Simultaneous confidence bands for nonlinear regression models with application to population pharmacokinetic analyses

Data-driven disease progression model of Parkinson's disease and effect of sex and genetic variants

As Parkinson's disease (PD) progresses, there are multiple biomarker changes, and sex and genetic variants may influence the rate of progression. Data-driven, long-term disease progression model analysis may provide precise knowledge of the relationships between these risk factors and progression and would allow for the selection of appropriate diagnosis and treatment according to disease progression. To construct a long-term disease progression model of PD based on multiple biomarkers and evaluate the effects of sex and leucine-rich repeat kinase 2 (LRRK2) mutations, a technique derived from the nonlinear mixed-effects model (Statistical Restoration of Fragmented Time course [SReFT]) was applied to datasets of patients provided by the Parkinson's Progression Markers Initiative. Four biomarkers, including the Unified PD Rating Scale, were used, and a covariate analysis was performed to investigate the effects of sex and LRRK2-related mutations. A model of disease progression over ~30 years was successfully developed using patient data with a median of 6 years. Covariate analysis suggested that female sex and LRRK2 G2019S mutations were associated with 21.6% and 25.4% significantly slower progression, respectively. LRRK2 rs76904798 mutation also tended to delay disease progression by 10.4% but the difference was not significant. In conclusion, a long-term PD progression model was successfully constructed using SReFT from relatively short-term individual patient observations and depicted nonlinear changes in relevant biomarkers and their covariates, including sex and genetic variants.

Identifying alert concentrations using a model-based bootstrap approach

The determination of alert concentrations, where a pre-specified threshold of the response variable is exceeded, is an important goal of concentration-response studies. The traditional approach is based on investigating the measured concentrations and attaining statistical significance of the alert concentration by using a multiple t-test procedure. In this paper, we propose a new model-based method to identify alert concentrations, based on fitting a concentration-response curve and constructing a simultaneous confidence band for the difference of the response of a concentration compared to the control. In order to obtain these confidence bands, we use a bootstrap approach which can be applied to any functional form of the concentration-response curve. This particularly offers the possibility to investigate also those situations where the concentration-response relationship is not monotone and, moreover, to detect alerts at concentrations which were not measured during the study, providing a highly flexible framework for the problem at hand.

Use of clinical tolerance limits for assessing agreement

In this study, we have further extended the methodology proposed, first, by Lin et al. (2002) and, later, extended by Stevens et al. (2017, 2018), on the coverage probability/probability of agreement, by relaxing the strong parametric assumptions regarding the distribution of the latent trait and developing inference methods allowing to compute both pointwise and simultaneous confidence bands. The methodology requires repeated measurements by at least one of the two measurement methods and accommodates heteroscedastic measurement errors. It performs often very well even when one has only one measurement by one of the two measurement methods and at least five repeated measurements from the other. It circumvents some of the deficiencies of the Bland & Altman limits of agreement method and provides a more direct assessment of the agreement level.

How can we justify grouping of nanoforms for hazard assessment? Concepts and tools to quantify similarity

The risk of each nanoform (NF) of the same substance cannot be assumed to be the same, as they may vary in their physicochemical characteristics, exposure and hazard. However, neither can we justify a need for more animal testing and resources to test every NF individually. To reduce the need to test all NFs, (regulatory) information requirements may be fulfilled by grouping approaches. For such grouping to be acceptable, it is important to demonstrate similarities in physicochemical properties, toxicokinetic behaviour, and (eco)toxicological behaviour. The GRACIOUS Framework supports the grouping of NFs, by identifying suitable grouping hypotheses that describe the key similarities between different NFs. The Framework then supports the user to gather the evidence required to test these hypotheses and to subsequently assess the similarity of the NFs within the proposed group. The evidence needed to support a hypothesis is gathered by an Integrated Approach to Testing and Assessment (IATA), designed as decision trees constructed of decision nodes. Each decision node asks the questions and provides the methods needed to obtain the most relevant information. This White paper outlines existing and novel methods to assess similarity of the data generated for each decision node, either via a pairwise analysis conducted property-by-property, or by assessing multiple decision nodes simultaneously via a multidimensional analysis. For the pairwise comparison conducted property-by-property we included in this White paper: The x-fold, Bayesian and Arsinh-OWA distance algorithms performed comparably in the scoring of similarity between NF pairs. The Euclidean distance was also useful, but only with proper data transformation. The x-fold method does not standardize data, and thus produces skewed histograms, but has the advantage that it can be implemented without programming knowhow. A range of multidimensional evaluations, using for example dendrogram clustering approaches, were also investigated. Multidimensional distance metrics were demonstrated to be difficult to use in a regulatory context, but from a scientific perspective were found to offer unexpected insights into the overall similarity of very different materials. In conclusion, for regulatory purposes, a property-by-property evaluation of the data matrix is recommended to substantiate grouping, while the multidimensional approaches are considered to be tools of discovery rather than regulatory methods.

Optimal Designs for Comparing Regression Curves: Dependence Within and Between Groups

We consider the problem of designing experiments for the comparison of two regression curves describing the relation between a predictor and a response in two groups, where the data between and within the group may be dependent. In order to derive efficient designs we use results from stochastic analysis to identify the best linear unbiased estimator (BLUE) in a corresponding continuous model. It is demonstrated that in general simultaneous estimation using the data from both groups yields more precise results than estimation of the parameters separately in the two groups. Using the BLUE from simultaneous estimation, we then construct an efficient linear estimator for finite sample size by minimizing the mean squared error between the optimal solution in the continuous model and its discrete approximation with respect to the weights (of the linear estimator). Finally, the optimal design points are determined by minimizing the maximal width of a simultaneous confidence band for the difference of the two regression functions. The advantages of the new approach are illustrated by means of a simulation study, where it is shown that the use of the optimal designs yields substantially narrower confidence bands than the application of uniform designs.

Testing for similarity of binary efficacy-toxicity responses

Clinical trials often aim to compare two groups of patients for efficacy and/or toxicity depending on covariates such as dose. Examples include the comparison of populations from different geographic regions or age classes or, alternatively, of different treatment groups. Similarity of these groups can be claimed if the difference in average outcome is below a certain margin over the entire covariate range. In this article, we consider the problem of testing for similarity in the case that efficacy and toxicity are measured as binary outcome variables. We develop a new test for the assessment of similarity of two groups for a single binary endpoint. Our approach is based on estimating the maximal deviation between the curves describing the responses of the two groups, followed by a parametric bootstrap test. Further, using a two-dimensional Gumbel-type model we develop methodology to establish similarity for (correlated) binary efficacy-toxicity outcomes. We investigate the operating characteristics of the proposed methodology by means of a simulation study and present a case study as an illustration.

Bio-equivalence tests in functional data by maximum deviation

We study the problem of testing equivalence of functional parameters, such as the mean or the variance function, in the two-sample functional data setting. In contrast to previous work where the functional problem is reduced to a multiple testing problem for the equivalence of scalar data by comparing the functions at each point, our approach is based on an estimate of a distance measuring the maximum deviation between the two functional parameters. Equivalence is claimed if the estimate for the maximum deviation does not exceed a given threshold. We propose a bootstrap procedure for obtaining quantiles of the distribution of the test statistic, and we prove consistency of the corresponding test in the large-sample scenario. As the methods proposed here avoid the use of the intersection-union principle, they are less conservative and more powerful than currently available approaches.

Assessing bias, precision, and agreement in method comparison studies

Recently, a new estimation procedure has been developed to assess bias and precision of a new measurement method, relative to a reference standard. However, the author did not develop confidence bands around the bias and standard deviation curves. Therefore, the goal in this paper is to extend this methodology in several important directions. First, by developing simultaneous confidence bands for the various parameters estimated to allow formal comparisons between different measurement methods. Second, by proposing a new index of agreement. Third, by providing a series of new graphs to help the investigator to assess bias, precision, and agreement between the two measurement methods. The methodology requires repeated measurements on each individual for at least one of the two measurement methods. It works very well to estimate the differential and proportional biases, even with as few as two to three measurements by one of the two methods and only one by the other. The repeated measurements need not come from the reference standard but from either measurement methods. This is a great advantage as it may sometimes be more feasible to gather repeated measurements with the new measurement method.

Lessons learned from IDeAl - 33 recommendations from the IDeAl-net about design and analysis of small population clinical trials

BACKGROUND

IDeAl (Integrated designs and analysis of small population clinical trials) is an EU funded project developing new statistical design and analysis methodologies for clinical trials in small population groups. Here we provide an overview of IDeAl findings and give recommendations to applied researchers.

METHOD

The description of the findings is broken down by the nine scientific IDeAl work packages and summarizes results from the project's more than 60 publications to date in peer reviewed journals. In addition, we applied text mining to evaluate the publications and the IDeAl work packages' output in relation to the design and analysis terms derived from in the IRDiRC task force report on small population clinical trials.

RESULTS

The results are summarized, describing the developments from an applied viewpoint. The main result presented here are 33 practical recommendations drawn from the work, giving researchers a comprehensive guidance to the improved methodology. In particular, the findings will help design and analyse efficient clinical trials in rare diseases with limited number of patients available. We developed a network representation relating the hot topics developed by the IRDiRC task force on small population clinical trials to IDeAl's work as well as relating important methodologies by IDeAl's definition necessary to consider in design and analysis of small-population clinical trials. These network representation establish a new perspective on design and analysis of small-population clinical trials.

CONCLUSION

IDeAl has provided a huge number of options to refine the statistical methodology for small-population clinical trials from various perspectives. A total of 33 recommendations developed and related to the work packages help the researcher to design small population clinical trial. The route to improvements is displayed in IDeAl-network representing important statistical methodological skills necessary to design and analysis of small-population clinical trials. The methods are ready for use.

Optimal designs for comparing curves

We consider the optimal design problem for a comparison of two regression curves, which is used to establish the similarity between the dose response relationships of two groups. An optimal pair of designs minimizes the width of the confidence band for the difference between the two regression functions. Optimal design theory (equivalence theorems, efficiency bounds) is developed for this non standard design problem and for some commonly used dose response models optimal designs are found explicitly. The results are illustrated in several examples modeling dose response relationships. It is demonstrated that the optimal pair of designs for the comparison of the regression curves is not the pair of the optimal designs for the individual models. In particular it is shown that the use of the optimal designs proposed in this paper instead of commonly used "non-optimal" designs yields a reduction of the width of the confidence band by more than 50%.

Comparison of asymptotic confidence sets for regression in small samples

In case of small samples, asymptotic confidence sets may be inaccurate, with their actual coverage probability far from a nominal confidence level. In a single framework, we consider four popular asymptotic methods of confidence estimation. These methods are based on model linearization, F-test, likelihood ratio test, and nonparametric bootstrapping procedure. Next, we apply each of these methods to derive three types of confidence sets: confidence intervals, confidence regions, and pointwise confidence bands. Finally, to estimate the actual coverage of these confidence sets, we conduct a simulation study on three regression problems. A linear model and nonlinear Hill and Gompertz models are tested in conditions of different sample size and experimental noise. The simulation study comprises calculation of the actual coverage of confidence sets over pseudo-experimental datasets for each model. For confidence intervals, such metrics as width and simultaneous coverage are also considered. Our comparison shows that the F-test and linearization methods are the most suitable for the construction of confidence intervals, the F-test - for confidence regions and the linearization - for pointwise confidence bands.