Gavin,
You say "The biological interpretation of the experiment will change
significantly depending on which way this goes!" so the first thing to
do is to use a model that can have some biological interpretation.
Statistical tests of differences are not biological interpretations -
they just help select the model that best describes the biology.
For the control group you need to describe the change in response with
time without treatment. You can try models such as 1) no change with
time 2) linear change with time (as Leonid suggests) 3) asymptotic
non-linear change with time (Leonid suggests an Emax model, I would also
consider an asymptotic exponential - see below) 4) rise and fall with
time (typically used for placebo responses which can be described with 2
exponentials (Bateman function) e.g. Holford 1992). These kinds of
models describing the time course of change in a control group are often
referred to as disease progression models.
You can find some introductory material on disease progression models
and how treatments affect them at
http://holford.fmhs.auckland.ac.nz/docs/disease-progression.pdf. Some
examples are provided showing how to code for NONMEM in that
presentation and also in
http://www.page-meeting.org/page/page2007/DiseaseProgressNONMEMFiles.zip
By fitting both the control group and test group data at the same time
you can then look for treatment effects on the parameters e.g. does the
treatment seem to change the baseline (intercept) parameter or does it
seem to change the slope parameter of a linear disease progression model
or even both types of effect? These distinctions can be helpful in
deciding if the treatment is just symptomatic or has a disease modifying
effect (e.g. Holford 2006, Vu et al. 2012).
In your example you describe the control group response increases and
perhaps approaches an asymptote so an asymptotic exponential model would
be expected to fit that. For the test group which you say increases then
decreases then that could be described by a treatment effect on the
baseline parameter with causes a decrease with time.
e.g.
Response = S0*exp(-TRT*kloss*time) + (Sss - S0)*(1-exp(-kprog*time))
or
Response = THETA(1)*exp(-TRT*THETA(2)*time) + (THETA(3) -
THETA(1))*(1-exp(-THETA(4)*time))
where S0 is the baseline response at time 0, kprog describes the time
course of increase to an asymptote (Sss), TRT is 0 for control and 1 for
test (but could also be used to introduce dose if the treatment was
given at different dose levels), kloss describes the decrease in
baseline with time. Note the use of an exponential model for the
treatment effect so that the response cannot become negative. This would
be a reasonable constraint for most biological responses which are
expected to have non-negative values.
You mention there are missing values. Do not be tempted to impute these
missing values using last observation carried forward. If missing
values are due to subject dropout then the hazard of dropout may depend
on the response (see Hu & Sale 2003 and
http://holford.fmhs.auckland.ac.nz/docs/dropout-models.pdf). If you want
to simulate from your model e.g. to evaluate it with a visual predictive
check then you may need a dropout model (see Vu et al 2012 for an example).
So my advice is to focus on the model first -- not how to show
statistical differences. The biological interpretation of the model and
its parameters will of course be determined by what you know about the
biology of the response.
Best wishes
Nick
1. Holford NH, Peace KE. Methodologic aspects of a population
pharmacodynamic model for cognitive effects in Alzheimer patients
treated with tacrine. Proc Natl Acad Sci U S A 1992; 89: 11466-70.
2. Holford NH, Chan PL, Nutt JG, Kieburtz K, Shoulson I. Disease
progression and pharmacodynamics in Parkinson disease - evidence for
functional protection with levodopa and other treatments. J
Pharmacokinet Pharmacodyn 2006; 33: 281-311.
3. Vu TC, Nutt JG, Holford NHG. Progression of motor and nonmotor
features of Parkinson's disease and their response to treatment. Br J
Clin Pharmacol 2012; 74: 267-83.
4. Hu C, Sale ME. A joint model for nonlinear longitudinal data with
informative dropout. J Pharmacokinet Pharmacodyn 2003; 30: 83-103.
On 10/01/2013 4:15 a.m., Leonid Gibiansky wrote:
Gavin,
I think polynomial is the wrong function to study this problem. It
would be better to use something more mechanistic that intrinsically
reflects the expected behavior. I would start with simple linear model
A = THETA(1)+ETA(1)
B = THETA(2)+ETA(2)
TEST = 0 for control, THETA(3) for test
Y=A+(B+TEST)*TIME
(assuming that at time zero both groups are identical).
Then move to Emax model
Y = A + (B+TEST)*TIME/(TIME50 + TIME), TIME50=THETA(4)
These two models test whether observations increase with time. If you
think that the response can go up and down, then you may try to use
more complex functions, like difference of two EMAX values where TEST
influences EMAX or TIME50 values
(see http://www.go-acop.org/sites/default/files/webform/Joy_Hsu.doc)
FOCEI should be as good as any other methods for this problem. To test
the quality of the fit it is better to plot DV versus PRED, not IPRED,
since with enough ETAs you can fit DV almost exactly, even if your
model fits only noise.
To get the idea of the appropriate function, you can compare means of
two groups (plots mean for each group by time versus time): the model
can help to quantify the dependence but if there is any trend, it
should be clearly visible on the plot.
Leonid
--------------------------------------
Leonid Gibiansky, Ph.D.
President, QuantPharm LLC
web: www.quantpharm.com
e-mail: LGibiansky at quantpharm.com
tel: (301) 767 5566
On 1/9/2013 5:19 AM, Gavin Jarvis wrote:
Dear NMusers
I am trying to analyse data from a study in which samples were taken
from each subject at 4 different time points (t=0,5,10,14). The problem
with the data is that there are many missing data points and there is
considerable variation between the subjects.
The subjects are in either a control or a test group, and I want to
determine whether there is any difference in the data values between
these groups.
Overall, it looks like the data values increase with time, but there is
a suggestion that in the test group the increase is not sustained but
returns to baseline levels by t=14, whereas the control group is either
levelled off or possibly still rising.
I have used a polynomial model to fit the data up to the 3rd power
(which I think is probably too much) and included additive parameters to
modify each of the coefficients from the polynomial model.
The problem I have is as follows:
When I use the FOCE method the ETA terms collapse towards zero. The
quality of the fit looks poor when judged by a plot of DV against
individual predicted values.
When I use the BAYES method, I get credible ETA values and a much better
fit (i.e., DV vs ipred clusters sensibly around a line of unity).
However, I cannot use the OBJV value from the BAYES method to carry out
hypothesis testing. The final reported parameter estimates following the
BAYES method are sensitive to initial starting values and the number of
iterations performed. If I use the parameter values obtained with the
BAYES method I can determine an accurate OBJV for those parameter values
using FOCE with just 1 evaluation. However, if I perform a minimisation
with FOCE starting with those values, the ETA values collapse and the DV
vs ipred plot looks awful again.
I hope this makes some sense to someone out there – I’m a bit of a
novice at NONMEM. I realise the data is far from ideal, but it would be
great to get some statistical information about the difference between
the two groups. If anyone had any suggestions I would be grateful. The
biological interpretation of the experiment will change significantly
depending on which way this goes!
Thanks
Gavin
__________________________________________________
*Dr Gavin E Jarvis MA PhD VetMB MRCVS*
University Lecturer
Department of Physiology, Development & Neuroscience
Physiological Laboratory
Downing Street
Cambridge
CB2 3EG
Tel: +44 (0) 1223 333745
Email: [email protected] <mailto:[email protected]>
--
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
tel:+64(9)923-6730 fax:+64(9)373-7090 mobile:+64(21)46 23 53
email: [email protected]
http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford
