New Contributor
Posts: 2

# Parameter Estimates for Mixing Probabilities in Proc FMM

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?

Lynn

SAS Super FREQ
Posts: 3,837

## Re: Parameter Estimates for Mixing Probabilities in Proc FMM

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.

New Contributor
Posts: 2

## Re: Parameter Estimates for Mixing Probabilities in Proc FMM

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?

SAS Super FREQ
Posts: 3,837

## Re: Parameter Estimates for Mixing Probabilities in Proc FMM

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.

Posts: 2,655

## Re: Parameter Estimates for Mixing Probabilities in Proc FMM

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

Valued Guide
Posts: 684

## Re: Parameter Estimates for Mixing Probabilities in Proc FMM

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.

Valued Guide
Posts: 684

## Re: Parameter Estimates for Mixing Probabilities in Proc FMM

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