# Dose proportionality **URL:** \ **Category:** Phoenix WNL basics\ **Created:** [May 14, 2016, 1:09pm UTC](https://forum.certara.com/t/dose-proportionality/2963 "2016-05-14T13:09:00Z")\ **Posts on this page:** 16\ **Page:** 1
**Author:** ![Helmut\_Schuetz](https://avatars.discourse-cdn.com/v4/letter/h/eb8c5e/32.png) [@Helmut\_Schuetz](https://forum.certara.com/u/Helmut_Schuetz)\ **Post date:** [May 14, 2016, 1:09pm UTC](https://forum.certara.com/t/dose-proportionality/2963/1 "2016-05-14T13:09:00Z")
Dear all, inspired by Angus’ [post in the BEBA-Forum](http://forum.bebac.at/mix_entry.php?id=16294) I tried to reproduce results of [Smith et al.](http://physiologie.envt.fr/spip/IMG/pdf/smithConfidence_Interval_Criteria_for_assessment_dose_proportionality_smith_Pharm_research.pdf) The estmated intercept and slope for Cmax are practically identical to the reported ones (obtained in SAS Proc Mixed), but not their 90% CIs (the ones by Phoenix are slightly _wider_). Questions: - Unfortunately Smith et al. didn’t give the SAS code. Maybe I screwed up defining the model. Any ideas ideas? - Until now I used the linear model 502 because I like the planar CI taking the generally high correlation between parameters into account. Drawback: Only the 95% CI is possible. - I’m currently planning a study with only three dose levels. In LME/Residuals I get the predicted values and their CIs. A plot will be ugly since estimates/CIs will be connected by straight lines which do not properly represent the power model. In 502 I got 1,000 values. That’s too much… Anyhow, I can calculate values from the estimates and some dummy doses but I have no idea how to calculate the predicted CI. [Smith et al 2000.phxproj](https://forum.certara.com/uploads/short-url/A3IbFF4UvE8pWbGRO3ou206qEPl.phxproj) (1.08 MB) ---
**Author:** ![sharmavineet](https://avatars.discourse-cdn.com/v4/letter/s/b487fb/32.png) [@sharmavineet](https://forum.certara.com/u/sharmavineet)\ **Post date:** [May 16, 2016, 12:55pm UTC](https://forum.certara.com/t/dose-proportionality/2963/2 "2016-05-16T12:55:26Z")
Thank you for this post: Yes; of course I too am wondering about the small difference in the confidence intervals when using the Phoenix LME model compared with Proc Mixed results (SAS) from Brian Smith. I wonder if there is any insight from Pharsight on this topic. Angus ---
**Author:** ![SimonDavis](https://avatars.discourse-cdn.com/v4/letter/s/d9b06d/32.png) [@SimonDavis](https://forum.certara.com/u/SimonDavis)\ **Post date:** [May 26, 2016, 4:35pm UTC](https://forum.certara.com/t/dose-proportionality/2963/3 "2016-05-26T16:35:52Z")
Sorry I looked at this last week but never got my head around the question being asked, looking back onthe original BEBAC thread it seems the answer was that Smith used ML ((usual maximum likelihood) in SAS versus REML (?r REstricted. Maximum Likelihood), is that right? i.e if you use the same settings in SAS the values match Is there anything else to answer? Simon. ---
**Author:** ![sharmavineet](https://avatars.discourse-cdn.com/v4/letter/s/b487fb/32.png) [@sharmavineet](https://forum.certara.com/u/sharmavineet)\ **Post date:** [May 27, 2016, 12:03pm UTC](https://forum.certara.com/t/dose-proportionality/2963/4 "2016-05-27T12:03:10Z")
_SIMON: the below option was favoured. Phoenix WinNonlin gives the same result as SAS and Rimer_ _SAS (REML/satterth) B0 1.9414 1.4968 2.3860_ _B1 0.7617 0.6696 0.8539_ _Angus_ ---
**Author:** ![llli](https://avatars.discourse-cdn.com/v4/letter/l/b9bd4f/32.png) [@llli](https://forum.certara.com/u/llli)\ **Post date:** [November 11, 2016, 3:23pm UTC](https://forum.certara.com/t/dose-proportionality/2963/5 "2016-11-11T15:23:53Z")
Dear All, I have some queations about DP and hope to get your answeres. 1. In Helmut’s project, 1/x was used as weighting. From the results, the weighting only changed the results of CIs a little bit, not the estimate. There was also no difference of residuals plots. I tried to use 1/(x_x) as weighting but only got information on intercept. Why? That is the reason that Helmut used 1/x, not 1/(x_x), as weighting? 2. Another question is about the decision rules. In Helmut’s unweighted example (also the example 1 in Smith’s paper), for the power model of ln(Cmax) vs ln(Dose), we only need to check if 90% CI is within the reference interval (which is dependent on r and H,L)? The paper used 0.8 and 1.25 becasue of the BE guideline? 1. Furthermore, I can’t understand the description of Rdnm in the paper. Did Simth use the BE method for Camx/Dose vs Dose? If yes, why there is only 1 Rdnm for 4 difference doses? I am confused. Any input is appreciated! Thank you! LLLi ---
**Author:** ![Helmut\_Schuetz](https://avatars.discourse-cdn.com/v4/letter/h/eb8c5e/32.png) [@Helmut\_Schuetz](https://forum.certara.com/u/Helmut_Schuetz)\ **Post date:** [November 11, 2016, 5:49pm UTC](https://forum.certara.com/t/dose-proportionality/2963/6 "2016-11-11T17:49:51Z")
Hi LLLi, > 1. In Helmut’s project, 1/x was used as weighting. From the results, the weighting only changed the results of CIs a little bit, not the estimate. There was also no difference of residuals plots. I tried to use 1/(x_x) as weighting but only got information on intercept. Why? That is the reason that Helmut used 1/x, not 1/(x_x), as weighting? I followed suggestions by Chow/Liu (Chow SC, Liu JP. Dose Proportionality Study. In: Chow SC, Liu JP, editors. Design and Analysis of Bioavail ­ability and Bioequivalence Studies. Boca Raton: Chapman & Hall / CRC Press; 2009. p. 563–573). Not only in Smith’s example I found no relevant difference between _w_=1, _w_=1/_x_, and _w_=1/_x_ ². The jury is out. 😉 If this is an exploratory study, pick out the best (minimum AIC). In a confirmatory study you have to specify the weighting scheme beforehand… > 1. Another question is about the decision rules. > In Helmut’s unweighted example (also the example 1 in Smith’s paper), for the power model of ln(Cmax) vs ln(Dose), we only need to check if 90% CI is within the reference interval (which is dependent on r and H,L)? The paper used 0.8 and 1.25 becasue of the BE guideline? Smith _et al._ give 0.8 and 1.25 as _an example_. Hummel _et al_. suggest a more relaxed range of 0.5 to 2 for _exploratory_ studies (Hummel J, McKendrick S, Brindley C, French R. Exploratory assessment of dose proportionality: review of cur ­rent approaches and proposal for a practical criterion. Pharmaceut Stat. 2009; 8(1): 38–49. [doi:10.1002/pst.326](http://dx.doi.org/10.1002/pst.326)). You have to transform these limits based on the dose range. More about that below. > 1. Furthermore, I can’t understand the description of Rdnm in the paper. Did Simth use the BE method for Camx/Dose vs Dose? If yes, why there is only 1 Rdnm for 4 difference doses? Let _r_ be the ratio of highest and lowest dose levels. Hence, only _one_ value; in Smith’s example 250/25=10). The parameter of interest is _rβ–1_ or the ratio of dose-normaized means _Rdnm_. Since _Rndm_ depends on the estimated _β_, you will get _different_ values for each PK metric. We are interested in the confidence interval of _β_ (the slope of the linearized power-model). In DP we have to modify the original acceptance range [_θL_, _θU_] based on _r_ (the wider the dose range, the more strict the criterion gets). Note also that the modified acceptance range [_θ_’_L_, _θ_’_U_] is symmetrical around 1. Formulas: _θ_’_L_,=[1+ln(_θL_)/ln(_r_ )] and _θ_’_U_=[1+ln(_θU_)/ln(_r_)]. From [0.8, 1.25] and _r_ 10 we get [0.9031, 1.0969] and from [0.5, 2] and _r_ 10 [0.6990, 1.3010]. Hope that helps. ---
**Author:** ![llli](https://avatars.discourse-cdn.com/v4/letter/l/b9bd4f/32.png) [@llli](https://forum.certara.com/u/llli)\ **Post date:** [November 11, 2016, 9:27pm UTC](https://forum.certara.com/t/dose-proportionality/2963/7 "2016-11-11T21:27:30Z")
Hi Helmut, Thank you for your reply! It helps a lot. In your reply, you mentioned “Formulas: _θ_’_L_,=[1+ln(_θL_)/ln(_r_ )] and _θ_’_U_=[1+ln(_θU_)/ln(_r_)]. From [0.8, 1.25] and _r_ 10 we get [0.9031, 1.0969] and from [0.5, 2] and _r_ 10 [0.6990, 1.3010]”. While in Smith’s paper, it is said that “the CI of Rdnm completely outside (0.8, 1.25), indicating a disproportionate increase”. I am confused about the difference between the reference interval. Which reference interval we should use to evalute whether DP or not? Furthermore, the paper also said that the Rdnm value of 1 would denote ideal dose-proportionality. Rdnm = PK/corresponding dose? For power model of Rdnm, what is Y and what is x? Thank you! LLLi ---
**Author:** ![Helmut\_Schuetz](https://avatars.discourse-cdn.com/v4/letter/h/eb8c5e/32.png) [@Helmut\_Schuetz](https://forum.certara.com/u/Helmut_Schuetz)\ **Post date:** [November 12, 2016, 1:06am UTC](https://forum.certara.com/t/dose-proportionality/2963/8 "2016-11-12T01:06:34Z")
Hi LLLi, > In your reply, you mentioned “Formulas: _θ_’_L_,=[1+ln(_θL_)/ln(_r_ )] and _θ_’_U_=[1+ln(_θU_)/ln(_r_)]. From [0.8, 1.25] and _r_ 10 we get [0.9031, 1.0969] and from [0.5, 2] and _r_ 10 [0.6990, 1.3010]”. > While in Smith’s paper, it is said that “the CI of Rdnm completely outside (0.8, 1.25), indicating a disproportionate increase”. I am confused about the difference between the reference interval. Which reference interval we should use to evalute whether DP or not? You are quoting the footnote _b_ below Table 2. I guess this is just a typo. Smith used a slightly different terminology (compared to Chow/Liu and Hummel _et al_.). He starts from 0.80 and 1.25 (ΘL and ΘU; page 1279, first paragraph). The transformed acceptance range (he calls it _“the critical region”_ and later _“the reference interval”_) is derived in Eq. (4). That’s the same one I used above. Now look at page 1282, left column, second paragraph, which reads: The corresponding 90% CI (0.679, 0.844) fell outside the reference interval (0.903, 1.097) defined by Eq. (4) for r = 10 and ΘU = 1/ΘL = 1.25, indicating a disproportionate change in Cmax across the dose range studied. In other words from the original range [0.80, 1.25] he gets the transformed one [0.903, 1.097] and _this_ is what you should use (of course depending on the actual _r_ in the study). _q.e.d._ > Furthermore, the paper also said that the Rdnm value of 1 would denote ideal dose-proportionality. Correct. Let _β_ ≡ 1 and _r_ ⊂ ℝ. Then _Rdnm =_ _r β–1_ = _r_ 0 = 1. Less mathematical: If the slope is _exactly_ 1 then for _any_ possible ratio of dose levels _Rdnm_ will be _exactly_ 1. > Rdnm = PK/corresponding dose? No. _r_ = the ratio of the highest/lowest dose and _Rdnm_ = _r β–1_. You do not dose-normalize in this model. > For power model of Rdnm, what is Y and what is x? In my project I used the _linearized_ power model, which is ln(_Y_j) = _α_ + _β_ · ln(_xj_), where _Y_ is he respective PK metric (AUC, Cmax, …) and _x_ the dose; both at level _j_. Most people prefer the linearized model over the original one – which is _Y_j = _α_ · _xj_ _β_ because the latter requires nonlinear fitting. If you have a Phoenix/NLME license go ahead with the “pure” model. Anyhow, I would not recommend that because in a regulatory setting the former is more easy to assess than the latter. However, there is a situation which demands nonlinear fitting: A power model with an _intercept, i.e._, _Y_j = _α_ + _λ_ · _xj_ _β_. You would need this model for dosing an endogenous compound and measurable basal levels. ---
**Author:** ![mittyright](https://avatars.discourse-cdn.com/v4/letter/m/e9a140/32.png) [@mittyright](https://forum.certara.com/u/mittyright)\ **Post date:** [November 12, 2016, 5:46pm UTC](https://forum.certara.com/t/dose-proportionality/2963/9 "2016-11-12T17:46:23Z")
Dear Helmut and LLLi, thank you for this very useful and interesting discussion! > If you have a Phoenix/NLME license go ahead with the “pure” model. Anyhow, I would not recommend that because in a regulatory setting the former is more easy to assess than the latter. BTW with Phoenix model we can get some desired results. Attached you can find PHX template. Please imrort it to the project from 1st message above and map the data I tried to implement the power model without log transformation. I also tried to build a log model as described by Smith, but there are some difficultes with plots and residuals due to exponent of residual error, so we need some trick to do it in log domain, I’ll show you if you need it. Please note that PHX cannot transfer Omega values from one model to another (or may be I don’t know how to do it). So in case of other data usage you need to copypaste the omegas from main model to vpc model. Or delete the main model and use the VPC model for estimation and VPC check (may be more robust) I also derived the bounds for non-transformed PK metrics from the equation (3). I’m using 0.5-2.0 here because otherwise the bounds are too strict. Please let me know if something is wrong or should be updated, I’ll appreciate that. BR, Mittyright **PS: I deleted the template because it does not work properly, please see the project in the next message** ![DP.jpg](https://us1.discourse-cdn.com/certara/original/2X/0/0ac73f43a7c69bfc326eb24fda1752f385ed90c2.jpeg) ---
**Author:** ![Helmut\_Schuetz](https://avatars.discourse-cdn.com/v4/letter/h/eb8c5e/32.png) [@Helmut\_Schuetz](https://forum.certara.com/u/Helmut_Schuetz)\ **Post date:** [November 13, 2016, 1:41am UTC](https://forum.certara.com/t/dose-proportionality/2963/10 "2016-11-13T01:41:06Z")
Hi Mittyright, PHX7 showed me the finger when trying to import the template. Errors: 1 File: NLME\_Dose\_Prop.wnlt Description: File “C:\Users\HS\AppData\Local\DataService\_dbXXXX.xml” could not be found …where XXXX is a long string a characters. 😉 No hurry; I’ll be in Berlin next week. ---
**Author:** ![mittyright](https://avatars.discourse-cdn.com/v4/letter/m/e9a140/32.png) [@mittyright](https://forum.certara.com/u/mittyright)\ **Post date:** [November 13, 2016, 9:36am UTC](https://forum.certara.com/t/dose-proportionality/2963/11 "2016-11-13T09:36:25Z")
Dear Helmut, shame on me, I didn’t check the template in other project. You are right, template export/import feature is still a dark side of our life 😉 I added the template to your project, please see attached (checked it twice, everything is working now). BR, Mittyright [Smith et al 2000\_PHX\_Model.phxproj](https://forum.certara.com/uploads/short-url/dRxfG5JoT5VsE4yO0FxWB48tkqA.phxproj) (2.13 MB) ---
**Author:** ![llli](https://avatars.discourse-cdn.com/v4/letter/l/b9bd4f/32.png) [@llli](https://forum.certara.com/u/llli)\ **Post date:** [November 14, 2016, 3:50pm UTC](https://forum.certara.com/t/dose-proportionality/2963/12 "2016-11-14T15:50:00Z")
Hi Helmut, Thank you very much for your reply! Another question is for the intercept parameter _β_0 from power model. What is the meaning of this parameter? Thank you! LLLi ---
**Author:** ![llli](https://avatars.discourse-cdn.com/v4/letter/l/b9bd4f/32.png) [@llli](https://forum.certara.com/u/llli)\ **Post date:** [November 14, 2016, 3:54pm UTC](https://forum.certara.com/t/dose-proportionality/2963/13 "2016-11-14T15:54:54Z")
Hi Mittyright, I am still using Phoenix 6.4 and can not open your project. Is there any way to transfer your project into a 6.4 version? Thank you! LLLi ---
**Author:** ![mittyright](https://avatars.discourse-cdn.com/v4/letter/m/e9a140/32.png) [@mittyright](https://forum.certara.com/u/mittyright)\ **Post date:** [November 14, 2016, 4:14pm UTC](https://forum.certara.com/t/dose-proportionality/2963/14 "2016-11-14T16:14:50Z")
Hi LLLi, you can do it by yourself: download Phoenix7, install it on VM or other machine (without license you cannot execute the project, but you can see everything), open and review the project and make the same objects in PHX6.4 Or maybe it’s time for update 😉 BR, Mittyright **PS: Please see the project in the next post** ---
**Author:** ![mittyright](https://avatars.discourse-cdn.com/v4/letter/m/e9a140/32.png) [@mittyright](https://forum.certara.com/u/mittyright)\ **Post date:** [November 15, 2016, 1:33pm UTC](https://forum.certara.com/t/dose-proportionality/2963/15 "2016-11-15T13:33:13Z")
Hi LLLi, here’s a VPC model from the project above I made it in 6.4, may be not so nice, but the steps are the same BR, Mittyright [Smith et al 2000\_forLLLi\_64.phxproj](https://forum.certara.com/uploads/short-url/5VMU0wRAB5xCMrsb57lSzYEUxKV.phxproj) (692 KB) ---
**Author:** ![llli](https://avatars.discourse-cdn.com/v4/letter/l/b9bd4f/32.png) [@llli](https://forum.certara.com/u/llli)\ **Post date:** [January 13, 2017, 4:56am UTC](https://forum.certara.com/t/dose-proportionality/2963/16 "2017-01-13T04:56:37Z")
Hi, I have some questions about the paper by Smith. In this paper, the authors mentioned 4 models for dose proportionality. Is there any rule to choose the model for different studies? Can someone provide more information about the saturable elimination model vs power model? If we don’t know whether the nonlinear PK exists or not, is it safer to use the saturable elimination model than to use power model? Thank you! LLLi