## Multinomial logistic & VPC

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pimverbunt
Posts: 11
Joined: Wed May 06, 2015 9:22 am

### Multinomial logistic & VPC

Dear all,

I am modelling a multinomial logistic regression and I came across some problems while calculating the Variance Partition Coefficient (VPC). In the binary logistic multilevel case, the VPC is easily calculated through the simulation method, written out on page 133 of the manual. The macro does not work in the case of the multinomial model (as mlwin models it as a three level model). I am wondering if Goldstein, Browne and Rasbash (2002) simulation method can be extended for the multinomial case through the following method:

1. use the higher level variances (of the categories of the dependent variable) to simulate a large number m of higher level residuals from N(0,s²(k)) for each category of the dependent variable. As the dependent variables has k categories, I will do this k-1 times.
2. For each category of the dependent variable, calculate the probability (Pk) for some values of the independent variables. Similarly, calculate the corresponding level 1 variance for each category: V1k=Pk(1-Pk). Is the formula for the variance correct in the multinomial case?
3. From here on it is straightforward to calculate the VPC for each category of the dependent variable.

Would this be a correct approach to model the VPC for each category? Is it sensible to do? And does it also hold if I’m modeling in MCMC?

Kind regards,

Pim Verbunt
pimverbunt
Posts: 11
Joined: Wed May 06, 2015 9:22 am

### Re: Multinomial logistic & VPC

Please note that I simulate k-1 times a large number of m, based on the variance-covariance matrix that was estimated in my model. The country level residuals, which I use to calculate the probability of belonging to the different categories of the dependent variable, are, in my specific case, correlated with each other.
Harvey Goldstein
Posts: 49
Joined: Sun Sep 06, 2009 5:30 pm

### Re: Multinomial logistic & VPC

What you suggest seems quite sensible to me and would apply whether you used IGlS or MCMC estimation. There is an analogous procedure for ordered data. You also might consider partitioning the covariances (correlations).