean Square BetweenIThe mean square will be the ratio of a sum of squares overthe corresponding degrees of freedom.MS(between) =SS(between)df(between)=PIi=1ni(¯yi·− ¯y··)2I − 1Total Sum of SquaresIIf we treated all observations as coming from a singlepopulation (which would be the case if all population meanswere equal and all population standard deviations were equalas well), then it would make sense to measure deviations fromthe grand mean.IThis is the total sum of squares.SS(total) =IXI =1niXj=1(yij− ¯y··)2IIt turns out that the total sum of squares can be decomposedinto the sum of squares within and the sum of squaresbetween.SS(total) = SS(within) + SS(between)Total Degrees of FreedomISimilarly, the total degrees of freedom would be n∗− 1.IThere is a similar decomposition.df(total) = df(within) + df(between)n∗− 1 = (n∗− I ) + (I − 1)The F StatisticIThe F statistic is the ratio of the mean square between overthe mean square within.F =MS(between)MS(within)IIf the populations are normal, the population means are allequal, the standard deviations are all equal, and allobservations are independent, then the F statistic hasan Fdistribution with I − 1 and n∗− I degrees of freedom.The F StatisticIAn F distribution is positive and skewed right like thechi-square distribution, but it has two separate degrees offreedom, thenumerator degrees of freedom and thedenominator degrees of freedom.IIf X1and X2are independent χ2random variables with k1andk2degrees of freedom respectively, thenF =X1/k1X2/k2has an F distribution with k1and k2degrees of freedom.ANOVA Table for the Cuckoo Example> fit = aov(eggLength ~ birdSpecies)> anova(fit)Analysis of Variance TableResponse: eggLengthDf Sum Sq Mean Sq F value Pr(>F)birdSpecies 5 42.940 8.588 10.388 3.152e-08 ***Residuals 114 94.248 0.827---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1IIn R, the columns are in an unconventional order and there isno row for totals.IR names the row corresponding to
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