Robust and Adaptive Two-stage Designs in Nonlinear Mixed Effect Models

Abstract

To get informative studies for nonlinear mixed effect models (NLMEM), design optimization can be performed based on Fisher Information Matrix (FIM) using the D-criterion. Its computation requires knowledge about models and parameters, which are often prior guesses. Thus, adaptive designs composed of several stages may be used. Robust approach can also be used to account for various candidate models. In the estimation step of a given stage, model selection (MS) or model averaging (MA) can be performed. In this work we propose a new two-stage adaptive design strategy, based on the robust expected FIM and MA over several candidate models. The methodology is applied to a clinical trial simulation in ophthalmology to optimize doses and time measurements. A set of dose-response candidate models is defined, and one-stage designs are compared to two-stage 50/50 designs (i.e., each stage performed with half of the available subjects), using either local optimal design or robust design, and performing analysis with one model, MS or MA. Performing a two-stage design with MS at the interim analysis can correct the choice of a wrong model for designing the first stage. Overall, starting from a robust design (1- or 2-stage) is valuable and leads to reasonable bias and precision. The proposed robust adaptive design strategy is a new tool to design longitudinal studies that could be used in different therapeutic areas.

Graphical abstract

Appendices

Appendix A: Multiplicative Algorithm for Robust Approach

The multiplicative algorithm is an iterative algorithm which can be used to computed the weights $\alpha _{q}$ to attribute to the individual designs in order to optimize the D-criterion or the CD-criterion in case of $\Xi _{Robust}$. [29]

The derivative of these criteria in case of one-stage and K-stage designs, which are required for the optimization process, are computed in the following.

CD-derivative for one-stage design

In the multiplicative algorithm, an expression for $\dfrac{\partial \phi _{D}(\Xi )}{\partial \alpha _{q}}$ is needed. Therefore let denote:

$$\begin{aligned} \phi _{D}(\Xi )= & {} Det\left( A(\alpha _{q}) \right) ^{1/P} \quad \text {with} \nonumber \\ \mathcal {M}_{F}(\Xi )= & {} A(\alpha _{q}) = N \sum _{q^{\prime }=1}^{Q} \alpha _{q^{\prime }} \mathcal {M}_{F}(\xi _{q^{\prime }}) \end{aligned}$$

(A1)

We have,

$$\begin{aligned} \dfrac{\partial \phi _{D}(\Xi )}{\partial \alpha _{q}} = \dfrac{1}{P} \left( \dfrac{d}{d\alpha _{q}}Det\left( A(\alpha _{q})\right) \right) Det(A(\alpha _{q}) )^{1/P-1} \end{aligned}$$

(A2)

According to Jacobi’s formula,

$$\begin{aligned} \dfrac{d}{d\alpha _{q}}Det\left( A(\alpha _{q})\right) = tr \left[ A(\alpha _{q})^{-1} \dfrac{dA(\alpha _{q})}{d\alpha _{q}} \right] Det\left( A(\alpha _{q})\right) \end{aligned}$$

(A3)

and we have: $\dfrac{dA(\alpha _{q})}{d\alpha _{q}} = N \mathcal {M}_{F}(\xi _{q})$, which leads to:

$$\begin{aligned} \dfrac{d}{d\alpha _{q}}Det\!\left( A(\alpha _{q})\right) \!=\! N tr\! \left[ \mathcal {M}_{F}(\Xi )^{\!-1} \mathcal {M}_{F}(\xi _{q}) \right] \! Det\!\left( \mathcal {M}_{F}(\Xi )\right) \end{aligned}$$

(A4)

Thus,

$$\begin{aligned} \dfrac{\partial \phi _{D}(\Xi )}{\partial \alpha _{q}}\!= & {} \! \dfrac{N}{P}\! \left( \dfrac{d}{d\alpha _{q}}\!Det\left( A(\alpha _{q})\right) \right) \!Det(A(\alpha _{q}) )^{1/P-1} \\= & {} \dfrac{N}{P} tr \left[ \mathcal {M}_{F}(\Xi )^{-1} \mathcal {M}_{F}(\xi _{q}) \right] Det(\mathcal {M}_{F}(\Xi ) )^{1/P} \nonumber \end{aligned}$$

(A5)

For the robust approach, we need expressions for derivatives of $\phi _{CD}(\Xi ) = \displaystyle \prod _{m=1}^{M} \left( \phi _{D,m}(\Xi ) \right) ^{w_{m}}$.

Recall that

$$\begin{aligned} \frac{\mathrm d}{\mathrm dx}\prod _{i=1}^{n}f_{i}(x)=\sum _{i=1}^{n}\left( \frac{\mathrm d}{\mathrm dx}f_{i}(x)\prod _{j\ne i}f_{j}(x)\right) \end{aligned}$$

(A6)

Thus,

$$\begin{aligned} \dfrac{\partial \phi _{CD}(\Xi )}{\partial \alpha _{q}}= & {} \sum _{m=1}^{M} \Bigg ( w_{m} \dfrac{ \partial \phi _{D,m}(\Xi )}{\partial \alpha _{q}} \left( \phi _{D,m}(\Xi ) \right) ^{w_{m}-1} \nonumber \\{} & {} \times \prod _{m^{\prime } \ne m} \left( \phi _{D,m^{\prime }}(\Xi ) \right) ^{w_{m^{\prime }}} \Bigg ) \end{aligned}$$

(A7)

which writes:

$$\begin{aligned} \dfrac{\partial \phi _{CD}(\Xi )}{\partial \alpha _{q}} = \phi _{CD}(\Xi ) \sum _{m=1}^{M} \left( w_{m} \dfrac{1}{\phi _{D,m}(\Xi )} \dfrac{ \partial \phi _{D,m}(\Xi )}{\partial \alpha _{q}} \right) \end{aligned}$$

(A8)

The robust version was implemented using PFIM and the new MultiplicativeAlgorithmRobust class.

CD-derivative for K -Stage Design In the general setting of robust and adaptive design with K stages, the first stage of the study is designed and conducted according to previous information and accounting for model uncertainty. Then each following stage is planned leveraging the data obtained in the previous stages. Indeed the FIM at stage k depends on the design of the $k^{th}$ stage, but also on those of all previous stages, from 1 to $k-1$. Thus, for a given model m (index m is omitted in the following equations for the sake of simplicity), the total FIM $\mathcal {M}_{F}^{(k)}(\hat{\psi }^{(k-1)}, \Xi )$ at stage k, for a design $\Xi$ and given estimated parameters $\hat{\psi }^{(k-1)}$ from stage $k-1$, writes:

$$\begin{aligned} \mathcal {M}_{F}^{(k)}(\hat{\psi }^{(k-1)}, \Xi )= & {} \mathcal {M}_{F}(\hat{\psi }^{(k-1)}, \Xi _{CD,1}) + \mathcal {M}_{F}(\hat{\psi }^{(k-1)}, \Xi _{CD,2}) \nonumber \\{} & {} + ... + \mathcal {M}_{F}(\hat{\psi }^{(k-1)}, \Xi _{CD,k-1}) \nonumber \\{} & {} + \mathcal {M}_{F}(\hat{\psi }^{(k-1)}, \Xi ) \end{aligned}$$

(A9)

where $\Xi _{CD,j}$ is the $\Xi _{Robust}$ computed at stage j.

Table IV Optimal First Stage Designs, $N_{1}=150$

Table V Relative D-Efficiencies between designs, $N_{1}=150$.

Thus, only $\mathcal {M}_{F}(\hat{\psi }^{(k-1)}, \Xi )$ depends on $\Xi$, the design on which optimization is performed. Therefore, at stage k, let denote the previous FIM:

$$\begin{aligned} \mathcal {M}_{F}^{(k), prev}(\hat{\psi }^{(k-1)})= & {} \mathcal {M}_{F}(\hat{\psi }^{(k-1)}, \Xi _{CD,1}) + \mathcal {M}_{F}(\hat{\psi }^{(k-1)}, \Xi _{CD,2}) \nonumber \\{} & {} + ... + \mathcal {M}_{F}(\hat{\psi }^{(k-1)}, \Xi _{CD,k-1}) \end{aligned}$$

(A10)

which leads to write:

$$\begin{aligned} \mathcal {M}_{F}^{(k)}(\hat{\psi }^{(k-1)}, \Xi )= & {} \mathcal {M}_{F}^{(k), prev}(\hat{\psi }^{(k-1)}) + \mathcal {M}_{F}(\hat{\psi }^{(k-1)}, \Xi ) \nonumber \\= & {} \mathcal {M}_{F}^{(k), prev}(\hat{\psi }^{(k-1)}) \nonumber \\{} & {} + N_{k} \sum _{q=1}^{Q} \alpha _{q} \mathcal {M}_{F}(\hat{\psi }^{(k-1)}, \xi _{q}) \end{aligned}$$

(A11)

As previously and with $A(\alpha _{q}) = \mathcal {M}_{F}^{(k)}(\hat{\psi }^{(k-1)}, \Xi )$, we have:

$$\begin{aligned} \dfrac{\partial \phi _{D}(\Xi )}{\partial \alpha _{q}} = \dfrac{1}{P} \left( \dfrac{d}{d\alpha _{q}}Det\left( A(\alpha _{q})\right) \right) Det(A(\alpha _{q}) )^{1/P-1} \end{aligned}$$

(A12)

which writes:

$$\begin{aligned} \dfrac{\partial \phi _{D}(\Xi )}{\partial \alpha _{q}}= & {} \dfrac{N_{k}}{P} tr \left[ \mathcal {M}_{F}^{(k)}(\hat{\psi }^{(k-1)}, \Xi )^{-1} \mathcal {M}_{F}(\xi _{q}) \right] \nonumber \\{} & {} \times Det(\mathcal {M}_{F}^{(k)}(\hat{\psi }^{(k-1)}, \Xi ) )^{1/P} \end{aligned}$$

(A13)

Thus, in robust approach, the derivative of $\phi _{CD}$ keeps the same expression:

$$\begin{aligned} \dfrac{\partial \phi _{CD}(\Xi )}{\partial \alpha _{q}} = \phi _{CD}(\Xi ) \sum _{m=1}^{M} \left( w_{m} \dfrac{1}{\phi _{D,m}(\Xi )} \dfrac{ \partial \phi _{D,m}(\Xi )}{\partial \alpha _{q}} \right) \end{aligned}$$

(A14)

Appendix B: Two-stage design: designs for the first stage and relative efficiency