Estimating Multinomial Multilevel Logistic Models by Binomial models

Question

I would like to fit a multinomial multilevel logistic Model. Unfortunately I couldn't find a package that implements this. I tried Stata's gsem but it is very very slow and does not converge. This is why I was wondering if it is possible to estimate contrasting binomial models, combine them and approximate the multinomial-model.

So I came up with an approach but got stuck at the "combining" part. Two questions:

1. Is my approach correct/doable?
2. If yes, how do I calculate predictions for the base category

Approach:
The Data $ D $ has an dependent variable $ y \in \{1,2,3\} $.
I split my data in two subsets: $ D_{1,2} $ and $ D_{1,3} $. Where $D_{1,2}$ consists of all rows with $ y \in \{1,2\};$
$D_{2,3}$ of all where $y \in \{1,3\}$

Now I calculated the model $$ logit(p_i) = \ln\left(\frac{P(Y_i = 1)}{1-P(Y_i = 1)}\right)=\dots $$

With $i \in \{1,2\}$ corresponding to the Dataset $D_{1,i}$ and $\dots$ symbolizes the model-specification. Predicting this model I get predictions $\hat{p_i}$ meaning the likelihood of taking alternative $i$ over $1$.
Denoting $\hat{p_i}$ = pi^ the resulting Data-Set would look as follows:

y  p2^  p3^
1  0.2  0.1
1  0.7  0.8
2  0.6  .
3  .    0.1 
...

This leads to my second question:
If I want to predict $\hat y$, what do I have to do? And how do I calculate the probability of p1^ meaning the likelihood of taking 1 over 2 & 3.

I am thankful for any help!

Answer 1

Before giving up on the existing estimator I would first try to estimate the model with gsem after centering all your explanatory/independent/right-hand-side/x-variables. A multilevel model includes an extra model for the intercept (a normal distribution). You can imagine that estimating such a thing is a lot easier when the intercept is a quantity that can actually occur within your data.

If you want to investigate an alternative estimator you can look into:

Peter Haan and Arne Uhlendorf (2006) "Estimation of multinomial logit models with unobserved heterogeneity using maximum simulated likelihood" The Stata Journal, 6(2): 229-245. http://www.stata-journal.com/article.html?article=st0104

Answer 2

I used gsem to fit a multilevel multinomial logistic and worked perfectly. I use the MP version so this helps a lot. The hardest model took about 45 to run. The trick I used was to specify the option difficult, this savind about 10-15% of computing time. Another option is to change the integration method and the number of integration points. You may want to define starting levels as well. The manual has a lot of examples and you will surely find it helpful.