# RE: SAEM and IMP

From: Bob Leary <Bob.Leary>
Date: Thu, 15 May 2014 14:22:48 -0400

Hi Emmanuel,
While I am a strong advocate of using quasi-random rather than pseudo- rand=
om sequences for importance sampling in EM methods like IMP, there is a the=
oretical (and very real) problem with their use in the context you suggest=
ed in your message, namely with a multivariate t distribution as the import=
ance sampling distribution. The 3S2 option implies you are using a Sobol q=
uasi-random sequence, while
the DF=7 implies the use of a multivariate T-distribution with 7 degrees =
of freedom. The standard way of generating
a p-dimensional multivariate t -random variable with DF degrees of freedo=
m is to generate a p-dimensional multivariate normal and then divide by an =
additional independent random variable which is basically the square root =
of a 1-d chi square random variable with DF degrees of freedom. Thus to g=
enerate a p-dimensional importance sample, you actually need to use p+1 in=
dependent random variables. If you simply use a p+1 dimensional Sobol vec=
tor as the base quasi-random draw, the nonlinear mapping from p+1 dimension=
s to the final p dimensional result destroys the low discrepancy property =
of the final sequence in the p-dimensional space and in fact introduces a =
significant amount of bias in the final result. The problem arises directl=
y from the p+1 vs p dimensional mismatch.

There is no problem if the final p-dimensional result can be generated from=
a p-dimensional quasi-random sequence, which is the case for multivariate =
normal
Importance samples. So quasi random sequences should really only be used=
for the DF=0 multivariate normal importance sampling distribution case, =
not the multivariate DF>0 multivariate t case.

I ran across this effect in testing the Sobol-based importance sampling EM =
algorithm QRPEM in Phoenix NLME. It is very real and the net effect is to =
introduce a significant bias. There is a partial fix that works but gives=
up some of the benefit of using low-discrepancy sequences - namely use a =
p-dimensional quasi-random vector to generate the p-dimensional multivariat=
e normal, but
then use a 1-d pseudo-random sequence to generate the chi-square random var=
iable.

From: owner-nmusers
Behalf Of Emmanuel Chigutsa
Sent: Thursday, May 15, 2014 1:03 PM
To: Pavel Belo; nmusers
Subject: Re: [NMusers] SAEM and IMP

Hi Pavel
I have experienced a similar problem. In my case, the following code for IM=
P after SAEM (using NM7.3) greatly reduced the Monte Carlo OFV noise from v=
ariations of about +/- 60 points to variations of +/- 6 points (though stil=
l not good enough for covariate testing):
\$EST METHOD=IMP LAPLACE INTER NITER=15 ISAMPLE=3000 EONLY=1 DF=7 =
IACCEPT=0.3
ISAMPEND=10000 STDOBJ=2 MAPITER=0 PRINT=1 SEED=123456 RANMETHOD=
=3S2
The settings are explained in the NM7.3 guide. If you are using NM7.3, you =
can also try IACCEPT=0.0 whereupon "NONMEM will determine the most approp=
riate IACCEPT level for each subject". Of course the settings for DF and IA=
CCEPT in the above code will depend on the type of data you have. Which bri=
ngs me to my own question. If I have both continous and categorical DVs in =
the dataset (which would mean different optimal settings) and I am using F_=
FLAG accordingly, what would the 'right' values of DF and IACCEPT be? I hav=
e noticed that the DF automatically chosen by NONMEM for individuals in the=
dataset can vary from 0-8 and this appears to be random. NOTICE: T=
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Received on Thu May 15 2014 - 14:22:48 EDT

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