# MANOVA multiple comparisons with equivalence testing

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?

1. Is multiple Hotelling $T^2$ tests along with Bonferroni correction or FDR/pFDR appropriate? FDR/pFDR q values would be preferable as the $\beta$ value is important here.

2. Any suggestions for R packages to do the same? (Particularly for MANOVA post-hoc multiple comparisons}

3. 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)

• I think for the equivalence tests you mean $H^{j}_{0}: |\mu^{j}_{1}-\mu_{2}^{j}| \ge \delta$? As in $H^{j}_{01}: \mu^{j}_{1}-\mu_{2}^{j} \ge \delta$ –or– $H^{j}_{02}: \mu^{j}_{1}-\mu_{2}^{j} \le -\delta$ along the lines of two one-sided tests, yes? Mar 12 '15 at 19:45
• @Alexis you are right. Have edited the question accordingly. Mar 13 '15 at 7:34

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 the MOats dataset that accompanies the package.
It also supports equivalence tests via providing a delta argument in summary or test. A section of the vignette (see vignette("using-lsmeans")) covers equivalence testing.
• @rvl But there is an option to do pairwise comparison of least square means derived from a multivariate model in lsmeans. See the edit to the question for the code. Mar 13 '15 at 7:37