Which one to choose Type-I, Type-II, or Type-III ANOVA? I don't understand what the difference is between TypeI and TypeIII?
Since my background is not very mathematics, it's very difficult for me to understand this mathematical notation;
Type I SS: SS(A) SS(B|A) SS (AB|A, B) 
Type II SS: SS(A|B) SS(B|A) SS (AB|A, B) 
Type III SS: SS (A|B, AB) SS (B|A, AB) SS (A*B|A, B)?
I want to check the interaction between my two factors (genotype*FuncBIN), but my output comes as Type II ANOVA with interaction. Is this normal?
structure( list(genotype = structure(c(3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L), .Label = c("WT", "Mut2", "Mut1" ), class = "factor"), FuncBIN = structure(c(1L, 1L, 1L, 2L, 2L, 2L, 3L, 3L, 3L, 1L, 1L, 1L, 2L, 2L, 2L, 3L, 3L, 3L, 1L, 1L, 1L, 2L, 2L, 2L, 3L, 3L, 3L), .Label = c("Mitochondria ", "Golgi", "ER"), class = "factor"), k = c(0.2, 0.3, 0.4, 0.5, 0.3, 0.2, 0.5, 0.6, 0.7, 0.4, 0.6, 0.9, 0.2, 0.3, 0.5, 0.4, 0.2, 0.3, 0.5, 0.6, 0.8, 0.5, 0.4, 0.6, 0.2, 0.3, 0.1)), .Names = c("genotype", "FuncBIN", "k"), class = "data.frame", row.names = c(NA, -27L) )


library(car)
library(lsmeans)
library(multcomp)
fit <- lm(k ~ genotype + FuncBIN + genotype*FuncBIN, df) 
plot(fit) shapiro.test(residuals(fit)) 
Anova(fit) 
lsmip(fit, FuncBIN ~ genotype) 
lsmeans(fit, pairwise ~ genotype)[[2]]
fit.tukey <- lsmeans(fit, pairwise ~ genotype | FuncBIN)[[2]]
fit.tukey
cld(fit.tukey)

How can I run an ANOVA with interaction followed my Tukey PostHOC?
 A: In a balanced design, all three of these model types will yield the same results. With unbalanced data, you will get different results for the main effects but not the interaction term, assuming you only have a two-way interaction (as in your example). The general advice is to use a Type II sums-of-squares model for unbalanced designs. This model tests the main effects without the interaction term. The difference comes down to how you weight each mean or cell of the factorial. In a Type III model, each cell of the factorial gets the same weight (even if some have smaller sample sizes). In a Type II model, each observation gets the same weight. In most situations (but not all!) the latter is more defensible.
If you are only interested in the interaction, then this decision doesn't matter. The sum-of-squares for the interaction will be the same for all three model types.
I just checked and your design is balanced. As such you can use Type III SS. Built-in R functions will work just fine here.
fit <- aov(k ~ genotype*FuncBIN, data=df)
summary(fit)
TukeyHSD(fit)

