RE: Rép. : RE: Objet : RE: MCSim beta

Hi Fredrik,

Yes, sending through help-mcsim is good. I can also put a copy there (I should think of it).

For question 2, I think that the answer would be the same. You have a mean and variance for
a variance term and you want to assign a prior for further updating of that variance. Using the formula 
below you could still 
derive a inv-shape and scale giving you an inverse-gamma of the desired mean and var.

Frederic

>>> "F Jonsson" <Fredrik.Jonsson <at> farmbio.uu.se> mardi 2 septembre 2003 10:27:04 >>>

Frédéric,

Thank you for that information. Of course, a few weeks have gone since I
posed my questions, and I suspected that a response from you would take
some time, so I sidestepped the problem by assigning straight lognormal
rather than inverse gamma priors. 

However, it seems that I did not really ge my point across with regards
to question 2: I was referring to the inverse gamma prior I wanted to
assign, not the fit of my posterior to an inverse gamma. But your
response still gave me some clues: I guess I could create a normal
distribution that corresponds to the values from the previous modeling,
fit that to an inverse gamma and then take the shape parameters from
that fit. Is that the way to do it?

By the way, would you rather like me to pose these questions to the
mcsim-list? Judging from the archives, it seems kind of dormant at the
moment. 

Kind regards, 

Fredrik 

-----Original Message-----
From: Frédéric BOIS [mailto:Frederic.Bois <at> ineris.fr] 
Sent: den 1 september 2003 16:49 
To: Fredrik.Jonsson <at> farmbio.uu.se 
Subject: Rép. : RE: Objet : RE: MCSim beta

Hi Fredrik,

I am just back from vacations.
1. I would use LogNormal_v for the distribution. This allows you to use
a variance (in logscale)
as the second parameter. The variance should be anything above 0.
2. Your prior is inverse gamma and your posterior is inverse gamma only
if you have a normal linear
model at the lower stages, which does not seem to be the case. So one
way to answer your question is :
forget the inverse gamma for the posterior and just report an histogram
and some summary statistics.
If you want to fit an inverse-gamma to your posterior distribution
(assuming it's good approximation), I
would simply use the relationships between shape and inverse-scale and
expectation and variance:
Inv-G(x | alpha, beta): 
E(x) = beta / (alpha -1)
V(x) = beta * beta / ( (alpha - 1) * (alpha - 1) * (alpha - 2) ) 

using the empirical mean and variances for E(x) and V(x). That gives

V = E * E / (alpha - 2) => alpha = 2 + E * E / V
and beta = E * (1 + E * E / V)

(check that, though, I did it on the fly)

I would still compare the "fitted" inv-g to the histogram to see whether
the fit is reasonnable.

F

==========================
Frederic Y. Bois,
Unite de Toxicologie Experimentale, responsable
INERIS
Parc ALATA, BP 2
5, rue Taffanel
60550 Verneuil en Halatte
FRANCE
tel: + 33 (0)3 44 55 65 96
fax: + 33 (0)3 44 55 66 05
email: frederic.bois <at> ineris.fr 
web: http://www.ineris.fr, http://toxi.ineris.fr 

>>> "F Jonsson" <Fredrik.Jonsson <at> farmbio.uu.se> jeudi 7 août 2003
15:42:52 >>>
Fréderic, 

I suppose you're on vacation right now, but just in case you are not, I
have a couple of questions:

My main objective at this stage of the modeling procedure is just to
reproduce the results from the previous, NONMEM-based modeling stages. 

Question 1: 
The final NONMEM model included two variability terms. The variability
was modeled in an exponential fashion, and I want to recreate this
variability model by sampling from a lognormal population distribution.
However, when I try to assign an inverse gamma prior to the variability
term, MCSim won't run, complaining that my variability is below 1. This
is not that surprising, given that MCSim want the lognormal variability
on a log scale, the null value being 1, and that it is not possible to
truncate the distribution. Still, if if want to do this, how do I do?
I'm trying the a Piecewise prior for now, but I want to do it properly.

Question 2 (related to the first one):
>From the previous modeling, I have estimates of the variability terms,
manmifested as a mean and a standard error. How do I translate these
values into shape and scale parameters for an inverse gamma? I managed
to find the equations for this distribution, but if I solve for sclae
and shape parameters, I get a third grade differential equation. Is
there no simpler way of doing it?

Grateful for any help, 

Fredrik