# Two-way linear model with interaction but without one main effect

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## Two-way linear model with interaction but without one main effect

 Hi, I know that the type of model described in the subject line violates the principle of marginality and it is rare in practice, but there may be some circumstances where it has sense. Let's take this imaginary example (not homework, just a silly made-up case for illustrating the rare situation): I'm measuring the energy absorption of sports footwear in jumping. I have three models (S1, S2, S3), that are known by their having the same average value of this variable for different types of ground, but I want to model the energy absorption for specific ground types (grass, sand, and pavement). To fit the model I take 90 independent measures (different shoes, different users for each observation), with 10 samples per footwear model and ground type. # Example data: shoe <- gl(3,30,labels=c("S1","S2","S3")) ground <- rep(gl(3,10,labels=c("grass","sand","pavement")),3) Y <- rnorm(90,120,20) My model may include a main effect of the ground type, and the interaction shoe:ground, but I think that in this peculiar case I could neglect the main effect of shoe, since my initial hypothesis is that the average energy absorption is the same for the three models. My first thought was fitting the following model (with effect coding, so that the interaction coeffs have zero mean.): mod1 <- lm(Y ~ ground + ground:shoe,     contrasts=list(shoe="contr.sum",ground="contr.sum")) But this model has the same number of coefficients as a full factorial, and actually represents the same model subspace, isn't it? In fact, the marginal means are not the same for the three types of shoes: # Marginal means for my (random) example data > tapply(predict(mod1),shoe,FUN=mean)       S1       S2       S3 116.3581 121.0858 118.3800 If I'm not mistaken, to create the model that I want I can start with the full factorial model and remove the part associated to the main shoe effect: # Full model and its model matrix mod1 <- lm(Y~shoe*ground,     contrasts=list(shoe="contr.sum",ground="contr.sum")) X <- model.matrix(mod1) # Split X columns by terms X1 <- X[,1] X.shoe <- X[,2:3] X.ground <- X[,4:5] X.interact <- X[,6:9] # New model without method main effect mod2 <- lm(Y~X.ground+X.interact) For this model the marginal means do coincide: > tapply(predict(mod2),shoe,FUN=mean)      S1      S2      S3 118.608 118.608 118.608 My questions are: Is this correct? And is there an easier way of doing this? Thanks Helios De Rosario -- Helios de Rosario Martínez    Researcher INSTITUTO DE BIOMECÁNICA DE VALENCIA Universidad Politécnica de Valencia • Edificio 9C Camino de Vera s/n • 46022 VALENCIA (ESPAÑA) Tel. +34 96 387 91 60 • Fax +34 96 387 91 69 www.ibv.org   Antes de imprimir este e-mail piense bien si es necesario hacerlo. En cumplimiento de la Ley Orgánica 15/1999 reguladora de la Protección de Datos de Carácter Personal, le informamos de que el presente mensaje contiene información confidencial, siendo para uso exclusivo del destinatario arriba indicado. En caso de no ser usted el destinatario del mismo le informamos que su recepción no le autoriza a su divulgación o reproducción por cualquier medio, debiendo destruirlo de inmediato, rogándole lo notifique al remitente. ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.