R help - Feb 2010 - Estimated Standard Error for Theta in zeroinfl()

Dear R Users,

When using zeroinfl() function to fit a Zero-Inflated Negative Binomial (ZINB)
model to a dataset, the summary() gives an estimate of log(theta) and its
standard error, z-value and Pr(>|z|) for the count component. Additionally,
it also provided an estimate of Theta, which I believe is the exp(estimate of
log(theta)).

However, if I would like to have an standard error of Theta itself (not the
SE.logtheta), how would I obtain or calculate that standard error?

Thank you very much for your time.

Best regards,
Tzeng Yih Lam

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PhD Candidate
Department of Forest Engineering, Resources and Management
College of Forestry
Oregon State University
321 Richardson Hall
Corvallis OR 97330 USA
Phone: +1.541.713.7504
Fax: +1.541.713.7504
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On 15/02/2010, at 12:49 PM, Lam, Tzeng Yih wrote:
> Dear R Users,
> 
> When using zeroinfl() function to fit a Zero-Inflated Negative Binomial
(ZINB) model to a dataset, the summary() gives an estimate of log(theta) and its
standard error, z-value and Pr(>|z|) for the count component. Additionally,
it also provided an estimate of Theta, which I believe is the exp(estimate of
log(theta)).
> 
> However, if I would like to have an standard error of Theta itself (not the
SE.logtheta), how would I obtain or calculate that standard error?

It's tricky.  The exp and log functions aren't linear!!!
So nothing really works very well.

To start with, if you have an unbiased estimate of log(theta),
say lt.hat, then exp(lt.hat) is NOT an unbiased estimate of theta.

If you are willing to assume that lt.hat is normally distributed
then the expected value of exp(lt.hat) is theta*exp(sigma^2/2)
where sigma^2 is the variance of lt.hat.

The variance of exp(lt.hat) is 

	(*) theta^2 * exp(sigma^2) * (exp(sigma^2) - 1).

You can ``plug in'' the SE of lt.hat for sigma into the foregoing and
get
an ***``approximately''*** unbiased estimate of theta:

	theta.hat = exp(lt.hat - SE^2/2)

and then an approximate estimate of the variance of this ``theta.hat''
(by plugging in theta.hat for theta and SE for sigma in (*)).  The results
won't be correct, but they'll probably be in the right ball park.  I
think!

This is all posited on the distribution of the estimate of log(theta) being
normal (or ``Gaussian'').  Whether this is a justifiable assumption in
your
setting is questionable.

Some simulation experiments might be illuminating.

	cheers,

		Rolf Turner

P. S.  The formulae I gave about are the result of quickly scribbled
calculations, and could be wrong.  They should be checked.

		R. T.
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On Sun, 14 Feb 2010, Lam, Tzeng Yih wrote:
> Dear R Users,
>
> When using zeroinfl() function to fit a Zero-Inflated Negative Binomial 
> (ZINB) model to a dataset, the summary() gives an estimate of log(theta) 
> and its standard error, z-value and Pr(>|z|) for the count component. 
> Additionally, it also provided an estimate of Theta, which I believe is 
> the exp(estimate of log(theta)).
As maximum likelihood estimation is employed, this does not matter for 
point estimation. theta is the ML estimator for theta and log(theta) is 
the ML estimator for log(theta). I don't think that there is an 
unibiasedness result for either one, but both are consistent (if the model 
is correctly specified).

What is done internally in zeroinfl() is that log(theta) is employed which 
is a standard approach for numeric optimization of positive parameters.
> However, if I would like to have an standard error of Theta itself (not 
> the SE.logtheta), how would I obtain or calculate that standard error?
You can do so by means of the delta method which is rather simple in this 
case: The standard error of theta is: theta * SE(logtheta). Thus, if obj 
is a fitted "zeroinfl" object:
   ## theta
   obj$theta
   ## associated standard error
   obj$theta * obj$SE.logtheta

Whether this is very useful is another story, of course...see also Rolf's 
remarks.
Z
> Thank you very much for your time.
>
> Best regards,
> Tzeng Yih Lam
>
>
------------------------------------------------------------------------------
> PhD Candidate
> Department of Forest Engineering, Resources and Management
> College of Forestry
> Oregon State University
> 321 Richardson Hall
> Corvallis OR 97330 USA
> Phone: +1.541.713.7504
> Fax: +1.541.713.7504
>
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