[R] Two-way linear model with interaction but without one main effect

[ONKELINX, Thierry]

Dear Helios,

I think you rather want a mixed model with shoe as random effect.

library(lme4)
lmer(Y ~ Ground + (1|Shoe)) #the effect of shoe is independent of the ground effect
or
lmer(Y ~ Ground + (0 + Ground|Shoe)) #the effect of shoe is different per ground.

Best regards,

Thierry

ir. Thierry Onkelinx
Instituut voor natuur- en bosonderzoek / Research Institute for Nature and Forest
team Biometrie & Kwaliteitszorg / team Biometrics & Quality Assurance
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-----Oorspronkelijk bericht-----
Van: r-help-bounces op r-project.org [mailto:r-help-bounces op r-project.org] Namens Helios de Rosario
Verzonden: dinsdag 12 juni 2012 13:35
Aan: r-help op r-project.org
Onderwerp: [R] 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

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