Poisson with Normal Residuals
Message/Author
 Paul Silvia posted on Friday, May 09, 2008 - 2:31 am
Good morning. The Web Talk for Mplus 5.1's new count options describes a Poisson model with normal residuals, created by defining a random slope fixed at 1 for all participants. Could you point me to places where this has been used or discussed? It's an interesting model. Cheers, Paul
 Bengt O. Muthen posted on Friday, May 09, 2008 - 5:58 am
I don't know that I've seen that explicitly laid out in the literature, but a general discussion related to this is in Land, McCall, Nagin (1996) A comparison of Poisson, Negative Binomial, and Semiparametric Mixed Poisson Regression Models in Sociological Methods & Research, 24, 387-442.
 Girish Mallapragada posted on Monday, October 06, 2008 - 6:54 am
Does Mplus 5.1 also include negative binomial regression models?
 Bengt O. Muthen posted on Monday, October 06, 2008 - 7:06 am
Yes. See Mplus Web Talks on our home page and also the Version 5.1 Addendum on our web site.
 Nick Roshon posted on Wednesday, October 26, 2011 - 4:19 pm
In my model I have two dependent variables-- one count and one binary. My count variable is not zero-inflated but has variance > mean, so I was thinking I should use negative binomial rather than poisson. However, I would like to include a covariance between the count and binary DVs (given that they may share non-specified predictors) using the following syntax:
cov BY binary countvar;
cov@1; [cov@0];
in which the loading for countvar would indicate the covariance. Is it appropriate to estimate this covariance, in addition to the variance (i.e. dispersion) parameter for the count variable?

The reason I ask is because I noticed that the loading (covariance estimate) is significant when I use poisson for countvar, but is not significant when I use NB (the dispersion parameter is significant). I am not sure what to make of this difference.

I am also considering using poisson with a normal residual as an alternative to NB for the count variable, but am having trouble understanding how to specify it (i.e., what is the Mplus "trick" that is referenced in the web talk on count variables). Do you have an example of syntax for poisson with a normal residual?
 Bengt O. Muthen posted on Wednesday, October 26, 2011 - 4:53 pm
The trick to get a normal residual for Poisson is at the bottom of this message. It uses a random slope s for a constant 1. I am not sure how reasonable it is to assume a normal residual. Negbin essentially assumes a residual but with a very different distribution, exp(residual) having a Gamma distribution.

Perhaps your residual covariance approach of using the "cov" factor giving a significant covariance for Poisson but not for negbin is due to the Poisson part thereby getting a zero-inflation from cov, while negbin doesn't need it. Perhaps there isn't a need for the residual covariance. It is a little strange to use negbin with the added cov factor because negbin already has a residual and the residual covariance is specified for a second, normal residual.

Here is that input for the marital affairs data:

count = naffairs(p);

usevar are naffairs kids vryhap hapavg
avgmarr vryrel smerel slghtrel notrel
yrsmarr3 yrsmarr4 yrsmarr5 yrsmarr6 one;

define:
if (yrsmarr==4) then yrsmarr3=1 else yrsmarr3=0;
if (yrsmarr==7) then yrsmarr4=1 else yrsmarr4=0;
if (yrsmarr==10) then yrsmarr5=1 else yrsmarr5=0;
if (yrsmarr==15) then yrsmarr6=1 else yrsmarr6=0;
one = 1;

data: file is affairs1.dat;
variances = nocheck;

analysis:
type = random;
estimator=ml;

model:

naffairs on kids-yrsmarr6 (p1-p12);
s| naffairs on one;
[s@0];
s*.5;