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Model-based dose finding under model uncertainty using general parametric models

José Pinheiro, Björn Bornkamp, Ekkehard Glimm, Frank Bretz

arXiv:1305.0889v2stat.ME

TL;DR

The paper addresses the limited applicability of traditional dose-finding methods to binary, count, time-to-event, repeated-measure, and crossover settings. It develops an extended MCPMod framework using general parametric models and provides computationally efficient dose-response fitting. The framework supports broader model-based dose selection and is illustrated across several endpoint and study-model settings.

  • Problem

    Traditional dose-finding methodology is mainly developed for normally distributed, homoscedastic single-timepoint responses in parallel-group studies, excluding important binary, count, time-to-event, longitudinal, and crossover settings.

  • Method

    The paper extends MCPMod to general parametric models by combining model-uncertainty-aware multiple comparisons, dose-response modeling, efficient fitting, and software implementation.

  • Results

    The extended approach handles dose-response analyses for general study designs and is illustrated for binary, overdispersed count, time-to-event, and longitudinal data.

  • Takeaways & Limitations

    Model-based dose finding can improve understanding of dose-response relationships and generally translate into more accurate dose selection for confirmatory trials.

  • Takeaways & Limitations

    Realizing the full potential of the methods requires addressing the larger number of parameters involved when extending the approach, including combinations such as dose and regimen.

Abstract

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Statistical methodology for the design and analysis of clinical Phase II dose response studies, with related software implementation, are well developed for the case of a normally distributed, homoscedastic response considered for a single timepoint in parallel group study designs. In practice, however, binary, count, or time-to-event endpoints are often used, typically measured repeatedly over time and sometimes in more complex settings like crossover study designs. In this paper we develop an overarching methodology to perform efficient multiple comparisons and modeling for dose finding, under uncertainty about the dose-response shape, using general parametric models. The framework described here is quite general and covers dose finding using generalized non-linear models, linear and non-linear mixed effects models, Cox proportional hazards (PH) models, etc. In addition to the core framework, we also develop a general purpose methodology to fit dose response data in a computationally and statistically efficient way. Several examples, using a variety of different statistical models, illustrate the breadth of applicability of the results. For the analyses we developed the R add-on package DoseFinding, which provides a convenient interface to the general approach adopted here.

1 Introduction

The paper addresses dose finding beyond normally distributed, homoscedastic parallel-group settings, where binary, count, time-to-event, longitudinal, and crossover data create practical challenges. It extends MCPMod into a general framework combining multiple comparisons, dose-response modeling, and software implementation for broader study designs.

  • Dose finding remains important throughout drug development, while insufficient efficacy and safety dose-response knowledge at Phase II contributes to pipeline problems.
  • Existing dose-response methodology largely targets normally distributed, homoscedastic endpoints measured once in parallel-group studies.
  • Practical studies often use binary, count, or time-to-event endpoints, repeated measurements, heteroscedasticity, non-normality, or crossover designs.
  • Extensions for new settings are typically situation-specific, and general-purpose software is unavailable, making case-by-case implementation a major coding effort.
  • The paper extends MCPMod to general parametric models and study designs by combining multiple comparisons and modeling for dose-response testing, selection, and estimation.
  • The approach pre-specifies plausible candidate dose-response models, tests for a dose-response signal with multiplicity adjustment, selects significant models, and estimates target doses.
  • Examples cover binary, overdispersed count, Cox proportional-hazards time-to-event, and longitudinal data with patient-specific random effects, supported by the R package DoseFinding.

2 Generalized MCPMod

The generalized MCPMod framework separates dose-response information from the broader response model, enabling model-uncertainty-aware testing and estimation across general parametric settings. It uses candidate dose-response shapes, asymptotic estimates and covariance information, optimal contrasts, multiplicity-adjusted testing, and subsequent model fitting.

  • The extension applies when a parametric response model contains a parameter that captures the dose-response relationship.
  • The framework decouples the dose-response model from the expected response by focusing on an interpretable dose-response parameter.
  • The method then supports model fitting, model selection or averaging, and dose estimation using the estimated dose-response vector and covariance matrix.
  • For time-to-event outcomes, the model can be reparameterized using median time to event rather than less clinically interpretable Weibull scale and shape parameters.
  • A set of candidate dose-response models is specified across doses, with separate ANOVA-style dose estimates obtained using methods such as maximum likelihood, GEE, or partial likelihood.
  • The method assumes the estimated dose-response vector is approximately multivariate normal with an estimated covariance matrix, supporting generalized linear, time-to-event, mixed-effects, and GEE models.
  • Candidate model shapes determine optimal contrasts for testing a flat dose-response profile, and the maximum model-specific statistic provides an overall dose-response test with multiplicity adjustment.
  • The proposed generalized approach has finite- and large-sample properties similar to full-likelihood approaches while offering general-purpose application based only on estimated dose-response parameters and covariance.

3 Non-linear dose response model fitting using a two-stage,

The paper fits nonlinear dose-response models through a two-stage procedure that first estimates dose-level means and covariance, then applies generalized least squares. Simulations show comparable GLS and ML performance, while GLS is more computationally efficient and its bootstrap variant improves small-sample coverage.

  • Two-stage fitting: The two-stage procedure first obtains ANOVA estimates of dose-level means and covariance, then fits the nonlinear dose-response model by generalized least squares.The resulting estimator is established as consistent and asymptotically normal under stated regularity conditions.
  • Two-stage fitting: The method fits f(x, θ) to the ANOVA estimates by minimizing a generalized least squares criterion with a symmetric positive definite weighting matrix.In practice, the weighting matrix is set to the estimated covariance matrix.
  • Computational properties: The approach is broadly applicable to general parametric models and is computationally efficient because its optimized target function is low-dimensional rather than based on the complete dataset.This speed advantage is relevant when clinical-trial simulations require repeated model fitting.
  • Simulation results: For asymptotic confidence intervals, GLS and ML behave similarly; nominal 90% coverage is reached earlier for quadratic and Emax models than for the exponential model.Coverage is achieved for Emax at roughly 50–100 patients per group, but only at very large sample sizes for the exponential model.
  • Simulation results: GLS-B achieves approximately nominal 90% coverage for all three dose-response models at sample sizes as small as 30 patients per group.The authors recommend GLS-B for small sample sizes; it is also computationally more efficient than a bootstrap based on maximum likelihood.
  • Simulation results: GLS and ML perform similarly across the investigated dose-response shapes, sample sizes, and data types for coverage probabilities, estimation error, and mean squared error.The two methods are reported as almost indistinguishable for dose-response estimation, even at small sample sizes.

4 Applications

This application illustrates MCPMod for longitudinal neurodegenerative-disease data using mixed-effects models, candidate dose-response shapes, and DoseFinding software. The analysis establishes a dose-response signal, favors the Emax model, and estimates a target dose near 2.1 mg.

  • Study design: The design includes placebo and four active doses—1, 3, 10, and 30 mg—with repeated functional-scale measurements over one year.Measurements occur at baseline and every three months thereafter, with 50 patients per arm.
  • Study setting and model: A mixed-effects representation includes patient-specific random intercepts and slopes, with linear and nonlinear dose-response forms yielding LME or NLME models.The dose-response function μ(x) determines whether the mixed-effects model is linear or nonlinear in dose.
  • Study setting and model: The study models functional-scale progression over one year, with treatment effects represented by changes in the linear progression slope.At one year, the slope effect equals the average change from baseline.
  • Data and implementation: The neurodegenerative-disease example uses simulated data because the actual trial data cannot be reported for confidentiality reasons.The simulated dataset imposes an Emax dose-response profile and is available in the neurodeg dataset.
  • MCPMod testing: The Emax, quadratic, and linear contrasts are significant, whereas the exponential contrast is not, establishing a dose-response signal.The analysis therefore proceeds to dose-response estimation and target-dose selection.
  • Model selection: The Emax model fits best, with gAIC values 10.66, 11.07, and 24.22 for Emax, quadratic, and linear models, respectively.The mixed-effects analysis similarly favors Emax, with AIC values 8352.60, 8353.10, and 8365.79.
  • Target-dose estimation: Estimated target doses are 2.13 under two-stage GLS and 2.15 under NLME for the clinically meaningful effect of 1.4.The target dose is defined as the smallest dose producing an effect at least as large as the target value.

5 Conclusions

The extended MCPMod methodology and its software broaden model-based dose finding across endpoint types and study settings. Its potential depends on more informative Phase II designs, while further extensions remain open.

  • Scope and contribution: The extended MCPMod methodology and software implementation broaden the original approach to most endpoint types and associated model-based analyses used in dose finding.The approach is presented as applicable across a broad range of endpoints and models.
  • Open issues: Exposure values rather than dose levels may complicate derivation of optimal models and contrasts, including the MCP step.Dose-time response modeling and model uncertainty are identified as areas needing further research.
  • Implications: Model-based dose finding can improve understanding of dose-response relationships and generally support more accurate dose selection for confirmatory trials.
  • Future directions: Realizing the full potential of model-based methods requires changes to traditional Phase II study design, and further extensions remain of practical interest.Examples include regimen selection and modeling exposure-response relationships by combining dose and regimen into one covariate.
  • Design constraints: Phase II studies often use few doses and sample sizes based on detecting a dose-placebo difference, which can be inadequate for dose-response modeling.Such designs are commonly organized as mini Phase III trials and may not provide enough information to estimate target doses.
  • Design implications: Using four or five doses with larger sample sizes could enable model-based methods to improve Phase II dose selection and Phase III success probability.

A Derivation of Optimal Contrasts

The appendix derives optimal contrasts by maximizing the non-centrality parameter of a contrast test under a specified mean vector and covariance matrix. The resulting solution is invariant to scalar changes in the mean vector.

  • Optimality criterion: Optimality is defined as maximizing the power of a univariate contrast test when a specified mean vector and positive definite covariance matrix are true.
  • Constraint: The contrast is chosen to maximize the non-centrality parameter subject to its coefficients summing to zero.
  • Derivation: The constrained problem is equivalent to maximizing a generalized Rayleigh quotient involving the mean contrast and its covariance.
  • Matrix structure: Because the relevant mean outer-product matrix has rank one, it has only a single non-zero eigenvalue.
  • Invariance: The optimal solution is unchanged by adding or multiplying the mean vector by a scalar, so standardized mean vectors can replace the original means.

B Placebo-adjusted dose response modeling

The paper extends MCPMod to placebo-adjusted estimates, a setting relevant when models include additive covariates or when Cox PH models provide control-adjusted estimates.

  • Extension: The extension supports MCPMod analyses based on placebo-adjusted estimates rather than only unadjusted dose-group responses.
  • Applications: This setting is useful with additive covariates and Cox PH models, where only control-adjusted estimates may be available.
  • Procedure: The section first establishes equality of test statistics and then calculates optimal contrasts.

B.1 Test statistics and optimal contrasts

The appendix shows that testing contrasts of dose-group responses and testing placebo-adjusted responses can yield equivalent optimal test performance. It then gives the corresponding contrast construction.

  • Contrast representation: The placebo-adjusted formulation represents dose-group contrasts through a contrast matrix whose rows span the subspace orthogonal to the all-ones vector.
  • Construction: Starting from the ANOVA estimate containing the placebo response, multiplying by the contrast matrix produces the corresponding placebo-adjusted contrast estimate and covariance.
  • Equivalence: The placebo-adjusted and direct contrast formulations have equal maximum non-centrality parameters, so their optimal tests are equivalent.
  • Optimal coefficients: The optimal contrast coefficients can be calculated from the covariance structure and the maximizing vector in the transformed formulation.

B.2 Dose Response Model Fitting

The dose-response fitting procedure uses a two-stage generalized least-squares criterion, with a reduced model when only contrasts are available. Under a zero-at-dose condition, the reduced and full formulations coincide.

  • The two-stage generalized least-squares procedure estimates the dose-response model by minimizing a fitting criterion.
  • When only bµC is available, the procedure fits a model without the intercept θ0.The reduced form is written as fC(x, θ) = θ1f(x, θ0).
  • The reduced and full fitting criteria are equivalent when f 0(0, θ0) = 0.
  • The equivalence follows from the matrix identity C0(C0SC′0)−1C0 = S−1.

C Proof of the result in section 3.1

The proof establishes consistency and asymptotic normality of the dose-response parameter estimator under covariance, normality, identifiability, and differentiability assumptions. Its limiting distribution is expressed using derivative and covariance matrices of the model.

  • The assumptions require a positive-definite covariance estimate converging to Σ, a multivariate normal limit, and a twice-differentiable bijective model mapping.
  • Under the stated assumptions, bθ is a consistent estimator of θ.
  • The proof uses standard M-estimator consistency theory to establish convergence of the fitted parameter.
  • The proof shows that the relevant matrix converges in probability to a non-singular matrix.
  • The asymptotic distribution of √an(bθ−θ0) is N(0, B(θ0)′M(θ0)B(θ0)).Here M(θ) and B(θ) are constructed from the model derivative matrix F(θ), weighting matrix A, and covariance matrix Σ.
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