Chris Gast cmgast at gmail.com
Tue Mar 15 17:07:59 PDT 2011

```Perhaps more simply, if I add the function (as a sdreport_number variable)

g = exp(t1)

where t1 is a RE term, ADMB computes

g = exp(0.032993) = 0.0057166

and estimates (in the .std and .cor files)

SE(g) = 0.057166

where a standard delta-method approximation would yield

Var(g) \approx (dg/dt1)^2 * var(t1) = exp(2*0.032993) * (0.0070425)^2 =
5.297957e-05 --> SE(g) = 0.007278707

So clearly, I must be doing something wrong, or have some misconception
about what is reported in the .std and .cor files for RE terms.  Any
thoughts?

Thanks,

Chris Gast

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

On Tue, Mar 15, 2011 at 12:41 PM, Chris Gast <cmgast at gmail.com> wrote:

> Hello again,
>
>>
> I'm still not sure what I think ADMB is doing and what it is actually doing
> are the same thing, and I think it's related to the second term on the rhs
> of eqn (3.1) in the ADMB-RE manual.
>
>>
> As an example, I have a function
>
>>
> p1 = 1-exp((cmu+t1)*f1)/(1+exp((cmu+t1)*f1))
>
>>
> where cmu is a parameter, f1 is a datum, and t1 is a random effect. I have
> added the function p1 as a sdreport_number in ADMB, and am trying to
> recompute the variance estimate via the estimates in the .cor file myself (I
> need to know what ADMB is doing, so I can use these estimates in additional
> approximations). I cannot seem to recreate ADMB's variance estimate for p1.
>  I have attached a trimmed version of the .cor file (trimmed to relevant
> portions to keep file size down) and some commented R code wherein I'm
> trying to verify the SE estimate for p1.  Would anyone who be willing to
> check my variance approximation, and see what I'm leaving out with respect
> to what ADMB is doing? Either I've made a dumb mistake, or ADMB is not doing
> what I think it's doing.
>
>
> Calculations like this seem to work (ADMB output can be verified) if RE
> terms are omitted, such as the meanp parameter function (noted in the R and
> .cor file), which is the same as p1 but with t1=0 and f1=1.4; in addition,
> my variance estimate for functions with REs tend to be lower than those from
> ADMB. These two facts may indicate that I probably don't handle the second
> term in eqn (3.1) of the ADMB-RE manual correctly. My understanding is that
> the SE value for a RE term (such as t1) is already computed as in eqn (3.1)
> in the ADMB-RE manual, and that is what is presented in the .std and .cor
> files. Perhaps this is not the case? Or perhaps there is something wrong
> with backwards-computing Cov(cmu,t1) = corr(cmu,t1)*SE(cmu)*SE(t1), where
> the three terms on the rhs are all found in the .cor file? I could do some
> testing of these theories if I could obtain the individual VCV matrix
> components from Dave Fournier's previous messages on this subject, but I
> don't see how that can be done (besides L_xx^{-1}, of course).
>
>>
> Thanks very much,
>
>>
> Chris Gast
>
>
>
>
>
>
> -----------------------------
>
>> Chris Gast
>
>> cmgast at gmail.com
>
>>
>>
>> On Tue, Nov 16, 2010 at 4:21 AM, dave fournier <davef at otter-rsch.com>wrote:
>>
>>> Chris Gast wrote:
>>>
>>> I think this is discussed in the paper Hans and I wrote.
>>> anyway since  uhat(x) is the value of uhat which maximizes L(x,u)
>>> It follows that
>>>
>>>  L_u(x,uhat(x))=0  for all x so differentiating wrt x you get
>>>
>>>  L_xu + L_uu uhat'(x)=0   so that
>>>
>>>  uhat'(x) = - L_uu^{-1} * L_xu
>>>
>>>  If there are n x's and m u's then uhat'(x) is an n x m matrix.
>>>
>>>  Thanks again, Dave.  Could you clarify what the general entry of the
>>>> gradient uhat'(xhat) looks like?  I had thought this would be dx_i/du_j, but
>>>> I think these would all be 1's or 0's (since, for instance , Shat_1 = shat +
>>>> uhat_1).  It is likely that I am misunderstanding something.
>>>>
>>>>
>>>> Thanks,
>>>>
>>>> Chris
>>>>
>>>>
>>>>
>>>>
>>>> -----------------------------
>>>> Chris Gast
>>>> cmgast at gmail.com <mailto:cmgast at gmail.com>
>>>>
>>>>
>>>>
>>>> On Mon, Nov 15, 2010 at 7:00 PM, dave fournier <davef at otter-rsch.com<mailto:
>>>> davef at otter-rsch.com>> wrote:
>>>>
>>>>    OK,
>>>>
>>>>    there are inly ar efew thing to play with.  for notation let
>>>>
>>>>    F(x,u) be the likelihood for x and u.
>>>>
>>>>    let
>>>>
>>>>      L(x) = int F(x,u) du
>>>>
>>>>    be what we get after integrating out u either exactly or by the
>>>>    laplace approx.
>>>>
>>>>    and let xhat uhat(xhat) be the miximizing values.
>>>>
>>>>    We have   the Hessians L_xx(xhat), and F_uu(xhat,uhat(xhat)
>>>>    Then the covariance matrix for x,u is assumed to be
>>>>
>>>>        L_xx^{-1}                        L_xx^{-1} * uhat'(x)
>>>>
>>>>         uhat'(xhat)*L_xx^{-1}    F_uu^{-1} +
>>>>    uhat'(xhat)*L_xx^{-1}*uhat'(x)
>>>>
>>>>    The idea is that u-uhat(xhat) is independent of xhat  i.e the only
>>>>    correlation is between
>>>>    xhat and uhat(x)
>>>>
>>>>    Put in transposes where necessary.
>>>>
>>>>    For any function of x,u the covariance is computed by the delta
>>>>    method.
>>>>
>>>>
>>>>
>>>>
>>>>
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>>>>