For a MANOVA
with $n$ variables, I would like to do pairwise comparisons between $k$ levels for one of the variables.
What is the suitable method to adopt for this while adjusting $\alpha$ for the $k(k-1)$ multiple comparisons?
-
Is multiple
Hotelling
$T^2$ tests along withBonferroni
correction or FDR/pFDR appropriate? FDR/pFDR q values would be preferable as the $\beta$ value is important here. -
Any suggestions for
R
packages to do the same? (Particularly for MANOVA post-hoc multiple comparisons} -
How to test the null hypothesis $H_0^j:|\mu_1^j-\mu_2^j|\ge\delta$ instead of $H_0^j:\mu_1^j=\mu_2^j$ as in an equivalence test for the multiple comparisons?
Edit
Based on the answer and further comment by rvl, I was able to explore and come up with the following.
library(lsmeans)
# Use the `oranges` dataset in `lsmeans` package.
# multivariate linear model
oranges.mlm <- lm(cbind(sales1,sales2) ~ price1 + price2 + day + store,
data = oranges)
# Get the least square means
oranges.Vlsm <- lsmeans(oranges.mlm, "store")
# Multiple comparisons with fdr p value adjustment
test(contrast(oranges.Vlsm, "pairwise"), side = "=", adjust = "fdr")
# With threshold spcified
test(contrast(oranges.Vlsm, "pairwise"), side = "=", adjust = "fdr", delta = 0.25)
Best Answer
For an R package, you might take a look at lsmeans. For
mlm
models, it sets up the multivariate response as if it were a factor whose levels are the dimenstions of the response. Then you can do estimates or contrasts of those, with or without other factors being involved. See the example for theMOats
dataset that accompanies the package.It also supports equivalence tests via providing a
delta
argument insummary
ortest
. A section of the vignette (seevignette("using-lsmeans")
) covers equivalence testing.