From: Paolo Denti <*paolo.denti*>

Date: Thu, 10 Dec 2015 18:09:01 +0000

Dear Dennis,

there is probably a nice formula with linear approximation to convert the uncertainty in your parameters (the slopes of the regression lines) into the CI at different concentrations, but linearization is so "last century" :), it may be just easier to run a simulation with uncertainty.

I would suggest is that you

1. first obtain uncertainty in both your parameters and importantly the correlation between the two uncertainty terms.

2. you simulate out your response at all values of Cp including this uncertainty in the parameters. You do this a bunch of times (n=500 should do)

3. you obtain the CI at each concentration level just by using the empirical percentiles of the simulation (2.5th and 97.5th).

It is very important to include the correlation value because that would affect greatly your confidence interval, and it will prevent the generation of implausible values.

For the estimation of the joint uncertainty (2x2 covariance matrix), you can do things parametrically by assuming that your joint distribution is a multi-variate Gaussian distribution, and for you could even use the values of precision provided by the covariance step of NONMEM. However, I would strongly discourage you from doing this - and I am sure Nick Holford would agree :)

What I suggest is that you estimate your precision in the slope of the regression lines by using a bootstrap, and then you use each single set of parameter estimates from the bootstrap in your simulation. This way you will make no assumptions on the distributions, and you will automatically keep the correlation into account.

I think the SSE (stochastic simulation and estimation) script in Perl Speaks NONMEM has a tool to do this kind of simulations with uncertainty, and you can in fact use as an input values the output estimates from a bootstrap. Option -raw_res, or something like that.

Good luck and greetings from Cape Town,

Paolo

On 2015/12/10 18:23, Fisher Dennis wrote:

Colleagues

I have fit an exposure response model using NONMEM — the optimal model is a segmented two-part regression with Cp on the x-axis and response on the y-axis. The two regression lines intercept at the cutpoint.

The parameters are:

slope of the left regression

cutpoint between regressions

“intercept” — y value at the cutpoint

slope of the right regression (fixed at zero; models in which the value was estimated yielded similar values for the objective function)

I have been asked to calculate the confidence interval for the response at various Cp values.

Above the cutpoint, this seems straightforward:

a. if NONMEM yielded standard errors, the only relevant parameter is the y value at the cutpoint and its standard error

b. if NONMEM did not yield standard errors, the confidence interval could come from either likelihood profiles or bootstrap

My concern is calculating at Cp values below the cutpoint, for which both slope and intercept come into play. Any thoughts as to how to do this in the presence or absence of NONMEM standard errors?

The reason that I mention with / without presence of SE’s is that this model was fit to two different datasets, one of which yielded SE’s, the other not.

Any thoughts on this would be appreciated.

Dennis

Dennis Fisher MD

P < (The "P Less Than" Company)

Phone: 1-866-PLessThan (1-866-753-7784)

Fax: 1-866-PLessThan (1-866-753-7784)

www.PLessThan.com<http://www.plessthan.com/>

--

------------------------------------------------

Paolo Denti, PhD

Pharmacometrics Group

Division of Clinical Pharmacology

Department of Medicine

University of Cape Town

K45 Old Main Building

Groote Schuur Hospital

Observatory, Cape Town

7925 South Africa

phone: +27 21 404 7719

fax: +27 21 448 1989

email: paolo.denti za<mailto:paolo.denti

------------------------------------------------

Disclaimer - University of Cape Town This e-mail is subject to the UCT ICT policies and e-mail disclaimer published on our website at http://www.uct.ac.za/about/policies/emaildisclaimer/ or obtainable from +27 21 650 9111. If this e-mail is not related to the business of UCT it is sent by the sender in the sender's individual capacity.

Received on Thu Dec 10 2015 - 13:09:01 EST

Date: Thu, 10 Dec 2015 18:09:01 +0000

Dear Dennis,

there is probably a nice formula with linear approximation to convert the uncertainty in your parameters (the slopes of the regression lines) into the CI at different concentrations, but linearization is so "last century" :), it may be just easier to run a simulation with uncertainty.

I would suggest is that you

1. first obtain uncertainty in both your parameters and importantly the correlation between the two uncertainty terms.

2. you simulate out your response at all values of Cp including this uncertainty in the parameters. You do this a bunch of times (n=500 should do)

3. you obtain the CI at each concentration level just by using the empirical percentiles of the simulation (2.5th and 97.5th).

It is very important to include the correlation value because that would affect greatly your confidence interval, and it will prevent the generation of implausible values.

For the estimation of the joint uncertainty (2x2 covariance matrix), you can do things parametrically by assuming that your joint distribution is a multi-variate Gaussian distribution, and for you could even use the values of precision provided by the covariance step of NONMEM. However, I would strongly discourage you from doing this - and I am sure Nick Holford would agree :)

What I suggest is that you estimate your precision in the slope of the regression lines by using a bootstrap, and then you use each single set of parameter estimates from the bootstrap in your simulation. This way you will make no assumptions on the distributions, and you will automatically keep the correlation into account.

I think the SSE (stochastic simulation and estimation) script in Perl Speaks NONMEM has a tool to do this kind of simulations with uncertainty, and you can in fact use as an input values the output estimates from a bootstrap. Option -raw_res, or something like that.

Good luck and greetings from Cape Town,

Paolo

On 2015/12/10 18:23, Fisher Dennis wrote:

Colleagues

I have fit an exposure response model using NONMEM — the optimal model is a segmented two-part regression with Cp on the x-axis and response on the y-axis. The two regression lines intercept at the cutpoint.

The parameters are:

slope of the left regression

cutpoint between regressions

“intercept” — y value at the cutpoint

slope of the right regression (fixed at zero; models in which the value was estimated yielded similar values for the objective function)

I have been asked to calculate the confidence interval for the response at various Cp values.

Above the cutpoint, this seems straightforward:

a. if NONMEM yielded standard errors, the only relevant parameter is the y value at the cutpoint and its standard error

b. if NONMEM did not yield standard errors, the confidence interval could come from either likelihood profiles or bootstrap

My concern is calculating at Cp values below the cutpoint, for which both slope and intercept come into play. Any thoughts as to how to do this in the presence or absence of NONMEM standard errors?

The reason that I mention with / without presence of SE’s is that this model was fit to two different datasets, one of which yielded SE’s, the other not.

Any thoughts on this would be appreciated.

Dennis

Dennis Fisher MD

P < (The "P Less Than" Company)

Phone: 1-866-PLessThan (1-866-753-7784)

Fax: 1-866-PLessThan (1-866-753-7784)

www.PLessThan.com<http://www.plessthan.com/>

--

------------------------------------------------

Paolo Denti, PhD

Pharmacometrics Group

Division of Clinical Pharmacology

Department of Medicine

University of Cape Town

K45 Old Main Building

Groote Schuur Hospital

Observatory, Cape Town

7925 South Africa

phone: +27 21 404 7719

fax: +27 21 448 1989

email: paolo.denti za<mailto:paolo.denti

------------------------------------------------

Disclaimer - University of Cape Town This e-mail is subject to the UCT ICT policies and e-mail disclaimer published on our website at http://www.uct.ac.za/about/policies/emaildisclaimer/ or obtainable from +27 21 650 9111. If this e-mail is not related to the business of UCT it is sent by the sender in the sender's individual capacity.

Received on Thu Dec 10 2015 - 13:09:01 EST