Thoughts on Interaction

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1 Thoughts o Iteractio Roald Christese Departmet of Mathematics ad Statistics Uiversity of New Mexico November 16, 2016 Abstract KEY WORDS: 0

2 The first sectio examies iteractios i a ubalaced two-way ANOVA. The secod sectio uses the first to establish that the two-way ANOVA displays orthogoality if ad oly if the ubalaced umbers are proportioal. The third sectio looks at Boik s work o iteractio testig for proportioal umbers. 1. Characterizig the Iteractio Space I a two-way ANOVA with iteractio ad ubalaced umbers, the cell-meas parameterizatio is y ijk = µ ij + ε ijk, i = 1,..., a, j = 1,..., b, k = 1,..., N ij, which is really just a oe-way ANOVA with uequal umbers ad the pair of subscripts ij idetifyig the ab differet groups. The traditioal parameterizatio is µ ij = µ+α i +η j +γ ij. Write the cell-meas model i matrix form as Y = Xµ + e. X has ab colums ad the i j colum has the form X i j = [t ijk], with t ijk = δ (i,j)(i,j ) where for ay two symbols a ad b, the kroecker delta has 1 if a = b δ ab = 0 if a b. Note that C(X) = {v v = [v ijk ], with v ijk = µ ij for some µ ij }. Similar to the balaced case, we will see that the iteractio space is the set of vectors T = [t ijk ], with t ijk = q ij /N ij where q i = 0 = q j, for all i, j. A iteractio cotrast is T Xµ = ij q ijµ ij = ij q ijγ ij. Clearly, T C(X), so T MY = T Y where M is the perpedicular projectio operator (ppo) oto C(X). It follows that the 1

3 least squares estimate of T Xµ is T Y = ij q ijȳ ij. These are essetially the same results as for balaced ANOVA. To see that vectors T characterize the iteractio space, write the correspodig mai effects model Y = Jµ + X α α + X η η + e. The matrix X α has colums X i = [t ijk ], t ijk = δ ii i = 1,..., a The matrix X η has colums X a+j = [t ijk ], t ijk = δ jj j = 1,..., b By defiitio, the iteractio space is C(J, X α, X η ) C(X). Thus, the characterizatio of the iteractio space results from vectors of the form T spaig a space of sufficiet rak [(a 1)(b 1) whe N ij > 0 for all ij], havig T C(X), ad X h T = 0, h = 1,..., a + b [which follows from the defiitios ad arithmetic]. Note that to have orthogoal iteractio cotrasts we eed the T vectors correspodig to the iteractio cotrasts to be orthogoal. Note also that all iteractio cotrasts are cotrasts i the µ ij oe-way model, but ot all µ ij cotrasts are iteractio cotrasts. As i PA Chapter 4, a arbitrary elemet of the µ ij cotrast space is S = [s ijk ], with s ijk = s ij /N ij where s = 0. I additio to the iteractio space beig a subset of the cotrast space, there is a a 1 dimesioal subspace of the cotrast space cosistig of vectors T α = [t ijk ], with t ijk = c i /bn ij where c = 0 ad a b 1 dimesioal subspace of vectors T η = [t ijk ], with t ijk = d j /an ij where d = 0. 2

4 These defie cotrasts i the iteractio model of T αxµ = ij c i µ ij /b = ij c i µ i = i c i (α i + γ i ) with estimate T αy = i c iȳ i ad T ηxµ = ij d j µ ij /a = ij d j µ j = j d j (η j + γ j ) with estimate T η Y = j d jȳ j. Uder proportioal umbers, these three spaces are orthogoal but i geeral they itersect i the zero vector. For the mai effects model, the vectors T α ad T η typically are ot i C(J, X α, X η ), so although, for example, T α[jµ + X α α + X η η] = i c iα i, the estimate is ot T αy, it is T αm 0 Y where M 0 is the ppo oto C(J, X α, X η ). 2. Orthogoal Mai Effects iff Proportioal Numbers A iterestig sidelight from writig the ubalaced model this way is that correctig for the grad mea gives ad [I (1/)JJ ]X i = [t ijk ], t ijk = δ ii N i [I (1/)JJ ]X a+j = [t ijk ], t ijk = δ jj N j which implies that mai effects are orthogoal iff the data have proportioal umbers. To see this, X i [I (1/)JJ ]X a+j = 0 0 = N i j N i N j = i ( δ ii N ) ( ) i δ jj N j j k I do t remember if orthogoality implies proportioal umbers is prove i the PA Chap 7. The reverse certaily is prove. 3

5 3. Thoughts o Testig Iteractio This requires some kowledge of Multivariate Aalysis. We kow how to test iteractio, just test the µ ij model agaist the additive model. As discussed above ad i PA (Chap. 7), i the balaced case ad ow i the ubalaced case, we ca characterize the iteractio space. The iteractio space ivolves cotrasts i j q ijµ ij with q ij s defied as above. But the most iterpretable iteractio cotrasts have a special form (product iteractios), built by combiig a cotrast i the α i s with a cotrast i the η j, see below. Boik developed a test for iteractios of this specific form. However, it seems that the distributio theory was ugly. (Admittedly, I thik that all distributio theory is ugly.) At least i some special cases, oe ca rewrite Boik s approach so that stadard results from multivariate ANOVA equate to Boik s results. Boik s results apply whe oe has proportioal umbers. The results below are more restrictive tha proportioal umbers. 1. Boik, R. J. (1986), Testig the Rak of a Matrix with Applicatios to the Aalysis of Iteractio i ANOVA, Joural of the America Statistical Associatio, 81, Boik, R. J. (1993), The Aalysis of Two-Factor Iteractios i Fixed Effects Liear Models, Joural of Educatioal ad Behavioral Statistics, 18, Boik s is a test of product iteractios, i.e., q ij c i d j where c = 0 = d. Note that these provide a basis for the iteractio space by pickig a 1 liearly idepedet α cotrasts ad b 1 liearly idepedet η cotrasts. But vectors T with this form geerate the etire iteractio space, ot just vectors of product form. Product form is ot closed uder vector additio. Write y ijk = µ + α i + η j + γ ij + ε ijk, i = 1,..., a, j = 1,..., b, k = 1,..., N ij. Fix j ad write a series of oe-way ANOVA models y ijk = (µ + η j ) + (α i + γ ij ) + ε ijk, i = 1,..., a, k = 1,..., N ij. 4

6 Whe N ij = N ij = N i /b for all i, j, j, we ca treat this as a multivariate oe-way model [Y 1,, Y b ] = XB + e. The coditio o the N ij is a special case of proportioal umbers. Note that X has N rows, whereas X has N /b rows. Let Z be a matrix whose colums are b 1 liearly idepedet sets of cotrast coefficiets. Thik of fittig the model Y Z = X(BZ) + ez. Our test for product iteractios is based o the test for group ( α ) effects i this oe-way model. The test statistic is H = Z Y M α Y Z where M α is the ppo for group effects i the oe-way. We could just use a stadard multivariate liear model test, like Roy s, or we could try to icorporate the fact that the colums i this multivariate liear model are idepedet ad homoscedastic to develop a eve better test. Roy s maximum F test should be very similar to what is i Boik, especially the 93 paper. Boik showed that his results hold for arbitrary sets of proportioal umbers, ot just the special case cosidered here. Icidetally, if we reverse the roles of i ad j we should get the same results provided N ij = N j /a, givig us two ways to achieve the eeded balace. Ackowledgemet My thaks to Robert for discussig his results with me. If I have misrepreseted his work, the resposibility is etirely mie. 5