Find the linear model of a latin square design - r

Question

I want to get the equation of the linear model for the following experiment mat in latin square.
data <- c(12.5,11,13,11.4)
row <- factor(rep(1:2,2))
col <- factor(rep(1:2,each=2))
car <- c("B","A","A","B")
mat <- data.frame(row,col,car,data)
mat
# row col car data
# 1 1 1 B 12.5
# 2 2 1 A 11.0
# 3 1 2 A 13.0
# 4 2 2 B 11.4

Answer 1

I might recommend using a mixed model approach to this.
mat <- data.frame(data=c(12.5,11,13,11.4),
row=factor(rep(1:2,2)),
col=factor(rep(1:2,each=2)),
car=c("B","A","A","B"))
I'm using lmerTest because it will more easily provide you with (approximate) p-values
By default anova() uses the Satterthwaite approximation, or you can tell it to use the more accurate Kenward-Roger approximation. In either case you can see that the denominator df are exactly, or nearly zero, and the p-value is either missing or very close to 1, indicating that your model doesn't make sense (i.e. even using the mixed model it's overparameterized).
library("lmerTest")
anova(m1 <- lmer(data~car+(1|row)+(1|col),data=mat))
anova(m1,ddf="Kenward-Roger")
## Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
## car 0.0025 0.0025 1 9.6578e-06 2.0019 0.9999
Try for a bigger design:
set.seed(101)
mat2 <- data.frame(data=rnorm(36),
row=gl(6,6),
col=gl(6,1,36),
car=sample(LETTERS[1:2],size=36,replace=TRUE))
m2A <- lm(data~car+row+col,data=mat2)
anova(m2A)
## (excerpt)
## Df Sum Sq Mean Sq F value Pr(>F)
## car 1 1.2571 1.25709 1.6515 0.211
m2B <- lmer(data~car+(1|row)+(1|col),data=mat2)
anova(m2B)
## Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
## car 1.178 1.178 1 17.098 1.56 0.2285
anova(m2B,ddf="Kenward-Roger")
## Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
## car 1.178 1.178 1 17.005 1.1029 0.3083
It surprises me a little bit that the lm and lmerTest answers are so far apart here -- I would have thought this was an example where there was a well-formulated "classic" answer -- but I'm not sure. Might be worth following up on CrossValidated or Google.

Answer 2

fit <- lm(data~row+col+car,mat)
coef(fit)
# (Intercept) row2 col2 carB
# 12.55 -1.55 0.45 -0.05
So the effect of the row factor is -1.55, the effect of the col factor is 0.45, and the effect of the car factor is -0.05. The intercept term is the value of data expected when al the factors are at the first level (row=1, col=1, car=A).
Notice that your design is over-specified: you have only 4 pieces of data, which is enough to specify the effects of two factors and their interaction, but you have set it up so that car is the interaction. So there are no degrees of freedom left for error.