Dose Finding Method in Joint Modeling of Efficacy and Safety Endpoints in Phase II Studies

Determination of appropriate dose(s) to advance into Phase III trials is one of the most challenging and important tasks during drug development. Selecting a dose too high may result in unacceptable safety problems, while a too low dose may lead to ineffective drugs. Proper estimation of dose-response profiles for relevant safety and efficacy endpoints allows the reliable evaluation of the risk-benefit profile of a drug at the end of Phase II, as well as the selection of appropriate doses to be brought into confirmatory Phase III trials. Thus how to select dose(s) in Phase II trials by combining information about the efficacy and safety in a joint model setting may play a key role in drug development programs and can serve as a gate-keeper for large confirmatory Phase III trials with greater chance of success. Dose finding methods through joint modeling of both efficacy and safety endpoints are studied in this paper. To be more specific, we extend the popular MCP-Mod dose finding method (Bretz et al., 2005), which considered only the efficacy endpoint, to the method that incorporates both efficacy and safety endpoints through joint modeling. Method of parameter estimation for the extended models, and methods of selection of dose(s) to be brought into confirmatory Phase III trials based on Phase II study data are discussed in the paper. The performances of the proposed methods are evaluated through simulations.


Introduction
Selection of appropriate dose(s) to carry into confirmatory Phase III trials is one of the most difficult decisions that need to be considered during drug development.It is believed by many that the high attrition rate currently observed in Phase III is largely driven by inadequate dose selection (FDC report, 1991;Bornkamp et al., 2007).Most commonly used dose finding designs and methods today still focus on selection of a target dose out of a fixed, generally small number of dose levels via pairwise hypothesis testing, which is typically inefficient (FDA, 2004).Assessment of dose-response should be an integral component of drug development, with studies designed to assess dose-response as an inherent part of establishing the safety and effectiveness of the drug.Characterization of dose-response relationship can be built into the development process with minimal extra effort and no much loss of time compared to development plans that ignore dose-response estimation.Proper estimation of such doseresponse profiles for relevant safety and efficacy endpoints allows reliable evaluation of the risk-benefit profile of a drug at the end of Phase II, as well as the selection of appropriate doses to be brought into confirmatory Phase III trials.Thus how to select dose(s) in Phase II trials by combining information about the efficacy and safety in a joint model setting may play a key role in drug development programs and can serve as a gate-keeper for large confirmatory Phase III trials with greater chance of success.
The primary goals of dose-response studies are to establish the presence of a dose-response signal and to estimate a target dose Bretz et al., 2005;Dragalin et al. 2007, usually the minimum effective dose (MED).There are fixed and adaptive approaches for designing and analyzing dose-ranging studies (Bornkamp et al. 2007).Bretz et al. (2005) combined multiple comparison procedure and modeling (MCP-Mod), which has been used extensively for analyzing dose finding trials recently.It includes a PoC (proof-of-concept) assessment and a dose-selection step.
The clear advantage of this approach, compared to traditional multiple comparison dose finding methods, is its added flexibility in selecting an appropriate dose-response model for future drug development.Most approaches in literature so far are targeted only on efficacy for an optimal dose by assuming that a dose range with an acceptable toxicity has been previously determined.However, under a variety of circumstances safety problems may lead to early termination of the drug development in Phase II/III trials.Therefore it is important to address safety and efficacy simultaneously.
There has been some research on how to design dose-finding studies based on efficacy and toxicity responses jointly in early phase (I/II) (Thall and Cook, 2004;Dragalin, 2005;Thall et al., 2008) for oncology trials.These dose finding designs are based on oncology study are different from other dose finding clinical studies in that for oncology early phase studies subjects (exclusively only cancer patients) enter the studies sequentially while for other studies subjects (most of time healthy volunteers) enter the studies simulaneously.There are a few dose finding designs the incorporate the safety and efficacy responses (see for example, Bekele and Shen, 2005;Padmanabhan et al. 2010).But these methods do not take advantage of MCP-Mod approaches of Bretz et al. (2005).
In this paper we will extend the MCP-Mod approach to select the best joint model based on two continuous correlated efficacy and safety outcomes and to get the final optimum dose(s) from the best joint model for the Phase III study.We will address the extension of the MCP-Mod approach for mixed (continuous and discrete) types of correlated efficacy and safety outcomes in a sequel article.This paper is organized as follows.In Section 2, we propose dose finding methods in joint modeling of efficacy and safety endpoints in Phase II studies.This section includes the methods of finding the maximum safety dose (MS D), the method of joint modeling of continuous efficacy and safety responses, the procedures of finding the MED and the MS D, and the methods of recommending dose(s) to carry into Phase III program development.In Section 3, the proposed methods are evaluated through simulations based on the data from our motivating example.Concluding remarks and recommendations are given in Section 4. et al. (2005) combined multiple comparison procedure and modeling (MCP-Mod), which has been used extensively for analyzing dose finding trials.It includes a PoC (proof-of-concept) assessment and a dose-selection step.The clear advantage of this approach, compared to traditional multiple comparison dose finding methods, is its added flexibility in selecting an appropriate dose-response model for future drug development.Bretz et al. (2005) considered only efficacy to identify the MED assuming that all considered doses are within safety tolerance.Assuming the following model for efficacy

Bretz
where f (d i , θ Y ) is a non-linear function of dose d i and vector of the model parameters θ Y , Y i j is the efficacy response of the jth patient in the ith dose group with dose d i .Assuming that there are a total of k dose groups with where △ e is the clinically meaningful minimun efficacy difference.Let L d be the lower 1 − 2γ confidence limit of the predicted mean value f (d, θ Y ) at dose d.They proposed that MED be estimated by where θ Y is the estimated θ Y .
In this section, we first show how to estimate the target dose, which is called maximum safety dose (MS D), for the safety endpoint.Then we propose the joint modeling of the continuous efficacy and safety responses, This includes the model formulations, the parameter estimation method, and the procedures of finding the MED and the MS D for combined efficacy and safety data.After that, we propose two strategies for recommending dose(s) to carry into Phase III program development through the joint criteria of success or through utility functions.

Target Dose Estimation for Safety Endpoint
For a safety endpoint, the MED in the MCP-Mod from Bretz et al. (2005) can be replaced by the MS D. Consider the following model for safety where g(d i , θ Z ) is a non-linear function of d i and θ Z , Z i j is the safety response of the jth patient in the ith dose group, i = 1, ..., k, and j = 1, ...., n i , d i is the dose for ith dose group with d 1 for placebo, and θ Z refers to the vector of model parameters.The MS D is defined as the maximum safety dose which shows clinically acceptable toxicity.
To be more specifically, given the model g(., θ Z ), we define where ∆ s is the largest clinically acceptable safety response difference from placebo, this implies that, the safety response at the dose is not too worse than that at placebo.Following the above definition, two different rules are proposed to estimate the true MS D. Denote U d the upper 1 − 2γ confidence limit of the predicted mean value g(d, θ Z ) at dose d based on the model g(., θ Z ) (see Figure 1), where θ Z is the estimated θ Z and can be obtained based on the model g(., θ Z ) using the nonlinear least squares fitting algorithm nls function (Bates and Chambers, 1992) in R (Version 2.11, R Foundation for Statistical Computing, Vienna, Austria, 2010).Based on our simulation in Section 3.2, although MS D 2 has a better approximation than MS D 1 in terms of bias and coverage rate, MS D 1 is still reasonable and is more clinically appropriate to be used to estimate the MS D as a conservative approach since we want to select a dose which has less chance to have toxicity (see Section 3.2 for more details).We will denote MS D for MS D 1 in the sequel.

Joint Nonlinear Modeling of Continuous Efficacy and Safety Outcomes
In this section, a joint nonlinear modeling of continuous efficacy and safety outcomes is introduced, and methods of parameter estimation and strategies of finding MED and MS D based on the joint model are proposed.The performance of the proposed methods is evaluated through simulation in Section 3.3.

Joint model formulation, log-likelihood function, and estimation and computational methods
Let Y i j and Z i j be the efficacy and safety outcomes respectively for jth patient in ith dose group with dose d i , i = 1, ..., k, and j = 1, ...., n i , where We assume that the variance-covariance structure between Y i j and Z i j is the same within the same dose group and may be different across dose groups.This formulation results in a simplified version of the extended nonlinear joint regression model as follows: where ] , BV N(µ, Σ) is for the bivariate normal distribution with mean vector µ and variance-covariance matrix Σ, and refers to the vector of model parameters, i to the dose group (i = 1 corresponds to placebo), and j to the patient within dose group i.
For computational convenience, the response variables Y i j or Z i j can be combined into a single variable V i jt with the indicator variable t, i.e., then the model can be reformulated as where ϵ i j ∼ N(0, σ 2 Λ i ) and Λ i is a positive-definite matrix.The matrices Λ i are determined by fixed, generally a small set of parameters λ.
Estimation and inference under a single response nonlinear model have been studied in Pinheiro and Bates (2000).
When the Λ i matrices are known, this is referred to the generalized nonlinear least-squares (GNLS) model.The model in ( 1) can be re-expressed as a "classic" nonlinear model: where ϵ * i j ∼ N(0, σ 2 I 2 ), I 2 is a 2 × 2 identity matrix.The log-likelihood function for the GNLS model in (2) can be written as: where n i , N represents the total number of observations.
The estimation and inference methods discussed in Chapiter 7.5 of Pinheiro and Bates (2000) can be applied to find the MLE estimates based on the log-likelihood function in (3).The gnls function (Pinheiro and Bates (2000) in R can be used to fit the extended nonlinear regression model using maximum likelihood.It can fit the reformulated model in (2) that is equivalent to the model in (1).Efficacy or safety data for the same patient are correlated and variances are usually unequal across dose groups.These two components can be reflected in the within-patient error variance structure by using the weight and correlation arguments in gnls .
2 A. With separate model fitting: The next paragraph describe how to estimate the MED and MS D from separate model fittings when ignoring the correlation between the efficacy and safety responses: 1.The following set of candidate efficacy models are chosen for fitting the data by MCP-Mod (Bretz et al. 2005): Linlog, Emax, exponential, and quadratic, while the Linglog, linear, Emax and exponential models are selected for fitting safety data.
2. Fit the efficacy data separately to choose the best model with Type I error α = 0.05 and the lowest Akaike information criterion (AIC).If PoC of efficacy is established, fit safety data separately and choose best model with Type I error α = 0.2 and the lowest AIC.
3. Get the MED and MS D based on the models from Step 2.

B. With joint model fitting:
After PoC for both efficacy and safety are established, we first get all the significant efficacy and safety models, of which t-statistics are greater than the critical values (q Y , q Z ) while controlling the family-wise error rate (FWER) from separate fittings.Type I error α = 0.05 is pre-specified for PoC of efficacy and Type I error α = 0.2 is pre-specified for PoC of safety, then the following two strategies can be used to obtain the MED and MS D from the joint model fitting: Strategy I: Keep the most significant model obtained from separate efficacy and safety model fitting 1. Select the most significant efficacy and safety models based on the lowest AIC criteria from the set of significant efficacy and safety models obtained from separate fittings, keep the parameter estimates to use as start values for joint model fittings.
2. Fit Joint model by generalized nonlinear least squares model.
3. Obtain MED and MS D based on updated efficacy and safety model from joint fitting.
Strategy II: Keep all the significant models obtained from separate efficacy and safety models 1.The parameter estimates for all the significant efficacy and safety models obtained from separate fittings are kept and used as initial values for the joint model fitting.
2. Joint model fitting for all the combinations of efficacy and safety models selected from separate model fitting in Step 1, choose the best combination based on the lowest AIC from all the joint models fitting.
3. Obtain MED and MS D based on the updated efficacy and safety models from the best joint model.
We evaluate these two strategies in Section 3.3.4.

Suggested Dose(s) for the Phase III Program Development based on Joint Continuous Safety and Efficacy Responses
After MED and MS D are obtained with MS D ≥ MED, we propose two different methods to select an optimal dose or a dose range for Phase III.The first method will focus on the joint success criteria for the efficacy and safety in Phase III; the second method will use the utility function to identify the final dose(s) to carry into Phase III program.Another method to determine the recommended dose is based on the following utility function.
= g(d, θ Z ) σ Z , where the parameter estimates are based on the joint or separate model fitting.

Evaluation of Proposed Methods through Simulations
In this section, simulation studies are performed to evaluate the proposed methods in Section 2. The parameters used in the simulations are based on the data from our motivating example, a real clinical trial in which both safety and efficacy outcomes need to be considered simultaneously.

Motivating Example
Angiotensin-converting enzyme (ACE) inhibitors are used to treat hypertension and congestive heart failure.They have been clinically shown superior to other classes of drugs in the reduction of morbidity and mortality for cardiovascular disorders and hypertension (Thomas, 2000;Rossi, 2004).However, renal impairment is a significant adverse effect of all ACE inhibitors.The reason for this is still unknown.Some suggest that it is associated with their effect on angiotensin II-mediated homeostatic functions such as renal blood flow.Renal blood flow may be affected by angiotensin II because it vasoconstricts the efferent arterioles of the glomeruli of the kidney, thereby increases glomerular filtration rate (GFR).Hence, by reducing angiotensin II levels, ACE inhibitors may reduce GFR, a marker of renal function.
In one clinical trial an ACE inhibitor (Drug A) is used to treat hypertension.The efficacy endpoint is the change of sitting diastolic blood pressure from baseline.Decreasing GFR is the undesirable effect and the main safety measure is the change of GFR from baseline.Both response variables are assumed to be normally distributed.When the bivariate continuous outcomes are considered to find a suitable dose, joint model fitting that accounts for the correlation between efficacy and safety outcomes should be superior to the separate modeling that ignores the correlation.

Evaluation of Target Dose Estimation for Safety Endpoint
We evaluate the two proposed estimation methods discussed in Section 2.1 through simulation.We assume that the mean dose-response model g(d i , θ Z ) for the safety responses Z i j (change in GFR from baseline) is described by an exponential model with mean 0.163 + 0.037 exp(0.5912× dose).The standard deviation for the change in GFR from baseline is 8 ml/min/1.73m 2 , and the clinical relevance for MS D is the change in GFR from baseline less than 5 ml/min/1.73m 2 .The placebo effect for mean change of GFR from baseline is assumed to be 0.2 ml/min/1.73m 2 .The true MS D from the safety mean model is 0.829.
Let σ 2 Z = 64, n = 100 per dose group and dose = 0, 0.05, 0.2, 0.4, 0.6, 0.8 and 1.Data are simulated 1000 times from the above safety model.The performance of the two proposed estimates of MS D are evaluated.
Figure 2 (Boxplots) shows that, as expected, MS D 2 is less biased and less dispersed than MS D 1 .However, for safety endpoints, it is more desirable and more clinically appropriate to use more conservative approaches so that the dose selected has less chance to be toxic in practice.From the Figure, we see that MS D 1 is still reasonably well to be an estimate of MS D. Hence MS D 1 is used in our simulation and is denoted as MS D.

Evaluation of Joint Nonlinear Modeling of Safety and Efficacy Outcomes
We now evaluate the methods discussed in Section 2.2 through simulations.

Simulation Setups
We assume the joint model for the efficacy and safety responses in (1) with The joint model fitting is conducted as described in Section 2.2.1.
The parameters used in the simulations are based on the related values from the motivating example.For the simulation purpose, we assumed that the change in diastolic blood pressure (DBP) from baseline follows the Emax model f (d i , θ Y ) = e 0 + e max × d i /(e d50 + d i ) = 2.5 + 14.5 × d i /(0.2 + d i ), and that the change in GFR from baseline follows the exponential model g(d i , θ Z ) = e s0 + e 1 exp(d i /δ) = 0.163 + 0.037 × exp(3.3× log(6) × d i ).The standard deviations for the changes from baselines in diastolic blood pressure and GFR are 7 mmHg and 8 ml/min/1.73m 2 respectively.The clinical relevance for MED is more than 3 mmHg of the difference in the change in DBP from baseline between treatment and placebo group, and for MS D is less than 5 ml/min/1.73m 2 of the difference in the change in GFR from baseline between treatment and placebo group, respectively.The placebo effect for the mean change in DBP from baseline is usually around 2.5 mmHg and the mean change in GFR from baseline is 0.2 ml/min/1.73m 2 .Based on the above, the true MED from the efficacy mean model is 0.056 and the true MS D from the safety mean model is 0.829.
Let Ψ YY = 49, Ψ ZZ = 64, and dose = 0, 0.05, 0.2, 0.4, 0.6, 0.8 and 1.Data are simulated 1000 times with n = 50 per dose group for evaluation of parameter (θ Y and θ Z ) estimates and 500 times with n = 50 per dose group for evaluation of MED and MS D estimates for each ρ = 0, 0.4, 0.8, from the above joint efficacy and safety models.
After data are simulated, models are fitted separately and jointly with Emax model for efficacy response and Exponential model for safety response for evaluation of model parameter estimations.The mean, standard deviation, median, bias, percent of bias, MSE and relative efficiency for parameter estimates are computed.For evaluations of the estimates of MED and MS D, the strategies described in Section 2.2.2 are used with candidate models for efficacy and safety listed therein.Joint model fitting is performed using gnls function (Pinheiro and Bates, 2000) in R.

Evaluation Methods
The median, mean, percent bias, standard deviation, and mean squared error of the parameter (θ Y and θ Z ) estimates, and MED and MS D estimates from the separate models and the joint model are computed from the 1000 and the 500 simulations, respectively.The estimates of the correlation between bivariate responses and the variances of the efficacy and safety responses from the gnls output in R are used to verify the correctness of simulation and goodness of the covariance matrix estimation of the model.
The parameter (θ Y and θ Z ) estimates for the joint model and the separate models are compared by the percent of bias and the relative efficiency.In addition, the MED and MS D from the joint or the separate fitting are compared with the true mean from the selected dose-response models.The mean/median bias, percent bias, bootstrap 95% confidence interval coverage for MED and MS D are computed.Also the percentage of simulated trials for which: • true MED MED true MS D, • MED > true MS D, and • 0 < MED true MED are calculated.

Simulation Results for Parameter (θ Y and θ Z ) Estimates
Parameter estimates for Emax and exponential models either from the separate or the joint fitting are shown in Table 1.When the efficacy and safety responses are independent, the parameter estimates of Emax and exponential models are similar between separate and joint model fitting.This can be seen from the mean bias, percent of bias, mean squared error and relative efficiency of parameter estimates.When data are not correlated (ρ = 0), the mean bias, percent of bias, mean squared error are a little bit bigger for joint fitting than for separate fitting because the responses for the underlying model are independent and joint fitting has an additional redudant parameter to estimate.But the differences between the two types of model fittings are minimal.When data are correlated (ρ = 0.4, or 0.8), the mean bias, percent of bias, mean squared error are smaller for joint fitting than for separate fitting.In addition, the efficiencies for joint modelling increase when the correlation between the efficacy and safety responses increases.As seen in Table 1, efficiencies increase from 0.98 − 1.0 to 1.2 − 3.8 for all parameter estimates when the correlation increases from 0 to 0.8.Table 2 shows the results of MED and MS D from joint and separate fitting when correlations ρ of efficacy and safety responses are 0, 0.4, and 0.8.When ρ = 0 (independent), percent of bias and mean squared error for MED are similar between separate model fitting and joint model fitting methods described in Section 2.2.2; mean squared error and percent bias for MS D are slightly better for the joint model fitting (using the approach in Strategy I) than separate fitting.mean squared error decreases from 0.00323 to 0.00183.

Simulation Procedure
The following is the simulation setup: 1. σ y , σ z , ρ are from the previous joint model fitting (these values will be retained from the previously described Strategy II, which select the most significant joint model from all combinations).µ y and µ z will be derived from the mean models for efficacy and safety through the simulation results based on the joint model fitting.The true correlation ρ between efficacy and safety response is 0.8.In the top row of Figure 3, the first two plots show the probability of efficacy response (change from baseline) > 3 and the probability of safety response (change from baseline) < 6 from separate fittings, respectively.The right most plot shows the joint probability for efficacy response (change from baseline) > 3 and safety response (change from baseline) < 6.The simulated probability is very close to the true probability, which results in the similar final best dose satisfying the joint efficacy and safety criteria.The best dose that satisfies the joint criteria based on the true parameter value is 0.47 with 66.03% of success probability while the best simulated dose is 0.45 with 65.65% of success probability.The set of doses with probability > 0.6 of satisfying the joint criteria are from 0.21 to 0.71 based on true parameter, and from 0.21 to 0.69 based on the simulated parameter estimates.
The middle row of Figure 3 shows the utility index based on the probability of the efficacy response (change from baseline) > 3, the safety response (change from baseline) < 6 and different weights k for the safety response.As seen in Figure 3, the simulation shows that when k increases from 0.2 to 0.8, which means the efficacy is discounted more for safety, the maximum utility index increases from 1.074 to 1.508 and the best dose to satisfy this criteria decreases from 0.63 to 0.48.
Similarly, the simulated best dose for standardized efficacy and safety response is very similar to the true dose based on the true parameter value (third row of the plots in Figure 3).Furthermore, the simulation shows that when

Figure 1 .
Figure 1.Maximum Safety Dose (MS D) 2.3.1 Method I: Identify the dose(s) through the joint criteria of continuous efficacy and safety responses for Phase III programThe recommended dose(s) will be determined byargmax d∈[MED, MS D] P(Y > a, Z < b | d) ≥ c, or a dose range in [MED, MS D] such that P(Y > a, Z < b | d) ≥ c,where Y is the efficacy variable and Z is the safety variable, a and b are the criteria for Phase III success and c is the success probability.The probabilities are calculated based on the joint bivariate distribution estimated from the joint fitted model in Section 2.2.1.2.3.2Method II: Utility function based on trade off of continuous efficacy and safety responses for Phase III program The dose is determined by maximizing the utility function F(d), i.e., argmax d∈[MED, MS D] F(d).Here, k > 0 is some weight to discount efficacy for safety.When k increases, we put more weight to safety, thus expect to lower the dose.The functions eff(d)and saf(d) can represent either 1) P(Y > a | d) and P(Z < b | d) respectively, with estimated marginal density function derived from either the separate model fitting or the joint model fitting, a and b are criteria for Phase III success probabilities, or 2) Standardized mean responses: eff

Figure 2 .
Figure 2. Box Plot of Maximum Safety Dose (MS D)

2.
Criteria values a and b are given such that the DBP change from baseline is more than certain value (a = 3 mmHg) and GFR change from baseline is less than certain value (b = 6 ml/min/1.73m 2 ) in order to claim the success of the Phase III program.d e and d s are the doses which fit the mean efficacy and safety model to satisfy criteria values a and b respectively.3.d e and d s are 0.056 and 0.824 respectively, which are derived from the updated mean efficacy and safety model from the joint model fitting to satisfy the mean efficacy change of a and mean safety change of b.3.4.2 Simulation ResultsSimulation results are shown in Figure3.The simulated parameter estimates for efficacy and safety response models are based on the Emax and exponential model simulation results respectively.There are 4 lines in each plot.The upper and lower lines represent upper 97.5% and lower 2.5% of 500 values of the probabilities or utility indices based on the simulated parameter estimates.The middle 2 lines are the true probability or utility indices based on the true parameter value and the mean of simulated probability or utility indeces, respectively.

Figure 3 .
Figure 3. Suggested doses based on success probabilities (Method I) or utility functions (Method II) .2.2 Procedures of finding MED and MS D for combined efficacy and safety data In PoC clinical trials, efficacy is usually tested first.If a drug does not have the PoC for efficacy, then there is no continued assessment for the safety endpoint.The significance level α for safety dose-response testing can be relaxed.The following procedures provide the testing sequence when both efficacy and safety endpoints need to be considered for the drug development.Denote the discrete dose set D = {d 1 , ..., d k } under investigation and the entire dose range (d 1 ,d k ].Let d 1 < d 2 < ... < d k .The first step is to confirm whether PoC exists for the efficacy, say, with Type I error α = 0.05.If there is no PoC established for efficacy, which indicates no dose-response relationship exists for efficacy or MED > d k , then there will be no dose-finding continued for this drug, i.e., the drug will not be carried to Phase III development.When the PoC of efficacy is established and also MED ≤ d k is satisfied, the second step is to test PoC of the safety response, say, with Type I error α = 0.2.When there is no established PoC for the safety response, the efficacy response alone is studied further to identify the MED for the Phase III program.In other word, we only need to focus on the dose finding for efficacy when the dose-response curve for safety is flat and no dose-response relationship is present for safety.When there is established PoC for the safety response, if MS D < MED, there will be no dose-finding continued for this drug.On the other hand, when both PoC for efficacy and safety responses exist and MS D ≥ MED, joint modeling for efficacy and safety responses are performed.The methods of finding MED and MS D under separate model fitting and joint model fitting are as follows:

Table 1 .
Parameter estimates based on separate and joint modeling -Emax for efficacy and Exponential for safety

Table 3 .
Comparisons of MED, MS D to true mean based on separate and joint fittings Note: outputs are from 500 simulations and n=100 per dose group by separate and joint model fitting.