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: <mailto:[email protected]> [email protected]
Minimisation problem...
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
Quoted reply history
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]>
Dear Gavin,
It would seem the first place to start is with your structural model. A 3rd
order polynomial with IIV on the parameters should give a perfect fit to every
subject subject with <= 3 sampling points. How about a more mechanistic
approach? It sounds like you are modelling the change in some kind of
biomarker that returns to baseline, so maybe a turnover model is appropriate.
Have a look at:
Mager DE, Wyska E, Jusko WJ. Diversity of mechanism-based pharmacodynamic
models. Drug Metab Dispos. 2003 May;31(5):510-8
A covariate on production or inhibition rate which varies between treatment and
control would tell you if there is a difference between groups.
BW,
Joe
Joseph F Standing
MRC Fellow, UCL Institute of Child Health
Antimicrobial Pharmacist, Great Ormond Street Hospital
Tel: +44(0)207 905 2370
Mobile: +44(0)7970 572435
Quoted reply history
________________________________________
From: [email protected] [[email protected]] On Behalf Of
Gavin Jarvis [[email protected]]
Sent: 09 January 2013 10:19
To: [email protected]
Subject: [NMusers] Minimisation problem...
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]>
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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.
Quoted reply history
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
Dear Leonid, Joe and Nick
Thank you for your helpful replies. It seems like a consensus view that I
should give some attention to the structural model, so I shall do just that.
All best
Gavin
Quoted reply history
-----Original Message-----
From: [email protected] [mailto:[email protected]] On
Behalf Of Nick Holford
Sent: 09 January 2013 19:26
To: nmusers
Subject: Re: [NMusers] Minimisation problem...
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