What are the proper assumptions of Multinomial Logistic Regression? And what are the best tests to satisfy these assumptions using SPSS 18?
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The key assumption in the MNL is that the errors are independently and identically distributed with a Gumbel extreme value distribution. The problem with testing this assumption is that it is made a priori. In standard regression you fit the least-squares curve, and measure the residual error. In a logit model, you assume that the error is already in the measurement of the point, and compute a likelihood function from that assumption.
An important assumption is that the sample be exogenous. If it is choice-based, there are corrections that need to be employed.
As far as assumptions on the model itself, Train describes three:
The first assumption you mostly just have to defend in the context of your problem. The third is largely the same, because the error terms are purely random.
The second is testable to a certain extent, however. If you specify a nested logit model, and it turns out that the inter-nest substitution pattern is entirely flexible ($\lambda = 1$) then you could have used the MNL model, and the IIA assumption is valid. But remember that the log-likelihood function for the nested logit model has local maxima, so you should make sure that you get $\lambda =1$ consistently.
As far as doing any of this in SPSS, I can't help you other than suggest you use the
One of the most important practical assumptions of multinomial logistic is that the number of observations in the smallest frequency category of $Y$ is large, for example 10 times the number of parameters from the right hand side of the model.
gmacfarlane has been very clear. But to be more precise, and I assume you perform a cross section analysis, the core assumption is the IIA (independence of irrelevant alternatives). You can not force your data fit into the IIA assumption, you should test it and hope for it to be satisfied. Spss could not handle the test until 2010 for sure. R of course does it, but it might me easier for you to migrate to stata and implement the IIA tests provided by the mlogit postestimation commands.
If the IIA does not holds, mixed multinomial logit or nested logit are reasonable alternatives. The first one can be estimated within the gllamm, the second with the far more parsimonious nlogit command.