topic Re: Parameter Estimates for Mixing Probabilities in Proc FMM in Statistical Procedures

Parameter Estimates for Mixing Probabilities in Proc FMM

lynnlaurie — Thu, 23 May 2013 04:22:48 GMT

Greetings,

Does anyone know if it is possible to force SAS to show you all of the estimates for mixing probabilities in Proc FMM? It defaults to omitting the last estimate, and I'm interested in this because I would like to know the standard error for all of the estimates for mixing probabilities. Alternatively, does anyone know how SAS calculates the standard error for the parameter estimates for mixing probabilities?

Thank you in advance,

Lynn

Re: Parameter Estimates for Mixing Probabilities in Proc FMM

Rick_SAS — Thu, 23 May 2013 10:00:26 GMT

Since the sum of the mixing probabilities is 1, the last estimate is 1 - sum_of_other_estimates. If you fit a model with k components, there are only (k-1) parameters.

Re: Parameter Estimates for Mixing Probabilities in Proc FMM

lynnlaurie — Thu, 23 May 2013 13:47:40 GMT

Hi Rick. Thanks for the response. I understand how to find the mixing probability (p) of the omitted value, but I'm interested in the standard error for that value. Do you know how it is calculated or how I can force Proc FMM to show it?

Re: Parameter Estimates for Mixing Probabilities in Proc FMM

Rick_SAS — Fri, 24 May 2013 15:06:05 GMT

This is an interesting question. Unfortunately, I do not know the answer. I do not think that there is a way to get FMM to show the stderr directly. I am not knowledgable enough to know how to compute it from the output that FMM provides. Perhaps someone like or has thought about this.

Re: Parameter Estimates for Mixing Probabilities in Proc FMM

SteveDenham — Thu, 30 May 2013 18:54:40 GMT

Not sure, but since the parameterization is of a generalized logit, with the last predictor set to zero, try fitting with the NOINT option in the model statement. If there is a single independent variable, that should address the issue.

(I hope)

Steve Denham

Re: Parameter Estimates for Mixing Probabilities in Proc FMM

lvm — Thu, 30 May 2013 22:09:52 GMT

If you have only two mixing probabilities, p and 1-p, then you are luck. There is symmetry in the SEs because FMM models the mixing process using the symmetric logit link. For instance, if p = 0.2, then 1-p=0.8 (obviously). Logit(p) = -logit(1-p). With p=0.2, logit(.2) = -1.386, and logit(.8) = +1.386. The variances carry through in the same way. I just checked this by writing a NLMIXED program for a two-component mixture and getting p and 1-p from logit(p) and logit(1-p). The SEs come from the inverse Hessian matrix in FMM (the mixing probability is part of the likelihood).

You can get a hint of all of this by comparing FMM with GENMOD for a zero-inflated Poisson (a simple mixing problem). Both PROCs fit this model, but FMM represents the mixing probability by p, whileGENMOD represents the mixing probability by 1-p. But the estimates and SEs all agree.

If you have more than two mixing probabilities, then you can't take advantage of the symmetry. You would have to get 1-(p1+p2+...) by hand; the logit of this is easy, but use of the delta method to get the variance (and SE) would be tricky.

Re: Parameter Estimates for Mixing Probabilities in Proc FMM

lvm — Thu, 30 May 2013 22:12:00 GMT

And I have not seen any way to force FMM to print 1-p or logit(1-p).

Re: Parameter Estimates for Mixing Probabilities in Proc FMM

SteveDenham — Fri, 31 May 2013 13:03:59 GMT

Agreed. I have to say that I think PROC FMM is a great program, but very frustrating. Hopefully the next update will hit on some of these issues.

Steve Denham