TWO WAY ANOVANext we consider the case when we have two factors, categorizations, e.g. lab and manufacturer.If there are I levels in the first factor and J levels in the second factor then we can think of thissituation as one where there are I × J levels of the combined factors.NotationNotation-wise, we simply add another subscript to the response, that is, y now has a triple subscript,where yi,j,krepresents the measurement on the kth subject that belongs to both the ith group (lab)of the first factor and the j group (manufacturer) of the second factor, i = 1, . . . I, j = 1, . . . , J,and k = 1, . . . , nij.For simplicity, we will only work with the special case of ni,j= K, i.e. all subgroups have the samenumber of responses. Then we write the model as follows:yi,j,k= α + ηi,j+ Ei,j,kSo, when i = 1 and j = 1, y1,1,k= + E1,1.kand when i = 2 and j = 1, y2,1,k= + E2,1.kAgain, we need a constraint because our model is over-parameterized. We add the constraint thatPiPjηi,j= 0.A Simpler Sub-modelIn our example of the study of the measurement process, we find that with 7 labs and 4 manufac-turers, we have 28 levels. If the effect of the lab is the same, regardless of which manufacturer thetablets are coming from, and if the effect of the manufacturer is the same regardless of which lab ismeasuring the tablets then we could express the model asyi,j,k= α + βi+ γj+ Ei,j,k1Note that now we have only I + J levels, rather than I × J. This model is called an additive model.It puts structure on the levels. That is the difference between measurements at LAb 1 and Lab 2 oftablets from Manufacturer A is β2− β1, and this difference is the same for the measurements at Labs1 and 2 for tablets from Manufacturer B, i.e. there is no interaction between lab and manufacturer.Degrees of FreedomTo see that the additive model is a sub-model of the full model, we can we express the full modelas follows:yi,j,k= α + βi+ γj+ νi,j+ Ei,j,kNow again, we need to put constraints on the parameterization. If we think about it from thegeometric perspective, we see that the 1 vector lies in both the space spanned by the lab indicators(the ei) and the space spanned by the manufacturer indicators (the uj). So, the 1 vector, and I − 1of the eivectors and J − 1 of the ujvectors are all that is needed for the additive part of the model.As for the rest, suppose we have vectors vi,jthat indicate whether a response belongs in group i, jor not.Note thatPjvi,j= , andand thatPivi,j= .So we need only of these I × J vectors.All together that gives us 1 + (I − 1) + ( ) + ( ) = ( ) of the 1 + I + J + IJvectors.If we are to put all of the parameters in then we must add constraints. Traditionally these areXiβi= 0Xjγj= 02Xi= 0, forXj= 0, forHow many constraints do we have?Sums of SquaresThe Anova table of the sums of squared deviations helps us assess whether the simple additivemodel is adequate to describe the variation in the means, and whether there is a lab effect or amanufacturer effect (i.e. whether all of the βi= 0 or all of the γj= 0).The decomposition of the sums of squares is a bit more complex here. First we need to introducesome more notation,¯y..=1IJKXiXjXkyijk¯yi.=1JKXjXkyijk, for i = 1 . . . I¯y.j=¯yij=Now let’s look at the sums of squares:XiXjXk(yijk− ¯y..)2To begin, let’s add and subtract the IJ means ¯yij.3XiXjXk(yijk− ¯y..)2=XiXjXk(yijk− ¯yij)2+XiXjXk(¯yij− ¯y..)2Show that the cross product term is 0.We call the first sum on the right-hand side of the equation the error sum of squares, or SSE. Wewant to further decompose the second term.XiXjXk(¯yij− ¯y..)2What do we add and subtract – ¯yi.or ¯y.j? Both:XiXjXk(¯yij− ¯yi.− ¯y.j+ ¯y..+ ¯yi.− ¯y..+ ¯y.j− ¯y..)2=XiXjK(¯yij− ¯yi.− ¯y.j+ ¯y..)2+XiJK(¯yi.− ¯y..)2+XjIK(¯y.j− ¯y..)2The three terms on the right-hand side of the equality are called, the interaction sum of squares,or SSLM, the sum of squares due to Lab, or SSL, and the sum of squares due to Manufacturer, or4SSM.Show that the cross products are all 0.ANOVA TableArrange the sum of squares into an ANOVA table.Source DF Sum of Squares Mean Square F-statisticLabsManufacturer 3Interaction 8Error 60Total 85The first F statistic is used to test whether there is a difference between labs, i.e. whether thereis a lab effect. The second F statistic is used to test whether there is a difference between man-ufactureres. The third is to tes t the additive model, i.e. is there an interaction between lab