# Re: [NMusers] Incorporating standard deviation (SD) on fitted mean values

From: Denney, William S. <William.S.Denney_at_pfizer.com>
Date: Thu, 19 Nov 2015 14:19:14 +0000

I agree with Nick, you will want to weight your precision by the inverse st=
andard error.

More generally, you are doing a model-based meta-analysis. When I was firs=
t learning about it, a book that I found very informative and readable was =
"Introduction to Meta-Analysis" by Borenstein, et al. It focuses on standa=
rd meta-analysis and not model-based, but the foundation it lays is necessa=
ry for both.

Thanks,

Bill

On Nov 19, 2015, at 4:43, "Nick Holford" <n.holford_at_auckland.ac.nz> wrote:

Paul,

Ahmad says he is fitting "means of a parameter X". I suspect he really mean=
s the "means of a variable X".

A = THETA(1)*EXP(ETA(1)) ;ETA1 is bet=
ween STUDY variability on A
ALPHA = THETA(2)*EXP(ETA(2)) ;ETA2 is between STU=
DY variability on ALPHA
IPRED = (A)*exp(-ALPHA*TIME)
Y = IPRED *(1+EPS(1)/SQRT(NSUB))

I would say that the random effect ETA is describing the between study vari=
ability in the parameters (A and ALPHA) while EPS is describing the random =
unexplained variability (RUV) in the prediction of the DV (mean of X) usin=
g an exponential function of A, ALPHA and TIME.

Some of the RUV arises from within study between subject variability in A a=
nd ALPHA and some from the usual sort of RUV (model misspecification, measu=
rement error, stochastic noise, etc).

The SD covariate in the data set

ID

NSUB

TIME

DV

SD

1

10

0.083333

4.776667

0.230317

is described by Ahmad as "the standard deviation of the observations in the=
subjects at TIME=t."
The random contributions to SD seem to be similar to those contributing to =
RUV as described above.

Therefore it seems to me that SD could be used in the prediction of X as yo=
u suggested:

Y = IPRED + SD*EPS(1)/SQRT(NSUB)

The variance of EPS(1) should be fixed to 1 like this:
\$SIGMA 1 FIX

Best wishes,

Nick

> On 17-Nov-15 20:43, Paul Matthias Diderichsen wrote:
> In your aggregate data, ETA describes between-study variability while
> EPS describes the between-subject variability. As such, EPS is not
> "unexplained" (as in RUV) but rather "explained" in the data.
>
> You can interpret the residual error in NONMEM as a weight of your data.
> If you have small sample size or large BSV for a given outcome, then you
> should not put as much weight on that data point = larger variance.
>
> Precision is a different beast altogether: this relates to the standard
> error of your estimates (= variance-covariance matrix), and depends
> (everything else being equal) on how much data you have.
>
> (I'm looping this back into NMUsers; maybe somebody else has comments)
>
>> On 11/17/2015 0:34, Abu Helwa, Ahmad Yousef Mohammad - abuay010 wrote:
>> Hi Paul,
>>
>> Thank you for your input on this. However, in the case you presented,=
the SD in the error model will then informs about the precision rather tha=
n between subject variability? In my case, the parameter I am modelling (ga=
stric pH) is measured in X number of subjects and the mean and SD are repor=
ted. So, the SD is not the precision of the measurement within a subjects =
(the measurement in each subject was performed one time), rather, it is bet=
ween subjects. The large SDs for some of the reported means is due to the f=
act that BSV in gastric pH is high.
>>
>> -----Original Message-----
>> From: Paul Matthias Diderichsen [mailto:pmdiderichsen_at_wequantify.com]
>> Sent: Monday, 16 November 2015 6:16 PM
>> To: Abu Helwa, Ahmad Yousef Mohammad - abuay010 <ahmad.abuhelwa_at_mymail.u=
nisa.edu.au>
>> Subject: Re: [NMusers] Incorporating standard deviation (SD) on fitted m=
ean values
>>
>>
>>> On 11/15/2015 23:46, Abu Helwa, Ahmad Yousef Mohammad - abuay010 wrote:
>>> Y = IPRED *(1+EPS(1)/SQRT(NSUB))
>>> 5) Is there any way where I can incorporate the SDs that I have to
>>> inform about the between SUBJECT variability in the model fitting?
>> Include the reported SD (REPSD) in your residual error variance and fix
>> the sigma to 1 (the variance is defined in your data). I would probably
>> describe the mean as a normal distributed variable, so:
>>
>> Y = IPRED + EPS(1)*REPSD/SQRT(NSUB)
>> \$SIGMA
>> 1 FIX
>>
>>
>>
>> Kind regards,

--
Nick Holford, Professor Clinical Pharmacology
Dept Pharmacology & Clinical Pharmacology, Bldg 503 Room 302A
University of Auckland,85 Park Rd,Private Bag 92019,Auckland,New Zealand
office:+64(9)923-6730 mobile:NZ+64(21)46 23 53
email: n.holford_at_auckland.ac.nz
http://holford.fmhs.auckland.ac.nz/

Holford SD, Allegaert K, Anderson BJ, Kukanich B, Sousa AB, Steinman A, Pyp=
endop, B., Mehvar, R., Giorgi, M., Holford,N.H.G. Parent-metabolite pharmac=
okinetic models - tests of assumptions and predictions. Journal of Pharmaco=
logy & Clinical Toxicology. 2014;2(2):1023-34.
Holford N. Clinical pharmacology = disease progression + drug action. Br =
J Clin Pharmacol. 2015;79(1):18-27.

Received on Thu Nov 19 2015 - 09:19:14 EST

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