The SAEM algorithm for group comparison tests in longitudinal data analysis based on non-linear mixed-effects model

Non-linear mixed-effects models (NLMEMs) are used to improve information gathering from longitudinal studies and are applied to treatment evaluation in disease-evolution studies, such as human immunodeficiency virus (HIV) infection. The estimation of parameters and the statistical tests are critical issues in NLMEMs since the likelihood and the Fisher information matrix have no closed form. An alternative method to numerical integrations, in which convergence is slow, and to methods based on linearization, in which asymptotic convergence has not been proved, is the Stochastic Approximation Expectation-Maximization (SAEM) algorithm. For the Wald test and the likelihood ratio test, we propose estimating the Fisher information matrix by stochastic approximation and the likelihood by importance sampling. We evaluate these SAEM-based tests in a simulation study in the context of HIV viral load decrease after initiation of an antiretroviral treatment. The results from this simulation illustrate the theoretical convergence properties of SAEM. We also propose a method based on the SAEM algorithm to compute the minimum sample size required to perform a Wald test of a given power for a covariate effect in NLMEMs. Lastly, we illustrate these tests on the evaluation of the effect of ritonavir on the indinavir pharmacokinetics in HIV patients and compare the results with those obtained using the adaptative Gaussian quadrature method implemented in the SAS procedure NLMIXED.     

Design in nonlinear mixed effects models: optimization using the Fedorov-Wynn algorithm and power of the Wald test for binary covariates

We extend the methodology for designs evaluation and optimization in nonlinear mixed effects models with an illustration of the decrease of human immunodeficiency virus viral load after antiretroviral treatment initiation described by a bi-exponential model. We first show the relevance of the predicted standard errors (SEs) given by the computation of the population Fisher information matrix using the R function PFIM, in comparison to those computed with the stochastic approximation expectation-maximization algorithm, implemented in the Monolix software. We then highlight the usefulness of the Fedorov-Wynn (FW) algorithm for designs optimization compared to the Simplex algorithm. From the predicted SE of PFIM, we compute the predicted power of the Wald test to detect a treatment effect as well as the number of subjects needed to achieve a given power. Using the FW algorithm, we investigate the influence of the design on the power and show that, for optimized designs with the same total number of samples, the power increases when the number of subjects increases and the number of samples per subject decreases. A simulation study is also performed with the nlme function of R to confirm this result and show the relevance of the predicted powers compared to those observed by simulation.     

Design evaluation and optimization for models of hepatitis C viral dynamics

Mathematical modeling of hepatitis C viral (HCV) kinetics is widely used for understanding viral pathogenesis and predicting treatment outcome. The standard model is based on a system of five non-linear ordinary differential equations (ODE) that describe both viral kinetics and changes in drug concentration after treatment initiation. In such complex models parameter estimation is challenging and requires frequent sampling measurements on each individual. By borrowing information between study subjects, non-linear mixed effect models can deal with sparser sampling from each individual. However, the search for optimal designs in this context has been limited by the numerical difficulty of evaluating the Fisher information matrix (FIM). Using the software PFIM, we show that a linearization of the statistical model avoids most of the computational burden, while providing a good approximation to the FIM. We then compare the precision of the parameters that can be expected using five study designs from the literature. We illustrate the usefulness of rationalizing data sampling by showing that, for a given level of precision, optimal design could reduce the total number of measurements by up 50 per cent. Our approach can be used by a statistician or a clinician aiming at designing an HCV viral kinetics study.     

Stochastic approximation EM for large-scale exploratory IRT factor analysis

A stochastic approximation EM algorithm (SAEM) is described for exploratory factor analysis of dichotomous or ordinal variables. The factor structure is obtained from sufficient statistics that are updated during iterations with the Robbins-Monro procedure. Two large-scale simulations are reported that compare accuracy and CPU time of the proposed SAEM algorithm to the Metropolis-Hasting Robbins-Monro procedure and to a generalized least squares analysis of the polychoric correlation matrix. A smaller-scale application to real data is also reported, including a method for obtaining standard errors of rotated factor loadings. A simulation study based on the real data analysis is conducted to study bias and error estimates. The SAEM factor algorithm requires minimal lines of code, no derivatives, and no large-matrix inversion. It is programmed entirely in R.     

Design evaluation and optimisation in crossover pharmacokinetic studies analysed by nonlinear mixed effects models

Bioequivalence or interaction trials are commonly studied in crossover design and can be analysed by nonlinear mixed effects models as an alternative to noncompartmental approach. We propose an extension of the population Fisher information matrix in nonlinear mixed effects models to design crossover pharmacokinetic trials, using a linearisation of the model around the random effect expectation, including within-subject variability and discrete covariates fixed or changing between periods. We use the expected standard errors of treatment effect to compute the power for the Wald test of comparison or equivalence and the number of subjects needed for a given power. We perform various simulations mimicking crossover two-period trials to show the relevance of these developments. We then apply these developments to design a crossover pharmacokinetic study of amoxicillin in piglets and implement them in the new version 3.2 of the r function PFIM.     

A linearization approach for the model‐based analysis of combined aggregate and individual patient data

The application of model‐based meta‐analysis in drug development has gained prominence recently, particularly for characterizing dose‐response relationships and quantifying treatment effect sizes of competitor drugs. The models are typically nonlinear in nature and involve covariates to explain the heterogeneity in summary‐level literature (or aggregate data (AD)). Inferring individual patient‐level relationships from these nonlinear meta‐analysis models leads to aggregation bias. Individual patient‐level data (IPD) are indeed required to characterize patient‐level relationships but too often this information is limited. Since combined analyses of AD and IPD allow advantage of the information they share to be taken, the models developed for AD must be derived from IPD models; in the case of linear models, the solution is a closed form, while for nonlinear models, closed form solutions do not exist. Here, we propose a linearization method based on a second order Taylor series approximation for fitting models to AD alone or combined AD and IPD. The application of this method is illustrated by an analysis of a continuous landmark endpoint, i.e., change from baseline in HbA1c at week 12, from 18 clinical trials evaluating the effects of DPP‐4 inhibitors on hyperglycemia in diabetic patients. The performance of this method is demonstrated by a simulation study where the effects of varying the degree of nonlinearity and of heterogeneity in covariates (as assessed by the ratio of between‐trial to within‐trial variability) were studied. A dose‐response relationship using an Emax model with linear and nonlinear effects of covariates on the emax parameter was used to simulate data. The simulation results showed that when an IPD model is simply used for modeling AD, the bias in the emax parameter estimate increased noticeably with an increasing degree of nonlinearity in the model, with respect to covariates. When using an appropriately derived AD model, the linearization method adequately corrected for bias. It was also noted that the bias in the model parameter estimates decreased as the ratio of between‐trial to within‐trial variability in covariate distribution increased. Taken together, the proposed linearization approach allows addressing the issue of aggregation bias in the particular case of nonlinear models of aggregate data. Copyright © 2014 John Wiley & Sons, Ltd.     

Multiple‐objective response‐adaptive repeated measurement designs in clinical trials for binary responses

A multiple‐objective allocation strategy was recently proposed for constructing response‐adaptive repeated measurement designs for continuous responses. We extend the allocation strategy to constructing response‐adaptive repeated measurement designs for binary responses. The approach with binary responses is quite different from the continuous case, as the information matrix is a function of responses, and it involves nonlinear modeling. To deal with these problems, we first build the design on the basis of success probabilities. Then we illustrate how various models can accommodate carryover effects on the basis of logits of response profiles as well as any correlation structure. Through computer simulations, we find that the allocation strategy developed for continuous responses also works well for binary responses. As expected, design efficiency in terms of mean squared error drops sharply, as more emphasis is placed on increasing treatment benefit than estimation precision. However, we find that it can successfully allocate more patients to better treatment sequences without sacrificing much estimation precision. Copyright © 2013 John Wiley & Sons, Ltd.     

群体药代动力学参数估计的SAEM算法

目的 建立在群体药代动力学参数估计问题中的SAEM算法.方法 本文基于房室模型的非线性混合效应模型,根据SAEM算法讨论群体药代动力学模型的参数估计问题,给出算法过程,并应用于计算机模拟C-肽释放实例分析.结果 计算机模拟证实了该方法的有效性,通过研究C-肽释放实例分析也表明SAEM算法估计群体药代动力学参数误差较小.结论利用SAEM算法估计群体药代动力学参数是一个比较好的方法.     

Model-based analyses of bioequivalence crossover trials using the stochastic approximation expectation maximisation algorithm

In this work, we develop a bioequivalence analysis using nonlinear mixed effects models (NLMEM) that mimics the standard noncompartmental analysis (NCA). We estimate NLMEM parameters, including between‐subject and within‐subject variability and treatment, period and sequence effects. We explain how to perform a Wald test on a secondary parameter, and we propose an extension of the likelihood ratio test for bioequivalence. We compare these NLMEM‐based bioequivalence tests with standard NCA‐based tests. We evaluate by simulation the NCA and NLMEM estimates and the type I error of the bioequivalence tests. For NLMEM, we use the stochastic approximation expectation maximisation (SAEM) algorithm implemented in monolix . We simulate crossover trials under H₀ using different numbers of subjects and of samples per subject. We simulate with different settings for between‐subject and within‐subject variability and for the residual error variance. The simulation study illustrates the accuracy of NLMEM‐based geometric means estimated with the SAEM algorithm, whereas the NCA estimates are biased for sparse design. NCA‐based bioequivalence tests show good type I error except for high variability. For a rich design, type I errors of NLMEM‐based bioequivalence tests (Wald test and likelihood ratio test) do not differ from the nominal level of 5%. Type I errors are inflated for sparse design. We apply the bioequivalence Wald test based on NCA and NLMEM estimates to a three‐way crossover trial, showing that Omnitrope®; (Sandoz GmbH, Kundl, Austria) powder and solution are bioequivalent to Genotropin®; (Pfizer Pharma GmbH, Karlsruhe, Germany). NLMEM‐based bioequivalence tests are an alternative to standard NCA‐based tests. However, caution is needed for small sample size and highly variable drug. Copyright © 2011 John Wiley & Sons, Ltd.     

Fisher information matrix for non-linear mixed-effects models: evaluation and application for optimal design of enoxaparin population pharmacokinetics

We address the problem of the choice and the evaluation of designs in population pharmacokinetic studies that use non-linear mixed-effects models. Criteria, based on the Fisher information matrix, have been developed to optimize designs and adapted to such models. We optimize designs under different constraints and evaluate them for a population pharmacokinetics study, within a new phase III trial of enoxaparin, a low molecular weight heparin. To do this, we approximate the expression of the Fisher information matrix for non-linear mixed-effects models including the residual error variance as a parameter to be estimated. We use the Fedorov-Wynn algorithm to minimize the inverse of the determinant of this matrix as required by the D-optimality criterion. Two optimal designs, as well as a design defined by pharmacologists, are evaluated by the simulation of 30 replicated data sets with NONMEM; all designs involve 220 patients with four measurements per patient. We also evaluate the relevance of the standard errors of estimation given from the Fisher information matrix by comparison with those given by NONMEM. The three designs provide more precise population parameter estimates; the optimal design gives the best precision and offers a simple clinical implementation. The expected standard errors given by the information matrix are close to those obtained by NONMEM on the simulation. Moreover, the proposed criterion of D-optimality appears to be a good measure to compare designs for population studies.     

A multiple‐model generalisation of updating clinical prediction models

There is growing interest in developing clinical prediction models (CPMs) to aid local healthcare decision‐making. Frequently, these CPMs are developed in isolation across different populations, with repetitive de novo derivation a common modelling strategy. However, this fails to utilise all available information and does not respond to changes in health processes through time and space. Alternatively, model updating techniques have previously been proposed that adjust an existing CPM to suit the new population, but these techniques are restricted to a single model. Therefore, we aimed to develop a generalised method for updating and aggregating multiple CPMs. The proposed “hybrid method” re‐calibrates multiple CPMs using stacked regression while concurrently revising specific covariates using individual participant data (IPD) under a penalised likelihood. The performance of the hybrid method was compared with existing methods in a clinical example of mortality risk prediction after transcatheter aortic valve implantation, and in 2 simulation studies. The simulation studies explored the effect of sample size and between‐population‐heterogeneity on the method, with each representing a situation of having multiple distinct CPMs and 1 set of IPD. When the sample size of the IPD was small, stacked regression and the hybrid method had comparable but highest performance across modelling methods. Conversely, in large IPD samples, development of a new model and the hybrid method gave the highest performance. Hence, the proposed strategy can inform the choice between utilising existing CPMs or developing a model de novo, thereby incorporating IPD, existing research, and prior (clinical) knowledge into the modelling strategy.     

Bayesian inference for unidirectional misclassification of a binary response trait

When assessing association between a binary trait and some covariates, the binary response may be subject to unidirectional misclassification. Unidirectional misclassification can occur when revealing a particular level of the trait is associated with a type of cost, such as a social desirability or financial cost. The feasibility of addressing misclassification is commonly obscured by model identification issues. The current paper attempts to study the efficacy of inference when the binary response variable is subject to unidirectional misclassification. From a theoretical perspective, we demonstrate that the key model parameters possess identifiability, except for the case with a single binary covariate. From a practical standpoint, the logistic model with quantitative covariates can be weakly identified, in the sense that the Fisher information matrix may be near singular. This can make learning some parameters difficult under certain parameter settings, even with quite large samples. In other cases, the stronger identification enables the model to provide more effective adjustment for unidirectional misclassification. An extension to the Poisson approximation of the binomial model reveals the identifiability of the Poisson and zero‐inflated Poisson models. For fully identified models, the proposed method adjusts for misclassification based on learning from data. For binary models where there is difficulty in identification, the method is useful for sensitivity analyses on the potential impact from unidirectional misclassification.     

Effects of ω‐3 Polyunsaturated Fatty Acids on the Homeostasis of CD4+ T Cells and Lung Injury in Mice With Polymicrobial Sepsis

Background: Sepsis is a common cause of death in critically ill patients. An overwhelming inflammatory response and imbalance of helper T (Th) cells and regulatory T (Treg) cells are thought to be involved in the progression of sepsis. ω‐3 Polyunsaturated fatty acids (PUFAs) were found to have anti‐inflammatory and immunomodulatory properties. This study investigated the effects of ω‐3 PUFAs on the balance of Th subsets, Treg cells, and the inflammatory response in septic mice. Methods: Mice were randomly assigned to soybean oil (SO) and fish oil (FO) groups. The 2 groups received an identical nutrient distribution except for the sources of the fat. The SO group was fed soybean oil, while part of the soybean oil was replaced by fish oil in the FO group. The FO group had an ω‐6/ω‐3 PUFA ratio of 2:1. After feeding the diets for 3 weeks, sepsis was induced by cecal ligation and puncture (CLP), and mice were sacrificed on days 0, 1, and 3. Results: Compared with the SO group, the FO group had lower inflammatory mediator levels in the plasma and peritoneal lavage fluid after CLP. Also, the FO group had lower Th1, Th2, and Th17 percentages and a higher Th1/Th2 ratio in blood. In lung tissues, neutrophil infiltration was reduced, whereas peroxisome proliferator–activated receptor γ expression was upregulated. Conclusions: A fish oil diet with an ω‐6/ω‐3 PUFA ratio of 2:1 may elicit more balanced Th polarization, alleviate inflammatory responses, and attenuate lung injury in CLP‐induced sepsis.     

A framework for the evaluation of patient information leaflets

Background The provision of patient information leaflets (PILs) is an important part of health care. PILs require evaluation, but the frameworks that are used for evaluation are largely under‐informed by theory. Most evaluation to date has been based on indices of readability, yet several writers argue that readability is not enough. We propose a framework for evaluating PILs that reflect the central role of the patient perspective in communication and use methods for evaluation based on simple linguistic principles. The proposed framework The framework has three elements that give rise to three approaches to evaluation. Each element is a necessary but not sufficient condition for effective communication. Readability (focussing on text) may be assessed using existing well‐established procedures. Comprehensibility (focussing on reader and text) may be assessed using multiple‐choice questions based on the lexical and semantic features of the text. Communicative effectiveness (focussing on reader) explores the relationship between the emotional, cognitive and behavioural responses of the reader and the objectives of the PIL. Suggested methods for assessment are described, based on our preliminary empirical investigations. Conclusions The tripartite model of communicative effectiveness is a patient‐centred framework for evaluating PILs. It may assist the field in moving beyond readability to broader indicators of the quality and appropriateness of printed information provided to patients.     

Non-compartmental estimation of pharmacokinetic parameters for flexible sampling designs

Pharmacokinetic (PK) studies aim to understand the kinetics of absorption, distribution, metabolism and elimination of a drug. Typically, such studies involve measuring the concentration of the drug in the plasma or blood at several time points after drug administration. In studying the PK behaviour, either the non-compartmental approach or alternatively a modelling approach can be utilized. Traditionally, the non-compartmental approach makes minimal assumptions about the data-generating process but requires the data to be collected in a very structured way. Conversely, the modelling approach depends heavily on assumptions about the data-generating process but does not impose a specific data structure. In this paper, we will discuss non-compartmental methods for estimating the area under the concentration versus time curve and other common PK parameters that use minimal assumptions about the data structure making it applicable to a wide range of PK studies. We will evaluate the methods using simulation and give an illustrative example.     

Early Fish Oil Supplementation and Organ Failure in Patients With Septic Shock From Abdominal Infections

Background: Fish oil (FO) has immunomodulating effects and may improve organ function and outcome in critically ill patients. This retrospective, propensity‐matched cohort study investigates the effects of early intravenous FO supplementation on organ failure in patients with septic shock from abdominal infection. Methods: A medical database was retrospectively searched for critically ill patients admitted because of septic shock from abdominal infection (n = 194). Demographic, clinical, and laboratory data; FO supplementation (10 g/d) (n = 42); rate, degree, and number of organ failures assessed by the Sequential Organ Failure Assessment (SOFA) score; and secondary outcome variables were recorded. A propensity score‐based model was used to establish 2 comparable groups (FO, n = 29; control, n = 29). Mann‐Whitney rank sum test, Fisher exact test, and logistic regression analyses were used to compare variables between groups. Results: There were no differences in the rate of single organ failures, the maximum SOFA score (median [interquartile range (IQR)], 12 [8‐15] vs 11 [9‐14]; P = .99), or the number of organ failures (median [IQR], 2 [1‐3] vs 2 [1‐3]; P = .54] between patients receiving FO supplementation and those not receiving supplementation. There were no group differences in the maximum C‐reactive protein levels (P = .1), duration of mechanical ventilation (P = .65) or hemofiltration (P = .21), intensive care unit–acquired infections, intensive care unit length of stay (P = .59), and intensive care unit (P = 1) or hospital mortality (P = 1). Conclusions: Early intravenous FO may not decrease the number and degree of organ failures in patients with septic shock from abdominal infection. Future trials are needed before FO supplementation in septic shock from abdominal infection can be recommended.     

Effects of fish oil on the neuro-endocrine responses to an endotoxin challenge in healthy volunteers

BACKGROUND & AIMS: Fish oil (FO) has been shown to modulate the acute and chronic inflammatory responses. Endotoxin (LPS) has been shown to mimic several aspects of sepsis. The study aimed at testing the effects of oral FO supplements in healthy subjects submitted to intravenous LPS on systemic and endocrine response. SUBJECTS AND METHODS: Fifteen healthy men (aged 26.0+/-3.1 years, BMI 23.8+/-1.9 kg/m2), were enrolled. Subjects were randomised to 3-4 weeks of oral FO supplementation (7.2 g/day, providing 1.1 g/day of 20:5 (n-3) and 0.7 g/day of 22:6 (n-3) fatty acids) or no supplementation and then submitted to endotoxin challenge: 2 ng/kg of LPS. All subjects were studied twice (placebo and LPS). Measurements: vital signs, energy expenditure (EE), glucose and lipid metabolism ((2)H2-glucose), plasma cytokines and stress hormones for 6 h after LPS or placebo. RESULTS: LPS caused cytokine release, fever, increases in heart rate, resting EE and substrate oxidation, plasma glucagon and glucose concentrations; the neuro-endocrine response was characterised by increased plasma stress hormones. FO significantly blunted fever, ACTH and cortisol plasma levels (no effect on cytokine release). FO blunted the peak norepinephrine after LPS. CONCLUSION: FO supplements blunted the endocrine stress response and the increase in body temperature, but had no impact on cytokine production after LPS. These findings conflict with the postulated anti-inflammatory effects of FO on arachidonic acid metabolism and cytokine release. These results suggest that FO may exert beneficial effects in sepsis though non-inflammatory which require further investigations.     

The effect of maternal fish oil supplementation during the last trimester of pregnancy on blood pressure, heart rate and heart rate variability in the 19-year-old offspring

Studies in experimental animals and human subjects have suggested that intake of n-3 fatty acids in early life can affect cardiovascular risk factors in adult life. Therefore, the aim of the present study was to investigate the effect of fish oil (FO) supplementation during the third trimester of pregnancy on blood pressure (BP), heart rate (HR) and HR variability (HRV) in the 19-year-old offspring. The study was based on follow-up of a randomised, controlled trial from 1990, in which 533 pregnant women were randomised to FO, olive oil (OO) or no oil (NO) during the last trimester of pregnancy. The offspring was invited to a physical examination including BP, HR and HRV measurements. A subgroup consisting of the offspring of mothers with a low baseline fish intake also had 24?h HRV determined. The OO group was used as reference and multiple linear regression modelling was used to compare the FO and OO groups. A total of 180 of the offspring from the FO and OO groups agreed to participate in the study (45?%). The adjusted difference between the FO and OO groups was 2 (95?% CI -?1, 4)?mmHg in systolic and 1 (95?% CI 0, 3)?mmHg in diastolic BP. The difference in HR was 1 (95?% CI -?2, 4). Also, HRV indices did not differ significantly between groups. Hence, FO supplementation during late pregnancy was not associated with offspring BP, HR and HRV during adolescence.     

An approximate approach to sample size determination in bioequivalence testing with multiple pharmacokinetic responses

The approval of generic drugs requires the evidence of average bioequivalence (ABE) on both the area under the concentration–time curve and the peak concentration C_max. The bioequivalence (BE) hypothesis can be decomposed into the non‐inferiority (NI) and non‐superiority (NS) hypothesis. Most of regulatory agencies employ the two one‐sided tests (TOST) procedure to test ABE between two formulations. As it is based on the intersection–union principle, the TOST procedure is conservative in terms of the type I error rate. However, the type II error rate is the sum of the type II error rates with respect to each null hypothesis of NI and NS hypotheses. When the difference in population means between two treatments is not 0, no close‐form solution for the sample size for the BE hypothesis is available. Current methods provide the sample sizes with either insufficient power or unnecessarily excessive power. We suggest an approximate method for sample size determination, which can also provide the type II rate for each of NI and NS hypotheses. In addition, the proposed method is flexible to allow extension from one pharmacokinetic (PK) response to determination of the sample size required for multiple PK responses. We report the results of a numerical study. An R code is provided to calculate the sample size for BE testing based on the proposed methods. Copyright © 2014 John Wiley & Sons, Ltd.     

Estimating within‐study covariances in multivariate meta‐analysis with multiple outcomes

Multivariate meta‐analysis allows the joint synthesis of effect estimates based on multiple outcomes from multiple studies, accounting for the potential correlations among them. However, standard methods for multivariate meta‐analysis for multiple outcomes are restricted to problems where the within‐study correlation is known or where individual participant data are available. This paper proposes an approach to approximating the within‐study covariances based on information about likely correlations between underlying outcomes. We developed methods for both continuous and dichotomous data and for combinations of the two types. An application to a meta‐analysis of treatments for stroke illustrates the use of the approximated covariance in multivariate meta‐analysis with correlated outcomes. Copyright © 2012 John Wiley & Sons, Ltd.