[R-sig-ME] Understanding and analysing split split design using lmer()

[Rabin KC]

Hello R mixed models community,

I am trying to fit a model for a split- split-plot design using lmer. I
know how to do this with aov and ssp.plot function in agricolae, but I need
to use lmer in this case.

I am interested in the response variable biomass of cover crops. I have the
main plots as crops (corn and soybean). The subplot is tillage, which is
randomized within the crops and has 3 levels (conventional-till, no-till,
and strip-till). Within tillage(sub plot), 3 cover crop strategies are
randomly assigned (AR, ARCC, ARCCFR). Therefore 1 rep (block) has 18
treatments. The whole experiment is replicated 4 times, therefore, total
experimental units equal to 72 units.

Data arranged in a csv file is attached in this email.

Now the analysis part:

I know how to do this using aov/ ssp.plot( agricolae). The code is as
follows:

model<-with(data,ssp.plot(rep,crop,tillage,cover,biomass)).
Same thing with aov:

model1<aov(cover_coverage~rep+crop*tillage*cover+Error(rep/crop/tillage),data=data)

For my  analysis using lmer, and lme,  I have the following code:

model2<-lmer(biomass~crop*tillage*cover+ (1|rep),data=data)

model3<-lme(fixed=biomass~crop*tillage*cover, data=data, random=~1|rep)

Now my confusion and questions are:
1. I have used crop, tillage, and cover as fixed effects. And only the rep
(block) as random. Is the random effect properly assigned for this design?

2. I have read a lot about crossed and nested design. Crawley and Oehlert
give particular examples about it, but I have trouble understanding if this
design has nested factors or crossed factors.

I would be very relieved to receive feedback with the matter in hand.

Looking forward, and thank  you  a lot,

Sincerely,
Rabin