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I want to simulate a multinomial logistic regression dataset. Suppose you have 1000 data points, the first 60% belong to the reference group 1, the next 30% belong to group 2 and the remaining 10% belong to group 3. How is it possible to recover this classification through the predicted probabilities of a multinomial logit?

My approach was to construct a specific combination of explanatory variables and coefficients that allow the classification probabilities to be more or less unambiguous. However, I did not succeed as the probabilities to belong to the reference group are too washed out, resulting in a lot of misclassified data-points.

My approach in R:

i <- 1000 #number of individuals
p <- c(0.6,0.3,0.1) #group proportions

#group 1 design matrix
X1 <- mvtnorm::rmvnorm(p[1]*i, c(0,0,0))
#group 2 design matrix
X2 <- rmvnorm(p[2]*i, c(10,0,0))
#group 3 design matrix
X3 <- rmvnorm(p[3]*i, c(0,10,0))
X = rbind(X1,X2,X3)

#true coefficient vectors
vCoef1 = rep(0,3) #reference group 1
vCoef2 = c(1,0,0) #group 2
vCoef3 = c(0,1,0) #group 3

# vector of probabilities
vProb = cbind(exp(X%*%vCoef1), exp(X%*%vCoef2), exp(X%*%vCoef3))

# standardize probabilities
vProb = vProb/rowSums(vProb)

# multinomial draws
Cluster <- apply(vProb, 1, function(x) sample(1:3, 1, prob = x))

# group sizes from multinomial logit predicted probabilities
tabulate(Cluster)/length(Cluster)
[1] 0.189 0.515 0.296

As you can see, the clustering from the multinomial logit is not even close to the group proportions as given in line 2. The problem is that the predicted probabilities for the reference group individuals are all over the place.

To clarify: I do not just want to recover the coefficients like in this thread (which works perfectly fine), I want to recover the coefficients AND the clustering.

Thank you very much!

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    $\begingroup$ I wonder if this question relates to clustering at all. In what sense do you use word "clustering". It seems to not cluster analysis (implied by the tag). $\endgroup$
    – ttnphns
    Nov 14, 2018 at 13:10

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