for a multinomial treatment and binary outcome, what is more appropriate, ATC or ATE? I need help to choose between ATC and ATE for my analysis with multinomial treatment and binary outcome.
In the example below taken from here, it seems that ATT does not sound well for multinomial treatments because I have several focal groups (treated groups), and my interest is in comparing each of these focal groups with the control group. For the binary case, the concept of ATT weight is straightforward, not certain here.
To deal with this, I think either ATC or ATE would probably be more appropriate for the case of multinomial treatments. I favour ATC over ATE because I see it as a generalization of ATT, which usually is deemed more desirable in observational studies. Also, I "might not have good overlap (common support) between the two treatment groups", making ATE less reasonable than ATT (and thus ATC).
Using ATC would mean making group 1 the focal group
and I could do: 
w_atc <- w*ps.mat[,"1"] #ATC
instead of
w[group == i] <- 1/ps.mat[group == i, i] #ATE
or
w_att <- w*ps.mat[,"2"]
w_att <- w*ps.mat[,"3"]  #ATT
sample data and code:
library("nnet")
library("cobalt")

set.seed(42)
group <- factor(sample(c(1,2,3), 100, #needs to be a factor
                       replace=TRUE))
demo1 <- rnorm(100,100,25)
demo2 <- rpois(100,10)
demo3 <- rbinom(100,1,0.67)

df <- data.frame(group, demo1, demo2,demo3)

fit <- multinom(group ~ demo1 + demo2 + demo3, data = df)
#> # weights:  15 (8 variable)
#> initial  value 109.861229 
#> iter  10 value 107.354113
#> final  value 107.353911 
#> converged

ps.mat <- predict(fit, type = 'probs')

w <- rep(0, nrow(df)) #inititalize weights

for (i in levels(group)) {
    w[group == i] <- 1/ps.mat[group == i, i]
}

bal.tab(group ~ demo1 + demo2 + demo3, data = df, 
        weights = w, un = TRUE)

Thank you in advance for any help.
 A: As McCaffrey et al. point out (Statist. Med. 32: 3388-3414, 2013), the average treatment effect among the treated (ATT) has useful meaning in the multiple treatments case. It does require some careful thought about the specific hypotheses you wish to test, however:

In the multiple treatments setting, the definition of ATT depends on what is meant by ‘the treated’. This language, which is borrowed from the treatment versus control literature, requires more careful explanation when extended to the multiple treatments settings...In our example, with three treatment groups, there exist six ATTs: one for each pair of treatments, t' and t'', and either the population receiving t' or t'' as the one of interest.

ATTs are appropriate if you recognize that there is no single ATT in the multiple-treatment setting and the number of pairwise ATTs will grow rapidly with the number of treatments, so you will need to specify carefully the particular comparisons among groups that you wish to make. For pairwise ATTs (as opposed to pairwise ATEs) you will also have to recalculate propensity weights for each new pairwise comparison.
I don't see how the counterfactual average treatment effect among the controls (ATC) represents a useful "generalization" of ATT, certainly not in a way that resolves the problems arising from multiple treatment groups.
So if lack of common support among treatment groups makes ATE inappropriate, then pairwise ATTs are a way to proceed. The reference cited above has much useful information for this type of work. Although the authors come from the perspective of using generalized boosted models for determining weights, much of what they write is independent of the ways that weights are determined.
