<font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">Hello ADMB Users,</span></font><div style="border-collapse: collapse; font-family: arial, sans-serif; font-size: 13px; ">
<br></div><div style="border-collapse: collapse; font-family: arial, sans-serif; font-size: 13px; ">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: <a href="http://students.washington.edu/cgast/Var-s.pdf" target="_blank" style="color: rgb(17, 37, 8); ">http://students.washington.edu/cgast/Var-s.pdf</a>), so it can be reconstructed if necessary.</div>
<div style="border-collapse: collapse; font-family: arial, sans-serif; font-size: 13px; "><br></div><div style="border-collapse: collapse; font-family: arial, sans-serif; font-size: 13px; ">Thanks in advance to anyone who is able to help out.</div>
<div style="border-collapse: collapse; font-family: arial, sans-serif; font-size: 13px; "><br></div><div style="border-collapse: collapse; font-family: arial, sans-serif; font-size: 13px; ">Chris Gast</div><div style="border-collapse: collapse; font-family: arial, sans-serif; font-size: 13px; ">
Quantitative Ecology and Resource Management</div><div style="border-collapse: collapse; font-family: arial, sans-serif; font-size: 13px; ">University of Washington, Seattle, WA</div><div style="border-collapse: collapse; font-family: arial, sans-serif; font-size: 13px; ">
<br></div><div style="border-collapse: collapse; font-family: arial, sans-serif; font-size: 13px; "><br></div><div style="border-collapse: collapse; font-family: arial, sans-serif; font-size: 13px; ">%%%%%%%%%%%%%%% begin LaTeX markup</div>
<div style="border-collapse: collapse; font-family: arial, sans-serif; font-size: 13px; "><br></div><div><div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">\documentclass{article}</span></font></div>
<div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">\usepackage{amsmath}</span></font></div><div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">\setlength{\parindent}{0pt}</span></font></div>
<div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">\setlength{\parskip}{2ex plus 0.5ex minus 0.2ex}</span></font></div><div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;"><br>
</span></font></div><div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">\begin{document}</span></font></div><div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;"><br>
</span></font></div><div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">\textbf{Problem}: I would like to better understand how ADMB estimates standard errors in random-effects models.</span></font></div>
<div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;"><br></span></font></div><div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">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. </span></font></div>
<div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;"><br></span></font></div><div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">\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</span></font></div>
<div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;"><br></span></font></div><div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">\[</span></font></div>
<div><span class="Apple-tab-span" style="white-space:pre"><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;"> </span></font></span><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">\hat{Var}(\hat{\mu}_s) = E(Var(\hat{\mu}_s | \mu_s)) + Var(E(\hat{\mu}_s | \mu_s))</span></font></div>
<div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">\]</span></font></div><div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;"><br>
</span></font></div><div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">and estimate this with</span></font></div><div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;"><br>
</span></font></div><div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">\[</span></font></div><div><span class="Apple-tab-span" style="white-space:pre"><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;"> </span></font></span><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">\hat{Var}(\hat{\mu}_s | \mu_s) + \hat{\sigma}_s^2 ?</span></font></div>
<div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">\]</span></font></div><div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;"><br>
</span></font></div><div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">(Is this what's reported in the .std file?)</span></font></div><div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;"><br>
</span></font></div><div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">\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</span></font></div>
<div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;"><br></span></font></div><div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">\[ \begin{split} & Var(\hat{N}_2) = Var(\hat{N}_1 (\hat{\mu}_s + \hat{\epsilon}_1)) \approx \\</span></font></div>
<div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">& \left( \hat{\mu}_s + \hat{\epsilon}_1 \right)^2 Var(\hat{N}_1) + \hat{N}_1^2 Var(\hat{\mu}_s) + \hat{N}_1^2 Var(\hat{\epsilon}_1) + \\</span></font></div>
<div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">& 2 \hat{N}_1 (\hat{\mu}_s + \hat{\epsilon}_1) Cov(\hat{N}_1, \hat{\mu}_s) + 2 \hat{N}_1 \hat{\mu}_s Cov(\hat{N}_1, \hat{\epsilon}_1) + 2 \hat{N}_1^2 Cov(\hat{\mu}_s, \hat{\epsilon}_1)</span></font></div>
<div><span class="Apple-tab-span" style="white-space:pre"><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;"> </span></font></span><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">\end{split}</span></font></div>
<div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">\]</span></font></div><div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;"><br>
</span></font></div><div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">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</span></font></div>
<div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;"><br></span></font></div><div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">\[\begin{split}</span></font></div>
<div><span class="Apple-tab-span" style="white-space:pre"><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;"> </span></font></span><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">& Cov(\hat{N}_1, \hat{\mu}_s) = E(Cov(\hat{N}_1, \hat{\mu}_s | \mu_s)) + Cov(E(\hat{N}_1 | s), E(\hat{\mu}_s | \mu_s )) =\\</span></font></div>
<div><span class="Apple-tab-span" style="white-space:pre"><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;"> </span></font></span><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">& \hat{Cov}(\hat{N}_s, \hat{\mu}_s) + Cov(N_1, \mu_s)</span></font></div>
<div><span class="Apple-tab-span" style="white-space:pre"><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;"> </span></font></span><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">\end{split}</span></font></div>
<div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">\]</span></font></div><div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;"><br>
</span></font></div><div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">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)?</span></font></div>
<div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;"><br></span></font></div><div><font class="Apple-style-span" face="arial, sans-serif"><span class="Apple-style-span" style="border-collapse: collapse;">\end{document}</span></font></div>
</div><div style="border-collapse: collapse; font-family: arial, sans-serif; font-size: 13px; "><br></div><div style="border-collapse: collapse; font-family: arial, sans-serif; font-size: 13px; ">%%%%%%%%%%%%%%% end LaTeX markup</div>
<br>-----------------------------<br>Chris Gast<br><a href="mailto:cmgast@gmail.com" target="_blank">cmgast@gmail.com</a><br>