# lect18.pdf - 3 4 Xij = µ τi #ij . Note how this...

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18.0 Two-Factor DesignsAnswer QuestionsRCBDConcrete Example118.1 RCBDTheRandomized Complete Block Designis also known as the two-wayANOVA without interaction. A key assumption in the analysis is that theeffect of each level of the treatment factor is the same for each level of theblocking factor.That assumption would be violated if, say, a particular fertilizer worked wellfor one stain but poorly for another; or if one cancer therapy were better forlung cancer but a different therapy were better for stomach cancer.In RCBD, there is one observation for each combination of levels of thetreatment and block factors.2RCBD notation:Iis the number of treatments;Jis the number of blocksXijis the measurement on the unit in blockjthat received treatmenti.n..=I*Jis the total number of experimental units.ni.=Jandn.j=I, the number of observations for a given treatmentlevel or block level, respectively.Xi.is the sum of all measurements for units receiving treatmenti, andX.jis the sum of all measurements for units in thejblock.¯Xi.is the average of all measurements for units receiving treatmenti, or1JJj=1Xij.¯X.jis the average of all measurements for units in thejth block, or1IIi=1Xij.¯X..is the average of all measurements.We continue to use the dot and bar conventions.3Some assumptions:The model for an RCBD (or two-way ANOVA without interactions) is:Xij=μ+τi+βj+ijwhereμis the overall mean of all experimental units,τiis the effect oftreatmenti,βjis the effect of blockj, and theijare random errors that are:normally distributed with mean zero and unknown standard deviationσindependent of the values for all other errors.Note how this generalizes the one-way ANOVA model:Xij=μ+τi+ij.

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