[ADMB Users] SE Estimates including process error in ADMB

[Chris Gast]

Hello ADMB Users,

I would like to better understand how ADMB estimates standard errors for
random-effects models.  Specifically, I have a nonlinear age-harvest
product-multinomial model where I have assumed process parameters to be
random.  I am interested in these parameter estimates, and also the
reconstruction of animal abundance based on the parameter estimates and the
variability in such reconstructions.  To be thorough, I have written my two
related questions in LaTeX and attached them to this email.  In case the
attachment is scrubbed from the email for whatever reason, I am pasting the
LaTeX markup below (and I uploaded it here:
http://students.washington.edu/cgast/Var-s.pdf), so it can be reconstructed
if necessary.

Thanks in advance to anyone who is able to help out.

Chris Gast
Quantitative Ecology and Resource Management
University of Washington, Seattle, WA


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\begin{document}

\textbf{Problem}:  I would like to better understand how ADMB estimates
standard errors in random-effects models.

If I have $n$ years of data, and assume annual survival probability (for
purposes of a simple example) $\mu_s + \epsilon_i, i=1, \ldots, n$ of a
group of animals to be distributed as $N(\mu_s, \sigma^2_s)$, and I fit the
model with ADMB, I obtain an estimate $\hat{\mu}_s$ of $\mu_s$ and its
associated standard error.  I also obtain an estimate $\hat{\sigma}_s^2$ of
$\sigma_s^2$, and its associated standard error.

\textbf{Question 1}:  When computing the standard error of $\hat{\mu}_s$,
does ADMB use the estimate of process error, $\hat{\sigma}_s^2$, as in

\[
\hat{Var}(\hat{\mu}_s) = E(Var(\hat{\mu}_s | \mu_s)) + Var(E(\hat{\mu}_s |
\mu_s))
\]

and estimate this with

\[
\hat{Var}(\hat{\mu}_s | \mu_s) + \hat{\sigma}_s^2 ?
\]

(Is this what's reported in the .std file?)

\textbf{Question 2}: Similarly, if I estimate abundance $N_1$ directly as
$\hat{N}_1$, does ADMB use the estimate of process error if I compute
$\hat{N}_2 = \hat{N}_1 ( \hat{\mu}_s + \hat{\epsilon}_1)$, as in

\[ \begin{split} & Var(\hat{N}_2) = Var(\hat{N}_1 (\hat{\mu}_s +
\hat{\epsilon}_1)) \approx \\
& \left( \hat{\mu}_s + \hat{\epsilon}_1 \right)^2 Var(\hat{N}_1) +
\hat{N}_1^2 Var(\hat{\mu}_s) + \hat{N}_1^2 Var(\hat{\epsilon}_1) + \\
& 2 \hat{N}_1 (\hat{\mu}_s + \hat{\epsilon}_1) Cov(\hat{N}_1, \hat{\mu}_s) +
2 \hat{N}_1 \hat{\mu}_s Cov(\hat{N}_1, \hat{\epsilon}_1) + 2 \hat{N}_1^2
Cov(\hat{\mu}_s, \hat{\epsilon}_1)
\end{split}
\]

where sampling error for $Var(\hat{N}_1)$ and $Var(\hat{\epsilon}_1)$ are
estimated from the inverse-Hessian, $Var(\hat{\mu}_s)$ is estimated as
above, and one of the covariance terms involving the random parameter is
computed as

\[\begin{split}
& Cov(\hat{N}_1, \hat{\mu}_s) = E(Cov(\hat{N}_1, \hat{\mu}_s | \mu_s)) +
Cov(E(\hat{N}_1 | s), E(\hat{\mu}_s | \mu_s )) =\\
& \hat{Cov}(\hat{N}_s, \hat{\mu}_s) + Cov(N_1, \mu_s)
\end{split}
\]

where the first term comes from the inverse-Hessian, and the second term
would appear to depend on the specific model begin studied (probably zero in
this case, since $N_1$ is assumed to be a fixed parameter)?

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-----------------------------
Chris Gast
cmgast at gmail.com
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