Journal of Biopharmaceutical Statistics Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/lbps20

Determining Causal Exposure-Response Relationships With Randomized Concentration-Controlled Trials a

Jixian Wang a

Novartis Pharma AG, Basel, Switzerland Accepted author version posted online: 03 Apr 2014.Published online: 14 May 2014.

Click for updates To cite this article: Jixian Wang (2014) Determining Causal Exposure-Response Relationships With Randomized Concentration-Controlled Trials, Journal of Biopharmaceutical Statistics, 24:4, 874-892, DOI: 10.1080/10543406.2014.901342 To link to this article: http://dx.doi.org/10.1080/10543406.2014.901342

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Journal of Biopharmaceutical Statistics, 24: 874–892, 2014 Copyright © Taylor & Francis Group, LLC ISSN: 1054-3406 print/1520-5711 online DOI: 10.1080/10543406.2014.901342

DETERMINING CAUSAL EXPOSURE-RESPONSE RELATIONSHIPS WITH RANDOMIZED CONCENTRATION-CONTROLLED TRIALS Jixian Wang

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Novartis Pharma AG, Basel, Switzerland Determining causal effects in exposure-response relationships is an important but also a challenging task since confounding factors that affect both drug exposure and response often exist and lead to confounding biases in causal effect estimation. Randomized concentration control (RCC) trials are designed to eliminate or to reduce the confounding bias. However, statistical issues in the design and analysis of these trials have not been examined closely in the literature. Analysis of dose-exposure relationship may also be affected by confounding factors if they affect dose adjustments. We examined these issues and suggest methodological and practical solutions. In particular, we proposed using instrumental variables (IV) for the estimation of causal effects in both exposure-response and dose-exposure relationships. We also examined the impacts of confounded treatment heterogeneity on the IV estimate for RCC trials. We illustrated these approaches with a trial design scenario showing the importance of considering multiple practical factors that may alter the performance of the IV estimate. The performance of the IV estimates was examined by simulations for a wide range of scenarios. The results showed clear advantages for the IV estimates over routine estimates. Some situations in which the IV estimates may fail were also identified. Key Words: Confounding bias; Dose-exposure model; Exposure-response model; Instrumental variable; Randomized concentration control.

1. INTRODUCTION The analysis of drug exposure-response relationship is often impaired by confounding factors that affect both drug exposure, as a pharmacokinetic measure such as drug concentration, and its response. In a clinical trial, the drug concentration is observed rather than randomized, and hence, patients having different concentration levels may have different characteristics relating to their response. These characteristics are known as confounding factors in exposure-response analyses. For example, elderly people may have lower drug clearance due to reduced hepatic and renal functions and, hence, may have higher drug exposure than young people. They may also be at higher risk of having adverse events (AE). Therefore, an apparent concentration–AE correlation may be observed even if the drug does not cause it. To eliminate the bias due to age, when modeling the concentration-response relationship, age has to be adjusted. Direct adjustment for confounding factors by including them as covariates in the model is an important tool, and it Received April 17, 2013; Accepted June 2, 2013 Address correspondence to Jixian Wang, Celgene International Sàrl, Route de Perreux 1, Boudry 2017, Switzerland; E-mail: [email protected]

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is simple and efficient. However, often there are unknown or unobserved confounding factors that make direct adjustment, or relying on some assumptions (such as no unobserved confounders) that are not verifiable, infeasible. The (asymptotic) bias in parameter estimates due to confounding factors is known as confounding bias. Here we use the term bias for asymptotic bias or inconsistency, for simplicity. The issue of confounding bias in exposure-response modeling has been emphasized in the Food and Drug Administration (FDA) guidance document for exposure-response relationship (FDA, 2003). The guidance recommended using randomized concentrationcontrolled (RCC) trials (Sanathanan and Peck, 1991; Levy, 1993; Kraiczi et al., 2003; Grahnen and Karlsson, 2001; Holford et al., 1993; Garcia et al., 2009; Lledo-Garcia et al., 2009; Karlsson et al., 2007) as one approach to deal with confounding factors. In an RCC trial patients are randomized into multiple groups, each with a predetermined concentration range. During the trial repeated concentration measures are taken from each patient and the dose is adjusted till the concentration is within the specified range. After the concentration is in the target range, safety and efficacy measures are also measured as the response to the exposure in the target range. The randomization makes patients in different concentration ranges comparable; hence comparison of efficacy and safety endpoints between the groups is free from confounding bias. However, often there is more useful information than the simple comparisons can give. For example, in a trial with target concentration ranges lower or higher than 100 ng/mL, suppose the response in the higher range is 1 unit higher than in the lower one. Although the result may suggest that a higher concentration leads to a higher response, it may not be sufficient to quantify the exposure-response relationship due to the wide concentration ranges. Although one may overcome this problem by using more dose ranges, it makes trial implementation more difficult. Therefore, RCC designs commonly have two or at most three ranges (Kraiczi et al., 2003). RCC trials have been increasingly used, particularly in the last 15 years. Kraiczi et al. (2003) reviewed a large number of RCC trials and also discussed the motivation, use, and limitation of RCC trials. More recent discussion on design and analysis issues can be found in Karlsson et al. (2007). They examined the analysis procedures used for a large number of published RCC trials and found that most analyses were simple comparisons between different exposure ranges rather than model-based approaches. They showed that model-based approaches with a correct model had considerable advantages over the simple comparisons, in particular on bias reduction. An alternative to RCC is the randomized dose controlled (RDC), trial in which patients are randomized into multiple dose groups so that the exposure differences between groups are controlled to some extent. Recent discussion on advantages and disadvantages of the two types of design can be found in, for example, Lledo-Garcia et al. (2009), although works in this area might be traced back to the 1990s (Sanathanan and Peck, 1991). Lledo-Garcia et al. showed with simulation the superiority of the RDC design over the RCC design in precision and bias reduction in the parameter estimation and consequently in the prediction of the optimal dose and its response. But none of the work was focused on confounding bias, and statistical consideration in this aspect was lacking. Although previous simulation studies provided useful information for choosing different design and analysis approaches, statistical considerations can provide more general insight into the comparison between designs and between analyses, as well as as guiding their implementations. Dose adjustment to achieve the target exposure and its impact on the estimation of dose-exposure relationship have also been investigated by Diaz et al. (2007) and O’Quigley et al. (2010), among others. They showed that under some

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conditions the mixed model estimates are consistent, but their approaches were not focused on a causal dose-response relationship. As shown later in this article, neither RCC nor RDC trials can completely eliminate the bias in a routine estimate, such as the least squares (LS) estimate for a linear exposureresponse model without adjusting for all confounding factors, because a large portion of the variation in the exposure may still be correlated to confounding factors. An approach to eliminate the bias when using RDC trials is to use the dose level as an instrumental variable (IV) for the exposure to estimate the treatment effect in an exposure-response model (Nedelman, 2005; Nedelman et al., 2007). The IV approach originated from the social sciences, particularly econometrics (Wooldridge, 2010), but recent applications in biostatistics have been seen in many areas, for example, to deal with noncompliance and treatment change that may relate to potential response in clinical trials (see, e.g., Angrist et al., 1996). The approach is very easy to implement for linear models and Poisson-type models (Wang, 2012). Determining causal effects has been a very active research area in statistics, where one can find recent developments on approaches dealing with issues such as noncompliance and treatment adaptions in clinical trials; the IV approach is among these. This research was motivated by a practical scenario in the design for a trial to determine the exposure-response relationship for a drug with highly variable exposure. A number of responses including efficacy and safety endpoints are of interest. Between these and drug exposures there are likely unknown or unmeasured confounding factors. Therefore, direct bias adjustment is infeasible, and alternative approaches to eliminate confounding biases are needed. Both RCC and RDC designs are possible candidates for this trial; the choice between them may depend on multiple factors, such as their efficiency in terms of the accuracy of the estimated exposure-response relationship, and feasibility. For RDC designs the main design parameters are the dose levels, while for RCC designs one has to decide the concentration ranges, dose adjustment algorithm, and starting doses. In addition, the analysis method to be used should also be taken into account. Suppose that a simple least-square estimate will be used to estimate the model parameter in a linear exposure-response model, and we would like to find out the bias in the estimate with an RCC and RDC design. Since the IV estimate can be used for RDC designs, we would also like to examine whether the approach can be used for RCC designs as well and how robust the approach is to some complications such as confounding in treatment effect heterogeneity. Here treatment effect heterogeneity refers to the variability in treatment effects among patients. The variability may relate to some factors that also have an impact on the exposure. For example, if the exposure causes a higher risk in males than in females, and the exposure is also higher in males, then gender is a confounding factor in treatment heterogeneity. As an illustration, consider a hypothetical example in which, in a population of equal numbers of males and females, the exposure levels are either low or high and the effect of high level versus low level is E = 2 on average, but for male it is Em = 3 and for female it is Ef = 1. For simplicity we assume everyone has effect level Eb = 0 under the low exposure. Therefore, if the exposure is independent of gender, then the effect difference between those receiving the high and low exposures is 0.5(Em − Eb ) + 0.5(Ef − Eb ) = 0.5 × (3 + 1) = 2. However, suppose that gender is a confounding factor so that all males received high exposure and females received low exposure. Then the effect difference becomes Em − Eb = 3 − 0 = 3 rather than 2. Therefore, gender as a confounding factor will cause bias in the average effect if it is estimated by the difference between high- and low-exposure groups, regardless of gender. Em and Eb are known as local effect, but here we concentrate on the average effect.

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In this article we examine statistical issues in design and analysis of RCC trials, particularly the use of IV estimates to determine causal drug effects. The next section introduces dose-exposure and exposure-response models for RDC and RCC trials. Section 3 introduces an IV estimate for RDC design and shows that the IV estimate is robust to confounding factors not only in patient baseline characteristics but also in treatment heterogeneity in a random coefficient model. In section 4 we show that the IV estimate can be used with an RCC design, but is not robust to confounding factors in treatment effect heterogeneity. In section 5 we come back to the practical scenario that motivated this research and examine the performance of each design and analysis with a simulation. To compare the designs and analysis approaches under more general scenarios, another simulation was conducted and results are reported in the following section.

2. EXPOSURE-RESPONSE AND DOSE-EXPOSURE MODELS AND DOSE ADJUSTMENT We introduce general exposure-response and dose-exposure models for RDC and RCC trials. Let yik and cik be the response and exposure of patient i in randomized group k, k = 1, ..., K, which may be an exposure range or a dose level, with nk subjects. We assume the following exposure-response model: yik = XTik γ + βcik + uik ,

(1)

where Xik is a set of covariates (nonconfounders) including the intercept, γ are corresponding parameters. β is the slope representing the exposure-response relationship, and uik is a random term including the effect of potential confounders. Therefore, we assume that uik and cik may be correlated, and hence the LS estimate βˆLS is biased. The correlation between uik and cik can be introduced in the following dose-exposure model. Let dik be the dose given to patient i in group k. We assume that the drug exposure cik follows a linear model, cik = α0 + αdik + vik ,

(2)

where α0 and α are coefficients and vik is a random term including inter- and intrapatient variabilities of cik for a given dose. For RDC trials, dik = dk , the kth dose level, while for RCC trials dik is a random variable. A typical example of model (2) is the power model, in which cik is the log-concentration and dik is the log-dose, commonly used in clinical pharmacology. For example, age and weight are common factors affecting drug exposure, and hence vik may represents these factors. If they also affect the response, they will also be correlated to uik and become confounding factors in the estimation of β. A more general model for cik and dik does not need the linear relationship in equation (2). We can assume that each dose results in a certain level of exposure: cik = α T dik + vik

(3)

where dik is vector of dose level indicators, with the kth element equal to 1 and other elements equal to 0, and α = (α1 , ..., αK ) are the corresponding mean exposure levels. The

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only essential restriction in this model is that vik is additive to α T dik . For RDC trials cik = αk + vik where αk is the mean exposure level for the kth dose. In RCC trials one needs to measure the exposure repeatedly in order to adjust the dose based on the observed exposure. For this we extend model (2) to a repeated measurement model, cijk = α0 + αdijk + vik + εijk , j = 1, . . . , r

(4)

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where cijk and dijk are the jth PK measure and dose for patient i in group k, respectively, and εijk is a zero mean measurement error with variance σe2 . Note that here, in contrast to equation (2), vik only include interpatient variability, as intrapatient variability is included in εijk . We consider a simple up-and-down algorithm: dij+1k =

dijk + D cijk < Lk dijk Lk ≤ cijk ≤ Uk dijk − D cijk > Uk

(5)

where Lk and Uk are the lower and upper bounds of the kth range, and D is the step size of dose adjustment, which may not necessarily be a constant. This algorithm is the most commonly used one in practice, although more complex ones using the information of doseexposure models are also available (Diaz et al., 2007; O’Quigley et al., 2010). At steady state, we may assume that based on model (3), cik = α T d∗ik + vik

(6)

where d∗ik is an indicator of the dose level subject i has. One can also calculate E(cik |k) = α T E(d∗ik |k) from the average dose level in group k. 3. RANDOMIZED DOSE-CONTROLLED TRIALS AND IV METHODS This section serves as an introduction to the approach of using the dose as an IV for RDC trials. Although an RDC trial randomizes patients to multiple dose groups so that patients in different groups have different mean exposure levels by randomization, simply using a regression for yik on cik without adjusting for confounders may result in confounding biases in the estimate of β. If the drug has very high intersubject variability while differences between designed dose levels are small, the exposure-level variation may still mainly be driven by confounding factors, and hence the confounding bias can be high. Using randomized dose levels as an IV, one can construct an unbiased estimate for β. An IV is a variable that correlates to the exposure but only correlates to the response through the exposure; that is, it has no direct correlation to the response. For doses in an RDC trial the first condition is satisfied due to the dose-exposure models such as equation (2), while the other is obvious since the doses are randomized and hence independent of factors that relate to the response. The correlation between cik and dik depends on the variations in dik s and vik s. When the variation in dik s is small and var(vik ) is large, for example, when the drug exposure is highly variable, the correlation may be weak, although not zero. In this case, the IV is known as a weak IV and should be used with care or may not be used. The dose level may be used directly as an IV; it may also be used to derive an IV. Often the best choice as a derived IV is E(cik |dik ), which can be estimated from model

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(2) or model (3). For model (2) using the dose directly as an IV and using E(cik |dik ) are equivalent. For simplicity, we do not distinguish them., except when it is necessary. For a linear and a few special nonlinear models, using E(cik |dik ) as the IV leads to the following two-stage IV (2SIV) approach, which is very easy to implement:

r Fit the dose-exposure model to the exposure data and obtain an estimated mean exposure ˆ ik |dik ) for each dose level. E(c

r Fit the exposure-response model with the exposure replaced by E(c ˆ ik |dik ) from the first

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step. The IV estimate βˆIV is the coefficient for the predicted exposure.

Although βˆIV does not have confounding bias, its small sample bias depends on how good the dose is as an IV. When the IV is very weak, βˆIV may be worse than the simple LS estimate, as sometimes happens in observational studies. The issue of weak IVs has been intensively investigated for analysis of observational studies. However, a properly designed RDC trial should make a large proportion of exposure variation attributed to between group difference to avoid the dose becoming a weak IV. In the following we review the basic concept and theory of IV-based methods for linear models, particularly exposure-response models with random coefficients, as they are less well known in the statistics literature. The content is mainly adapted from Wooldrige (2007, 2010). First we consider the linear model (1) with fixed coefficients. An LS estimate βˆLS for β can be obtained by regressing yik on Xik and cik regardless of the group k. Although the IV estimate βˆIV can be obtained by the preceding two-stage method, it is also the solution to the following estimating equation:

S (θ) =

nk K

ZTik yik − XTik γ − cik β = 0,

(7)

k=1 i=1

where θ = (γ , β) and Zik = (Xik , dik ) (or Zik = (Xik , E(cik |dik )). Here, we treat Xik as its own IV, so the solution to (7) γˆ is an IV estimate for γ . We further assume that E(Zik ) = 0, since otherwise we can centralize it as Zik − E(Zik ). For simplicity, one may consider nonconfounders Xik as their own IVs so we refer to Zik as a set of IVs. The estimating equation condition for (EE) (7) plays a key role in checking whether θ IV is unbiased since the key k E(Zik uik ) = 0, the unbiasness of βˆIV is E(S(θ )) = 0, which is satisfied here since Kk=1 ni=1 due to independence between dik and uik in RDC trials. Note that the standard error (SE) of θˆ IV given in the second-stage regression is not correct. The following sandwich variance estimate for θ IV is often used: Vˆ (θ IV ) = σˆ 2 (

nk K k=1 i=1

ZikT Uik )−1

nk K k=1 i=1

ZTik Zik (

nk K

ZTik Uik )−1 ,

(8)

k=1 i=1

where Uik = (Xik , cik ), and σˆ 2 is the sample variance of the residuals rik = yik − UTik θ IV . Although the estimate is robust, it is worthwhile to check its small sample properties, particularly when the IV is weak. Next we consider confounding in treatment heterogeneity. To introduce random coefficients in model (1) we assume that the model coefficients βi for each patient i in model (1) are random variables following

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θ ik = θ + sik ,

(9)

where θ are the population means and sik = (sxik , scik ) are the random components with E(sik ) = 0. This leads to a model

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yik = XikT (γ + sxik ) + cik (β + scik ) + uik ,

(10)

where scik represents treatment effect heterogeneity and may also be confounded with cik . The confounding component in sxik does not cause confounding bias in βIV since XTik sxik can be treated as a part of uik . However, the term cik scik may cause problem. First E(cik scik ) may not be zero although E(scik ) = 0. But this will only lead to a biased estimate for the intercept. We therefore can assume E(cik scik ) = 0. Note that here we assume no repeated measures from each subject are taken, and concentrate on the IV estimates of solving the EE (7). Based on model (10), equation (7) can be written as S (θ ) =

nk K

ZTik XTik sxik + cik scik + uik = 0

(11)

k=1 i=1

The condition of E(S(θ )) = 0 only relies on E (Zik cik scik ) = 0

(12)

since we have assumed that Xik and dik in Zik are independent of sxik and uik . For condition (12) the dose-exposure model is sufficient, since E (Zik cik scik ) = E (Zik scik (α0 + αdik + vik )) = E (Zik ) E (scik vik ) = 0

(13)

in which Xik is independent of scik and E(scik vik ) = 0 as we assumed. Here we used E(Zik ) = 0 and cov(scik vik |Zik ) = cov(scik vik ), which is satisfied when vik is additive in the model for cik . Note that E(scik cik ) = 0 is not sufficient for E(S(θ )) = 0 since cik might be Zik dependent. On the other hand, the linear dependence of cik on dik is not necessary since equation (12) holds for model cik = g(dik ) + vik as well. In summary, for RDC trials, doses can be used as instrumental variables for exposure-response relationship analysis using linear models with and without random coefficients. The approach does not need to estimate (scik ) separately, since it is not needed in the EE (7). The robust variance estimate (8) is also valid in the presence of variance heterogeneity, which leads to an extra random component cik scik . Although the estimating equation does not take this into account, the extra variation is in the residual rik and is accounted for in σˆ 2 = var(rik ). When model (3) is used, that is, dose is treated as a categorical variable, each element in dik can be used as an IV. Therefore, dik contains multiple IVs, which are more than needed to identify β. One approach to solve this issue is to use a summary of dik as the IV. In our T case, a natural choice is the predicted exposure cˆ ik = αˆ dik as the IV. This is an estimate of E(cik |dik ), the conditional exposure given dose level dik , which, under some conditions, is the optimal IV. A technical advantage of this choice is that it leads to the two-stage approach in section 2 too. The preceding argument for the unbiasness of βˆIV also applies to the IV estimate using model (3).

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4. IV APPROACHES FOR RCC TRIALS

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4.1. IV Estimate for β in Exposure-Response Models RCC and RDC trials are similar in the sense that patients in multiple groups with different mean exposure levels are exchangeable. Since the IV method can be used for RDC trials, it is natural to ask whether the randomized exposure ranges can be used as an IV in RCC trials. One important difference between RDC and RCC is that for the latter, randomized groups can only be treated as a categorical variable, even if model (3) is assumed. Formally we denote Ri , which takes values 1, ..., K, as the randomized ranges. In this case, a derived IV based on Ri is E(cik |Ri = k). We start from a relatively straightforward situation under which βˆIV for a RCC trial is also unbiased. It is easy to verify that for model (1) the two-stage IV estimate has no confounding bias using equation (7) together with Zik = (Xik , E(cik |Ri = k)), since yik − XTik γ − cik β = uik holds for both RCC and RDC trials. The condition E(S(β)) = 0 follows the fact that the independence between Ri and uik also holds with RCC. However, although confounding in treatment heterogeneity does not lead to bias in βˆIV with RDC trials, it causes a problem when RCC trials are used. In RDC trials the mean exposure in each dose group is well determined, while in RCC trials, although the exposure ranges are well defined, the mean exposure in each range group depends not only on the dose and subject’s characteristics, but also on the dose-adjustment process as well as the starting dose. One can rewrite, for example, model (2) as cik = E (cik |Ri = k) + v∗ik ,

(14)

with v∗ik = vik + α(dik − E(dik |Ri = k)). However, a fundamental difference is that vik is independent of the dose levels, but v∗ik may depends on Ri . Formally we can examine whether βˆIV is biased under this setting by checking whether the condition E(S(θ )) = 0 is met. Following equation (7), the EE with random coefficient βi can be written as S (θ)

= =

nk K k=1 i=1 nk K k=1 i=1

ZTik (cik scik + uik ) ZTik

μk + v∗ik scik + uik = 0,

(15)

To satisfy E(S(θ )) = 0, the key condition is E(Zik (μk + v∗ik )scik ) = 0, or more specifically E(E(cik |Ri )v∗ik scik ) = 0. However, E(E(cik |Ri )α(dik − E(dik |Ri = k))scik ) = 0 for an arbitrary scik , for example, when scik is an arbitrary function of (dik − E(dik |Ri = k)). Note that (dik − E(dik |Ri = k)) is the nonrandomized variation in dik , and hence scik as a confounder may depend on it. Therefore, the IV method cannot eliminate confounding bias caused by scik , as can be found in simulation results later. In contrast, for RDC trials, there is no term (dik − E(dik |Ri = k)) in the EE, and hence the problem does not occur there. 4.2. IV Estimation for Dose-Exposure Relationship Often it is also desirable to estimate α in the dose-exposure model (2) since it is the key parameter representing dose-exposure relationships. As estimating α with an RDC trial is straightforward, we concentrate on RCC trials. For an RCC trial one needs to take repeated PK measures for dose adjustment. From equation (4), cijk depends on

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(dij1 , ..., dij−1k ) and consequently on vik so that cijk is confounded; hence, fitting a mixed effect model generally results in a biased estimate for α. Note that the α estimate with a fixed-effect model approach treating vik s as unknown parameters does not suffer from confounding bias, since it is based on within-subject comparisons and vik is eliminated. A similar approach to the fixed effect model is to center both cijk and dijk within patient to eliminate vik directly, a common approach for panel data analysis in econometrics. Let c¯ i.k , d¯ i.k and ε¯ i.k be the individual means of cijk , dijk , εijk , respectively, and c¨ ijk = cijk − c¯ i.k , ∗ = εijk − ε¯ i.k are the the centered data following d¨ ijk = dijk − d¯ i.k , and εijk

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∗ c¨ ijk = α d¨ ijk + εijk .

(16)

The parameter α can be estimated by a simple LS together with a robust standard error estimate, or a mixed effect model, if there is serial correlation between repeated measures. We denote the estimate as αˆ w . However, the information of exposure change due to dose-level difference between patients has no contribution to this estimate. In general, the information is confounded due to vik , and hence the routine approach of using a mixed model approach to recover the information cannot apply here. However, in an RCC trial one can recover the information with randomization as the IV. Specifically, the following two-stage approach provides an estimate αˆ b for α, based on c¯ i.k and d¯ i.k without confounding bias:

r Fit a linear model with the randomized exposure range groups as a covariate to individual mean dose levels d¯ i.k and predict the expected mean dose for each group.

r Fit a linear model with predicted dose levels as a covariate to individual mean exposures. With the two unbiased estimates for α, we can combine them to obtain a more accurate estimate. A common choice of the estimate is the weighted average using inverse variances of the estimates as weights. Formally, we can use ∗ = αˆ w Ww + αˆ b Wb / (Ww + Wb ) , αˆ IV

(17)

∗ )= where Ww = var(αˆ w )−1 and Wb = var(αˆ b )−1 are the weights. This leads to var(αˆ IV 1/(Ww + Wb ), which can be easily verified to be the minimum variance estimate among all linear combinations of αˆ w and αˆ b . It is also possible to estimate α using a one-stage IV approach with randomization and within-patient dose change as IVs. However, we prefer the above separate modelling approach because of its simplicity. The estimates based on between- and within-patient variation also give more information than a single estimate.

5. A PRACTICAL EXAMPLE We use the practical scenario stated in the introduction as an example to illustrate the use of the proposed approach. The parameters presented here are not the true values, due to confidentiality, but reflect the real scenario. We would like to design the trial to identify the exposure-response relationship and to find the appropriate dose to achieve the target response level, with both RCC and RDC designs as candidates. Let cik be the logtransformed trough concentration with σv2 = 1 and σε2 = 0.32 . We assume that the efficacy

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measure yik follows the linear model (1). With safety concern, for the RCC design, conservative starting doses are preferred; so are equal starting doses for all groups. The drug has been tested up to 20 ng/mL median trough level, while the lowest efficacy level is unknown. Also unknown is the extent of confounding. The exposure is expected to be dose proportional, and model (2) with α = 1 can be used for design purpose, but it is also important to examine dose proportionality in the study population. With practical considerations we only consider two-group designs with target concentration range of 0–10 ng/mL and 10–20 ng/mL. To target the median exposure level for each group, for the RDC design the dose levels are 5 mg and 15 mg. For the RCC design we use the simple up-and-down adjustment, which is to reduce the next dose if cik is higher than the upper limit of the target range and to increase the next dose if cik is lower than the lower limit, with a dose adjustment step of 2 mg. The sample size is largely depending on feasibility and is likely to be between 50 and 100 patients. PK samples are taken every 2 weeks starting from week 2, and the response is assessed after 12 weeks with 6 potential dose adjustments. The dose-exposure model used here is cijk = α0 + αdijk + vik + εijk

(18)

where cijk is the log-trough level and dijk the log-dose of patient i in group k at visit j, vik ∼ N(0, σv2 ), and εijk ∼ N(0, σ 2 ) is a random term, with σv2 = 1 and σε2 = 0.32 . Note that this adjustment cannot ensure that after 12 weeks all cik are in the ranges as randomized. Here cˆ ik = αˆ T dik was used as the IV, where k is the kth randomized range regardless whether the patient is within the range or not. Visits j, j = 1, ...r, are introduced for the RCC design and r = 1 is assumed for the RDC design. The exposure-response model is yik = cik (β + φ2 vik ) + φvik +

1 − φ 2 eik

(19)

where cik is the log-trough level either generated by dose level dik with an RDC design or ci6k at visit 6 with the RCC design, and eik is a nonconfounding random term. Here we confounded part (φvik ) and a nonconfounded part split uik in model (1) into a completely eik , while keeping var(φvik + 1 − φ 2 eik ) = 1. In the model, vik induces confounding in baseline and treatment heterogeneity in yik and controlled by φ and φ2 , respectively. We evaluate the small-sample performance of LS estimate βˆLS and the two-stage IV estimate βˆIV with 10,000 simulation runs under a number of scenarios for both the RDC and RCC designs, programmed with R (Ihaka and Gentleman, 1996). The results for the RDC design with and without confounded heterogeneity are presented in Table 1 with sample sizes 50 and 100, and φ = φ2 = 0.5 and φ = φ2 = 0.25, respectively. With φ = φ2 = 0.5 there was a remarkable bias in βˆLS even using the RDC design, reflecting the difficulty of controlling the concentration when the exposure is highly variable. Without confounding in heterogeneity, the bias in βˆIV was negligible, but it increased when confounded heterogeneity was introduced, although still much lower than βˆLS under the same situation. Increasing the sample size reduced the bias in βˆIV considerably. The bias reduction in βˆIV was at the cost of its substantial variance increase. Note that increasing the sample size from 50 to 100 led to a variance reduction of more than 12 , suggesting with the small sample size that the variance change may not be proportional to the inverse of sample size.

884

φ2

0.50 0.50 0.25 0.25

φ

0.50 0.50 0.25 0.25

Design parameter

50 100 50 100

n 0.3567 0.3580 0.1776 0.1791

βˆLS1 1.1269 1.1314 0.5633 0.5664

βˆLS2

Bias βˆIV2 −0.1356 −0.0543 −0.0789 −0.0314

βˆIV1 −0.0391 −0.0177 −0.0197 −0.0093

0.0161 0.0077 0.0152 0.0076

βˆLS1

0.0602 0.0305 0.0257 0.0131

βˆLS2

Var

0.2363 0.0493 0.1150 0.0396

βˆIV1

1.1162 0.1756 0.3000 0.0736

βˆIV2

Table 1 Estimated bias and variance of the LS and IV estimates by simulation using the RDC design and model (19), where βˆLS(IV)1 and βˆLS(IV)2 denote the LS (IV) estimates without and with confounded treatment heterogeneity, respectively

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The small-sample performance of the two estimates with the RCC design was also examined with 10,000 simulation runs for each scenario. In addition to confounding strengths and samples sizes, design parameters also include the starting dose for each range. We consider one starting dose of 10 mg for both ranges targeting 10 ng/mL exposure: the boundary between the two regions, a conservative dose 5 mg, and two starting doses of 5 mg, and 15 mg for the low and high ranges, respectively. The simulation results are given in Table 2. Comparing with the RDC design, the RCC design reduced the bias in βˆLS substantially. The bias in βˆIV was also much lower when there was no confounding in treatment heterogeneity. The bias increased with this confounding, and under two scenarios it was even slightly higher than that of βˆLS . βˆIV had substantially higher variance than βˆLS , as in the case of the RDC design. Note that increasing the sample size from 50 to 100 had almost no impact on the bias, while the reduction on the variance was expected. Since the example is for illustration purpose, we do not disclose the real decision made. However, some general consideration under the scenarios investigated is in order. When an RDC design is used the unbiased βˆIV may have considerable advantage over βˆLS , although in practice there is often a need to make a compromise between the bias and variance of the estimates. The RCC design can substantially reduce bias in both estimates when confounding in treatment heterogeneity is not present. However, if such confounding occurs it may introduce bias in βˆIV ; hence RCC should be used with care if this situation is likely. 6. SIMULATION In order to assess small sample properties of the IV-based estimates under more general scenarios than the example, we conducted additional simulations with a number of combinations of different dose adjustment step size, starting dose, and exposure ranges, as well as the strength of confounding. We consider two dose levels for the RDC design, and for the RCC trial, two exposure ranges divided by exposure level c = 1, corresponding to dose level d = 1. D1 and D2 are starting doses (on log-scale) of the low- and high-range groups. Therefore, Dk = 0 is expected to produce a median exposure at c = 1. The adjustment step D was fixed at 0.3. The two models used in the previous section were used, but here we explore more general scenarios and examine additional properties of the estimates, such as the reliability of the variance estimates in equation (8). For the estimation of β, we use the exposure and response data after the dose adjustment ends. The exposure data were generated from model (16) with σε = 0.2, which is lower than that in the example, but the corresponding coefficient of variation is 20%, a typical value for a large number of drugs. Table 3 presents the bias and estimated and empirical variances of βˆLS and βˆIV with the RCC design when the confounding is in vik only (φ2 = 0), based on 5000 simulations for each scenario. When σv = 1 and φ = 0.5 or φ = 1, the bias in βˆLS was consistently high with all the parameter settings, while βˆIV was almost unbiased. The variance of βˆIV was much higher than that of βˆLS , and it was almost double under some scenarios with σv = 1. Note that these are the scenarios with large bias; hence, correcting the large bias had a high price to pay. When σv = 0.5 and φ = 0.5 the highest bias occurred with (D1 , D2 ) = (−0.5, 0.5), while the lowest was found when (D1 , D2 ) = (0, 0). With all the parameter settings, the estimated variances of βˆLS and βˆIV seemed unbiased, compared with the empirical ones.

886

φ2

0.50 0.50 0.50 0.50 0.50 0.50 0.25 0.25 0.25

φ

0.50 0.50 0.50 0.50 0.50 0.50 0.25 0.25 0.25

100 100 100 50 50 50 50 50 50

n

Design parameter

10 5 5 10 5 5 10 5 5

D1 10 5 15 10 5 15 10 5 15

D2 0.1238 0.1515 0.1069 0.1200 0.1466 0.1022 0.0630 0.0752 0.0500

βˆLS1 0.1626 0.1364 0.1778 0.1584 0.1289 0.1813 0.0769 0.0651 0.0890

βˆLS2

βˆIV2 −0.1262 −0.1872 −0.0999 −0.1355 −0.1977 −0.0925 −0.0723 −0.0945 −0.0515

βˆIV1 −0.0030 −0.0044 −0.0022 −0.0043 −0.0074 0.0000 0.0019 −0.0016 −0.0005

Bias

0.0234 0.0150 0.0273 0.0476 0.0303 0.0555 0.0391 0.0263 0.0455

βˆLS1

0.0626 0.0356 0.0747 0.1279 0.0759 0.1496 0.0577 0.0374 0.0713

βˆLS2

Var

0.0394 0.0250 0.0472 0.0805 0.0514 0.0964 0.0688 0.0436 0.0824

βˆIV1

0.1066 0.0650 0.1317 0.2200 0.1403 0.2654 0.1002 0.0668 0.1264

βˆIV2

Table 2 Estimated bias and variance of the LS and IV estimates by simulation using the RCC design and model (19), where βˆLS(IV)1 and βˆLS(IV)2 denote the LS (IV) estimates without and with confounded treatment heterogeneity, respectively, and D1 and D2 are the starting doses of low- and high-range groups

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φ2

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1.0 1.0 1.0 1.0 1.0 1.0

σv

1.0 1.0 1.0 1.0 1.0 1.0 0.5 0.5 0.5 0.5 0.5 0.5 1.0 1.0 1.0 1.0 1.0 1.0

0.0 −0.2 −0.5 0.2 0.5 0.5 0.0 −0.2 −0.5 0.2 0.5 0.5 0.0 −0.2 −0.5 0.2 0.5 0.5

D1

Design parameter

0.0 −0.2 −0.5 −0.2 −0.5 −0.5 0.0 −0.2 −0.5 −0.2 −0.5 −0.5 0.0 −0.2 −0.5 −0.2 −0.5 −0.5

D2 50 50 50 50 50 100 50 50 50 50 50 100 50 50 50 50 50 100

n 0.3379 0.3296 0.3175 0.3293 0.3045 0.3089 0.0967 0.1036 0.1162 0.1295 0.1461 0.1506 0.4759 0.4744 0.4459 0.4657 0.4295 0.4358

βˆLS

Bias βˆLS 0.0137 0.0135 0.0121 0.0109 0.0076 0.0038 0.0183 0.0175 0.0145 0.0150 0.0100 0.0050 0.0125 0.0122 0.0111 0.0097 0.0067 0.0033

βˆIV −0.0059 −0.0098 −0.0048 −0.0070 −0.0037 −0.0007 −0.0021 −0.0006 −0.0026 −0.0011 −0.0037 0.0005 −0.0093 −0.0067 −0.0114 −0.0092 −0.0083 −0.0032

Var (est)

0.0224 0.0221 0.0199 0.0182 0.0129 0.0064 0.0233 0.0224 0.0187 0.0193 0.0130 0.0065 0.0224 0.0219 0.0200 0.0182 0.0130 0.0065

βˆIV

0.0139 0.0141 0.0121 0.0106 0.0069 0.0033 0.0196 0.0190 0.0151 0.0155 0.0103 0.0050 0.0135 0.0125 0.0111 0.0087 0.0051 0.0025

βˆLS

Var (emp)

0.0225 0.0221 0.0197 0.0185 0.0134 0.0064 0.0230 0.0230 0.0187 0.0191 0.0134 0.0065 0.0232 0.0223 0.0209 0.0180 0.0133 0.0065

βˆIV

Table 3 Estimated bias and variance of the LS estimate βˆLS and the IV estimate βˆIV by simulation using the RCC design and model (19) without confounded treatment heterogeneity; where D1 and D2 are the starting doses of low- and high-range groups, and Var(est) and Var(emp) are mean of the robust variance estimates and the sample variance of βˆLS or βˆIV

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Table 4 presents the bias and estimated and empirical variances of βˆLS and βˆIV with the RCC design when the model may also contain confounding in treatment effect heterogeneity, based on 5000 simulations for each scenario. The first panel shows results when only treatment heterogeneity is confounded. The largest bias in both βˆLS and βˆIV occurred when D1 = D2 = −0.5, which led to the most asymmetric exposure distribution, although it was much smaller in βˆIV than in βˆLS . The bias in both estimates increased roughly proportionally when φ2 increased from 0.5 to 1. The variance of βˆIV was slightly higher than that of βˆLS . The estimated variance of βˆLS slightly underestimated the real (empirical) variance, while that of βˆIV had almost no bias. There was almost no bias in either estimate when the starting doses were symmetric to the dose ranges (D1 , D2 = (0, 0), (−0.2, 0.2) and (−0.5, 0.5)). When both φ and φ2 were non zero, the biases introduced by the two sources were roughly additive so that when D1 , D2 = (0, 0), (−0.2, 0.2), and (−0.5, 0.5) the bias was similar to the second panel of Table 3, and when D1 , D2 = (−0.2, −0.2), (−0.5, −0.5) a part of the bias was cancelled out. The impacts of increasing σv and φ2 (the third and fourth panels) seemed also independent and proportional, comparing with the first two panels and Table 3. βˆIV seemed unbiased with symmetric starting doses when φ2 = 0. Under some situations, mainly when σv = 1, the variance of βˆLS was severely underestimated, while the sandwich variance estimate for βˆIV seemed to have very low bias. In general, βˆIV performed very well with confounding in either uik or βik or both with the symmetric starting doses, while βˆLS not only was biased but also its variance estimate may also be unreliable. Finally we examined the small-sample properties of the α estimates with a simulation using the dose-exposure model and the RCC design. For the estimation of α we only consider RCC trials, since for RDC trials it is no more than a standard application of linear ∗ , were calculated from 5000 simulation runs mixed models. Three estimates, αˆ b , αˆ w , αˆ IV and their bias and estimated and empirical variances are summarized in Table 5, together with the mean exposure difference between the ranges. For simplicity, the results of standard mixed model estimates are not presented. When the starting dose is at D = (0, 0) the dose adjustment led to similar mean exposures in the two groups, with difference around 0.026, comparing with σv = 1 and σε = 1; hence, randomization was a very weak IV and led to very poor βˆIV . There was a slight improvement when the common starting dose was far away from D = (0, 0), while a significant improvement was seen when the starting doses were changed to (–0.2, 0.2) and (–0.5,0.5), corresponding to a large increase in the mean exposure difference between the two ranges. Reducing the number of measuring and adjustment visits from 4 to 3 (lines 9 compared with line 5) doubled the variance of the within-subject estimate, and increasing it from 4 to 6 significantly reduced the variance. In general, increasing the dose adjustment step reduced the variances. Apart from the scenarios when dose was a very weak IV, the biases of all the estimates were quite small. The combined estimate was the best under all the scenarios. In general the estimated variances showed a good consistency with the empirical variance from the simulation. 7. DISCUSSION We have investigated the issues of confounding bias in design and analysis of RCC trials, in comparison with RDC trials. We have proposed to use the IV approach to eliminate confounding bias in analyses of dose-exposure relationship and exposure-response relationship in combination with an RCC or RDC design. Our simulation showed that in general the IV approach can eliminate confounding bias for both RCC and RDC trials, but it is more robust to confounded treatment heterogeneity when an RDC is used. However,

889

φ

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

φ2

0.5 0.5 0.5 0.5 0.5 0.5 1.0 1.0 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1.0 1.0 1.0 1.0 1.0 1.0

0.5 0.5 0.5 0.5 0.5 0.5 0.2 0.2 0.5 0.5 0.5 0.5 0.5 0.5 1.0 1.0 1.0 1.0 1.0 1.0 0.5 0.5 0.5 0.5 0.5 0.5

σv

0.0 −0.2 −0.5 0.2 0.5 0.5 −0.2 −0.5 0.0 −0.2 −0.5 0.2 0.5 0.5 0.0 −0.2 −0.5 0.2 0.5 0.5 0.0 −0.2 −0.5 0.2 0.5 0.5

D1

Design parameter

0.0 −0.2 −0.5 −0.2 −0.5 −0.5 −0.2 −0.5 0.0 −0.2 −0.5 −0.2 −0.5 −0.5 0.0 −0.2 −0.5 −0.2 −0.5 −0.5 0.0 −0.2 −0.5 −0.2 −0.5 −0.5

D2 50 50 50 50 50 100 50 50 50 50 50 50 50 100 50 50 50 50 50 100 50 50 50 50 50 100

n 0.0024 −0.0547 −0.1081 0.0017 −0.0039 0.0011 −0.1044 −0.2223 0.0992 0.0490 0.0052 0.1292 0.1437 0.1466 0.3384 0.2177 0.0601 0.3308 0.2992 0.3097 0.0963 0.0010 −0.1051 0.1285 0.1459 0.1478

βˆLS

Bias

0.0025 −0.0251 −0.0492 0.0011 −0.0035 0.0008 −0.0465 −0.1013 0.0005 −0.0244 −0.0500 −0.0002 −0.0043 −0.0036 −0.0010 −0.0530 −0.1172 −0.0049 −0.0077 −0.0012 −0.0029 −0.0461 −0.1010 −0.0023 −0.0020 −0.0022

βˆIV 0.0304 0.0292 0.0242 0.0252 0.0173 0.0087 0.0318 0.0265 0.0192 0.0181 0.0147 0.0161 0.0109 0.0055 0.0188 0.0182 0.0159 0.0158 0.0122 0.0061 0.0220 0.0206 0.0166 0.0190 0.0138 0.0069

βˆLS

Var (est)

0.0385 0.0371 0.0311 0.0322 0.0221 0.0111 0.0405 0.0342 0.0245 0.0231 0.0190 0.0207 0.0142 0.0072 0.0302 0.0287 0.0247 0.0259 0.0198 0.0100 0.0280 0.0262 0.0214 0.0245 0.0179 0.0091

βˆIV

0.0316 0.0323 0.0248 0.0289 0.0205 0.0097 0.0411 0.0338 0.0226 0.0195 0.0153 0.0194 0.0136 0.0068 0.0456 0.0434 0.0354 0.0405 0.0298 0.0147 0.0322 0.0279 0.0208 0.0294 0.0227 0.0112

βˆLS

βˆIV 0.0381 0.0382 0.0303 0.0337 0.0232 0.0111 0.0420 0.0351 0.0251 0.0228 0.0193 0.0212 0.0143 0.0073 0.0293 0.0291 0.0254 0.0280 0.0205 0.0100 0.0286 0.0265 0.0217 0.0244 0.0182 0.0092

Var (emp)

Table 4 Estimated bias and variance of the LS estimate βˆLS and the IV estimate βˆIV by simulation using the RCC design and model (19) with confounded treatment heterogeneity; where D1 and D2 are the starting doses of low- and high-range groups, and Var (est) and Var (emp) are mean of the robust variance estimates and the sample variance of βˆLS or βˆIV

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D1

0.0 −0.2 −0.5 0.2 0.5 0.5 0.5 0.5 0.5 0.5 0.5

σε

0.5 0.5 0.5 0.5 0.5 0.5 0.7 0.7 0.5 0.5 0.5

0.0 −0.2 −0.5 −0.2 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5

D2

0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.5 0.3 0.3 0.2

D

Design parameter

50 50 50 50 50 100 50 50 50 50 50

n 4 4 4 4 4 4 4 4 3 6 6

r NA NA −0.0202 0.0027 0.0036 −0.0019 −0.0015 −0.0037 −0.0010 0.0025 0.0008

αˆ b

∗ αˆ IV

−0.0052 −0.0036 0.0009 −0.0030 −0.0091 −0.0128 −0.0188 −0.0339 −0.0029 −0.0178 −0.0119

αˆ w −0.0042 −0.0057 −0.0038 −0.0138 −0.0243 −0.0227 −0.0541 −0.0514 −0.0176 −0.0238 −0.0229

Bias

NA NA 0.3820 0.0725 0.0394 0.0197 0.0304 0.0417 0.0362 0.0519 0.0398

αˆ b 0.0130 0.0132 0.0132 0.0158 0.0224 0.0110 0.0465 0.0166 0.0521 0.0069 0.0152

αˆ w

Var (est)

0.0129 0.0130 0.0125 0.0128 0.0140 0.0070 0.0181 0.0117 0.0210 0.0060 0.0109

∗ αˆ IV

NA NA 2.4037 0.2469 0.0347 0.0169 0.0263 0.0284 0.0343 0.0378 0.0343

αˆ b

0.0124 0.0127 0.0129 0.0156 0.0226 0.0109 0.0458 0.0165 0.0544 0.0067 0.0153

αˆ w

Var (emp)

0.0131 0.0135 0.0168 0.0187 0.0137 0.0065 0.0166 0.0104 0.0213 0.0058 0.0104

∗ αˆ IV

0.0349 −0.0367 −0.1376 −0.3682 −0.9737 −0.9745 −0.9759 −0.9368 −0.9907 −0.9192 −0.9614

Mean experimental difference

∗ by simulation, with different combinations of starting doses (log-scale), size of dose Table 5 Estimated bias and estimated (est) and empirical (emp) variances of αˆ b , αˆ w , and αˆ IV adjustment (D), sample size (n), number of visits, and σε , where Var (est) and Var (emp) are mean of the robust variance estimates and the sample variance of the αs, ˆ and NA indicates one or more infinite αˆ b values due to a very small exposure difference between groups

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the approach should be used with care to avoid randomization becoming a weak IV; a good design of RDC and RCC trials can ensure a good IV estimate. In this article we have only considered linear models, to avoid technical details in using IV approaches for nonlinear models. The two-stage estimate can also be used for some specific nonlinear models such as the Poisson regression model for count data, a common model for risk of events. Wang (2012) showed its use in RDC trials, but the approach here for RCC trials can also be used when the exposure-response model is a Poisson model, or more generally a multiplicative model. For more general nonlinear models, although the IV approach can also be used, the two-stage algorithm is not applicable any more and special software or programming will be needed to calculate the IV estimate. Weak IVs have been a focus of research in the area of IV-based approaches. Weak IVs are often unavoidable in observational studies; hence, intensive investigation on the performance of weak-IV-based estimates has been reported. In our context, the best solution is to avoid randomized group becoming a weak IV at the design stage. For RDC trials this can be avoided by selecting dose-level differences relatively larger than exposure variation between patients. For RCC design one needs to consider multiple factors such as the exposure range, starting doses, and dose-adjustment algorithm. Simulation is a convenient way to explore how these factors affect the IV estimates and to confirm that a design will not make randomized groups as a weak IV, by comparing controlled exposure variation between groups and that between patients. The analysis of exposure-response relationship is also an area in pharmacometrics, in which mechanistic modeling is the main approach and confounding bias may be eliminated by simultaneous modeling of the dose-exposure and exposure-response models, including the joint distribution of uik and vik . Determining causal effects needs correct model specification. But often model verification is difficult. Therefore, a number of statistical approaches have been developed not relying on fully specified models. In addition to the IV approach, the dynamic treatment regimen approach (Murphy, 2003) provides a framework for causal effect inference with treatment changes potentially depending on the response. This approach has potential applications in analyses of a wide range of scenarios with exposure change and adjustment, but is also technically more complex. Another approach, the joint modeling method, has considerable overlap with the simultaneous modeling approach. A summary of issues and approaches for causal effect determination with response-dependent dose adjustment can be found in Wang (2013). Adapting these approaches for analysis of exposure-response relationship will provide powerful tools for drug development, as well as new topics for methodological research.

ACKNOWLEDGMENTS The author thanks the editor and the referee for their helpful comments and suggestions, which led to significant improvement to the article. The author also thanks Dr. E. Waldron for his proofreading.

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