Modeling intra-individual inter-trial EEG response variability in autism.

Autism spectrum disorder (autism) is a prevalent neurodevelopmental condition characterized by early emerging impairments in social behavior and communication. EEG represents a powerful and non-invasive tool for examining functional brain differences in autism. Recent EEG evidence suggests that greater intra-individual trial-to-trial variability across EEG responses in stimulus-related tasks may characterize brain differences in autism. Traditional analysis of EEG data largely focuses on mean trends of the trial-averaged data, where trial-level analysis is rarely performed due to low neural signal to noise ratio. We propose to use nonlinear (shape-invariant) mixed effects (NLME) models to study intra-individual inter-trial EEG response variability using trial-level EEG data. By providing more precise metrics of response variability, this approach could enrich our understanding of neural disparities in autism and potentially aid the identification of objective markers. The proposed multilevel NLME models quantify variability in the signal's interpretable and widely recognized features (e.g., latency and amplitude) while also regularizing estimation based on noisy trial-level data. Even though NLME models have been studied for more than three decades, existing methods cannot scale up to large data sets. We propose computationally feasible estimation and inference methods via the use of a novel minorization-maximization (MM) algorithm. Extensive simulations are conducted to show the efficacy of the proposed procedures. Applications to data from a large national consortium find that children with autism have larger intra-individual inter-trial variability in P1 latency in a visual evoked potential (VEP) task, compared to their neurotypical peers.

Bayesian pattern-mixture models for dropout and intermittently missing data in longitudinal data analysis.

Valid inference can be drawn from a random-effects model for repeated measures that are incomplete if whether the data are missing or not, known as missingness, is independent of the missing data. Data that are missing completely at random or missing at random are two data types for which missingness is ignorable. Given ignorable missingness, statistical inference can proceed without addressing the source of the missing data in the model. If the missingness is not ignorable, however, recommendations are to fit multiple models that represent different plausible explanations of the missing data. A popular choice in methods for evaluating nonignorable missingness is a random-effects pattern-mixture model that extends a random-effects model to include one or more between-subjects variables that represent fixed patterns of missing data. Generally straightforward to implement, a fixed pattern-mixture model is one among several options for assessing nonignorable missingness, and when it is used as the sole model to address nonignorable missingness, understanding the impact of missingness is greatly limited. This paper considers alternatives to a fixed pattern-mixture model for nonignorable missingness that are generally straightforward to fit and encourage researchers to give greater attention to the possible impact of nonignorable missingness in longitudinal data analysis. Patterns of both monotonic and non-monotonic (intermittently) missing data are addressed. Empirical longitudinal psychiatric data are used to illustrate the models. A small Monte Carlo data simulation study is presented to help illustrate the utility of such methods.

Alternative covariance structures in mixed-effects models: Addressing intra- and inter-individual heterogeneity.

Mixed-effects models for repeated measures and longitudinal data include random coefficients that are unique to the individual, and thus permit subject-specific growth trajectories, as well as direct study of how the coefficients of a growth function vary as a function of covariates. Although applications of these models often assume homogeneity of the within-subject residual variance that characterizes within-person variation after accounting for systematic change and the variances of the random coefficients of a growth model that quantify individual differences in aspects of change, alternative covariance structures can be considered. These include allowing for serial correlations between the within-subject residuals to account for dependencies in data that remain after fitting a particular growth model or specifying the within-subject residual variance to be a function of covariates or a random subject effect to address between-subject heterogeneity due to unmeasured influences. Further, the variances of the random coefficients can be functions of covariates to relax the assumption that these variances are constant across subjects and to allow for the study of determinants of these sources of variation. In this paper, we consider combinations of these structures that permit flexibility in how mixed-effects models are specified to understand within- and between-subject variation in repeated measures and longitudinal data. Data from three learning studies are analyzed using these different specifications of mixed-effects models.

The Impact of Functional Form Complexity on Model Overfitting for Nonlinear Mixed-Effects Models.

Nonlinear mixed-effects models (NLMEMs) allow researchers to model curvilinear patterns of growth, but there is ambiguity as to what functional form the data follow. Often, researchers fit multiple nonlinear functions to data and use model selection criteria to decide which functional form fits the data "best." Frequently used model selection criteria only account for the number of parameters in a model but overlook the complexity of intrinsically nonlinear functional forms. This can lead to overfitting and hinder the generalizability and reproducibility of results. The primary goal of this study was to evaluate the performance of eight model selection criteria via a Monte Carlo simulation study and assess under what conditions these criteria are sensitive to model overfitting as it relates to functional form complexity. Results highlighted criteria with the potential to capture overfitting for intrinsically nonlinear functional forms for NLMEMs. Information criteria and the stochastic information complexity criterion recovered the true model more often than the average or conditional concordance correlation. Results also suggest that the amount of residual variance and sample size have an impact on model selection for NLMEMs. Implications for future research and recommendations for application are also provided.

Disease Progression in Multiple System Atrophy-Novel Modeling Framework and Predictive Factors.

BACKGROUND: Multiple system atrophy (MSA) is a rare and aggressive neurodegenerative disease that typically leads to death 6 to 10 years after symptom onset. The rapid evolution renders it crucial to understand the general disease progression and factors affecting the disease course. OBJECTIVES: The aims of this study were to develop a novel disease-progression model to estimate a population-level MSA progression trajectory and predict patient-specific continuous disease stages describing the degree of progress into the disease. METHODS: The disease-progression model estimated a population-level progression trajectory of subscales of the Unified MSA Rating Scale and the Unified Parkinson's Disease Rating Scale using patients in the European MSA natural history study. The predicted disease continuum was validated via multiple analyses based on reported anchor points, and the effect of MSA subtype on the rate of disease progression was evaluated. RESULTS: The predicted disease continuum spanned approximately 6 years, with an estimated average duration of 51 months for a patient with global disability score 0 to reach the highest level of 4. The predicted continuous disease stages were shown to be correlated with time of symptom onset and predictive of survival time. MSA motor subtype was found to significantly affect disease progression, with MSA-parkinsonian (MSA-P) type patients having an accelerated rate of progression. CONCLUSIONS: The proposed modeling framework introduces a new method of analyzing and interpreting the progression of MSA. It can provide new insights and opportunities for investigating covariate effects on the rate of progression and provide well-founded predictions of patient-level future progressions. © 2022 The Authors. Movement Disorders published by Wiley Periodicals LLC on behalf of International Parkinson and Movement Disorder Society.

Practical identifiability in the frame of nonlinear mixed effects models: the example of the in vitro erythropoiesis.

BACKGROUND: Nonlinear mixed effects models provide a way to mathematically describe experimental data involving a lot of inter-individual heterogeneity. In order to assess their practical identifiability and estimate confidence intervals for their parameters, most mixed effects modelling programs use the Fisher Information Matrix. However, in complex nonlinear models, this approach can mask practical unidentifiabilities. RESULTS: Herein we rather propose a multistart approach, and use it to simplify our model by reducing the number of its parameters, in order to make it identifiable. Our model describes several cell populations involved in the in vitro differentiation of chicken erythroid progenitors grown in the same environment. Inter-individual variability observed in cell population counts is explained by variations of the differentiation and proliferation rates between replicates of the experiment. Alternatively, we test a model with varying initial condition. CONCLUSIONS: We conclude by relating experimental variability to precise and identifiable variations between the replicates of the experiment of some model parameters.

Mixed-effects models for censored data with autoregressive errors.

Mixed-effects models, with modifications to accommodate censored observations (LMEC/NLMEC), are routinely used to analyze measurements, collected irregularly over time, which are often subject to some upper and lower detection limits. This paper presents a likelihood-based approach for fitting LMEC/NLMEC models with autoregressive of order p dependence of the error term. An EM-type algorithm is developed for computing the maximum likelihood estimates, obtaining as a byproduct the standard errors of the fixed effects and the likelihood value. Moreover, the constraints on the parameter space that arise from the stationarity conditions for the autoregressive parameters in the EM algorithm are handled by a reparameterization scheme, as discussed in Lin and Lee (2007). To examine the performance of the proposed method, we present some simulation studies and analyze a real AIDS case study. The proposed algorithm and methods are implemented in the new R package ARpLMEC.

Nonlinear mixed-effects models with misspecified random-effects distribution.

Nonlinear mixed-effects models are being widely used for the analysis of longitudinal data, especially from pharmaceutical research. They use random effects which are latent and unobservable variables so the random-effects distribution is subject to misspecification in practice. In this paper, we first study the consequences of misspecifying the random-effects distribution in nonlinear mixed-effects models. Our study is focused on Gauss-Hermite quadrature, which is now the routine method for calculation of the marginal likelihood in mixed models. We then present a formal diagnostic test to check the appropriateness of the assumed random-effects distribution in nonlinear mixed-effects models, which is very useful for real data analysis. Our findings show that the estimates of fixed-effects parameters in nonlinear mixed-effects models are generally robust to deviations from normality of the random-effects distribution, but the estimates of variance components are very sensitive to the distributional assumption of random effects. Furthermore, a misspecified random-effects distribution will either overestimate or underestimate the predictions of random effects. We illustrate the results using a real data application from an intensive pharmacokinetic study.

Determination of plasma concentration reference ranges for oral aripiprazole, olanzapine, and quetiapine.

BACKGROUND: Schizophrenia is a common disease which is commonly managed using antipsychotic medications (APS). Inadequate response and lack of adherence often prevent optimal therapeutic effectiveness. Monitoring APS concentrations can be useful to help improve outcomes for the patient. AIMS: The aim of this study was to develop "reference ranges" for oral aripiprazole, olanzapine, and quetiapine to allow clinicians to understand expected variability in patients treated with APS. The reference ranges were constructed to account for different oral doses, sampling times, and variability both between, and within, subjects. METHODS: Population pharmacokinetic models were used to simulate plasma concentrations over time under different doses and population demographics. The references were validated against external data both numerically and graphically. RESULTS: Reference ranges for oral aripiprazole, olanzapine, and quetiapine were derived and successfully validated against the external data. The 80% reference range for aripiprazole following a 2-mg oral dose was 14.7-41.6 ng/mL 0-4 h post dose and 10.6-37.1 ng/mL 20-24 h post dose. These ranges increased to 221-624 ng/mL 0-4 h post dose following administration of a 30-mg dose, and 159-557 ng/mL 20-24 h post dose. The 80% reference range 0-4 h post dose was 22.5-67.1 ng/mL following a 15-mg dose once daily of oral olanzapine, and 179-768 ng/mL following a 150-mg dose once daily of oral quetiapine. CONCLUSIONS: Comparing individual patients' APS levels with reference ranges, along with a full clinical assessment, could provide important insights to help a clinician optimize APS therapy.

Modeling subject-specific nonautonomous dynamics.

We consider modeling non-autonomous dynamical systems for a group of subjects. The proposed model involves a common baseline gradient function and a multiplicative time-dependent subject-specific effect that accounts for phase and amplitude variations in the rate of change across subjects. The baseline gradient function is represented in a spline basis and the subject-specific effect is modeled as a polynomial in time with random coefficients. We establish appropriate identifiability conditions and propose an estimator based on the hierarchical likelihood. We prove consistency and asymptotic normality of the proposed estimator under a regime of moderate-to-dense observations per subject. Simulation studies and an application to the Berkeley Growth Data demonstrate the effectiveness of the proposed methodology.

Recursive Partitioning with Nonlinear Models of Change.

In this article, we introduce nonlinear longitudinal recursive partitioning (nLRP) and the R package longRpart2 to carry out the analysis. This method implements recursive partitioning (also known as decision trees) in order to split data based on individual- (i.e., cluster) level covariates with the goal of predicting differences in nonlinear longitudinal trajectories. At each node, a user-specified linear or nonlinear mixed-effects model is estimated. This method is an extension of Abdolell et al.'s (2002) longitudinal recursive partitioning while permitting a nonlinear mixed-effects model in addition to a linear mixed-effects model in each node. We give an overview of recursive partitioning, nonlinear mixed-effects models for longitudinal data, describe nLRP, and illustrate its use with empirical data from the Early Childhood Longitudinal Study-Kindergarten Cohort.

Quantitative Methods in Psychological Aging Research: A Mini-Review.

As research on psychological aging moves forward, it is increasingly important to accurately assess longitudinal changes in psychological processes and to account for their (often complex) associations with sociodemographic, lifestyle, and health-related variables. Traditional statistical methods, though time tested and well documented, are not always satisfactory for meeting these aims. In this mini-review, we therefore focus the discussion on recent statistical advances that may be of benefit to researchers in psychological aging but that remain novel in our area of study. We first compare two methods for the treatment of incomplete data, a common problem in longitudinal research. We then discuss robust statistics, which address the question of what to do when critical assumptions of a standard statistical test are not met. Next, we discuss two approaches that are promising for accurately describing phenomena that do not unfold linearly over time: nonlinear mixed-effects models and (generalized) additive models. We conclude by discussing recursive partitioning methods, as these are particularly well suited for exploring complex relations among large sets of variables.

Assessing robustness of designs for random effects parameters for nonlinear mixed-effects models.

Optimal designs for nonlinear models are dependent on the choice of parameter values. Various methods have been proposed to provide designs that are robust to uncertainty in the prior choice of parameter values. These methods are generally based on estimating the expectation of the determinant (or a transformation of the determinant) of the information matrix over the prior distribution of the parameter values. For high dimensional models this can be computationally challenging. For nonlinear mixed-effects models the question arises as to the importance of accounting for uncertainty in the prior value of the variances of the random effects parameters. In this work we explore the influence of the variance of the random effects parameters on the optimal design. We find that the method for approximating the expectation and variance of the likelihood is of potential importance for considering the influence of random effects. The most common approximation to the likelihood, based on a first-order Taylor series approximation, yields designs that are relatively insensitive to the prior value of the variance of the random effects parameters and under these conditions it appears to be sufficient to consider uncertainty on the fixed-effects parameters only.

Nonlinear mixed-effects models for pharmacokinetic data analysis: assessment of the random-effects distribution.

Nonlinear mixed-effects models are frequently used for pharmacokinetic data analysis, and they account for inter-subject variability in pharmacokinetic parameters by incorporating subject-specific random effects into the model. The random effects are often assumed to follow a (multivariate) normal distribution. However, many articles have shown that misspecifying the random-effects distribution can introduce bias in the estimates of parameters and affect inferences about the random effects themselves, such as estimation of the inter-subject variability. Because random effects are unobservable latent variables, it is difficult to assess their distribution. In a recent paper we developed a diagnostic tool based on the so-called gradient function to assess the random-effects distribution in mixed models. There we evaluated the gradient function for generalized liner mixed models and in the presence of a single random effect. However, assessing the random-effects distribution in nonlinear mixed-effects models is more challenging, especially when multiple random effects are present, and therefore the results from linear and generalized linear mixed models may not be valid for such nonlinear models. In this paper, we further investigate the gradient function and evaluate its performance for such nonlinear mixed-effects models which are common in pharmacokinetics and pharmacodynamics. We use simulations as well as real data from an intensive pharmacokinetic study to illustrate the proposed diagnostic tool.

An automated sampling importance resampling procedure for estimating parameter uncertainty.

Quantifying the uncertainty around endpoints used for decision-making in drug development is essential. In nonlinear mixed-effects models (NLMEM) analysis, this uncertainty is derived from the uncertainty around model parameters. Different methods to assess parameter uncertainty exist, but scrutiny towards their adequacy is low. In a previous publication, sampling importance resampling (SIR) was proposed as a fast and assumption-light method for the estimation of parameter uncertainty. A non-iterative implementation of SIR proved adequate for a set of simple NLMEM, but the choice of SIR settings remained an issue. This issue was alleviated in the present work through the development of an automated, iterative SIR procedure. The new procedure was tested on 25 real data examples covering a wide range of pharmacokinetic and pharmacodynamic NLMEM featuring continuous and categorical endpoints, with up to 39 estimated parameters and varying data richness. SIR led to appropriate results after 3 iterations on average. SIR was also compared with the covariance matrix, bootstrap and stochastic simulations and estimations (SSE). SIR was about 10 times faster than the bootstrap. SIR led to relative standard errors similar to the covariance matrix and SSE. SIR parameter 95% confidence intervals also displayed similar asymmetry to SSE. In conclusion, the automated SIR procedure was successfully applied over a large variety of cases, and its user-friendly implementation in the PsN program enables an efficient estimation of parameter uncertainty in NLMEM.

Energetic constraint of non-monotonic mass change during offspring growth: a general hypothesis and application of a new tool.

Post-natal growth is an important life-history trait and can be a sensitive indicator of ecological stress. For over 50 years, monotonic (never-decreasing) growth has been viewed as the predominant trajectory of post-natal mass change in most animal species, notably among birds. However, prevailing analytical approaches and energetic constraints may limit detection of non-monotonic (or multiphasic), determinate growth patterns, such as mass recession in birds (weight loss prior to fledging, preceded by overshooting adult mass), which is currently believed to be restricted to few taxa. Energetic surplus and shortfall are widespread conditions that can directly influence the degree of mass overshooting and recession. Thus, we hypothesize that in many species, prevailing energetic constraints force mass change away from a fundamental non-monotonic trajectory to instead follow a monotonic curve. We observed highly non-monotonic, mass change trajectories (overshooting adult mass by up to almost 20%) among common tern Sterna hirundo chicks, a well-studied species long-established as growing monotonically. We quantified the prevalence and magnitude of non-monotonic mass change prior to fledging for 313 common tern chicks that successfully fledged from two discrete populations in multiple years. We used a new approach for analysing non-monotonic curves to examine differences in mass change trajectories between populations under contrasting abiotic (freshwater vs. saltwater) and biotic stresses (low rates of food provisioning). Some degree of mass recession occurred in 73% of all study chicks. Overshooting adult mass followed by extensive mass recession was most prevalent at our freshwater colony, being detected among 34-38% of chicks annually. Non-monotonic trajectories were less marked in populations experiencing ecological stress and among lower quality individuals. Chicks that were provisioned at higher rates were more likely to both overshoot adult mass and experience subsequent mass recession. Our results in common terns provide strong support for the hypothesis that non-monotonic trajectories are the fundamental pattern of mass change but are constrained to be monotonic under energetic shortfall. This justifies future tests of the generality of this hypothesis across a broad range of taxa. We also demonstrate a recent analytical tool that prevents routine fitting of monotonic curves without prior investigation of non-monotonic trends.

Evaluation of estimation, prediction and inference for autocorrelated latent variable modeling of binary data-a simulation study.

Longitudinal models of binary or ordered categorical data are often evaluated for adequacy by the ability of these to characterize the transition frequency and type between response states. Drug development decisions are often concerned with accurate prediction and inference of the probability of response by time and dose. A question arises on whether the transition probabilities need to be characterized adequately to ensure accurate response prediction probabilities unconditional on the previous response state. To address this, a simulation study was conducted to assess bias in estimation, prediction and inferences of autocorrelated latent variable models (ALVMs) when the transition probabilities are misspecified due to ill-posed random effects structures, inadequate likelihood approximation or omission of the autocorrelation component. The results may be surprising in that these suggest that characterizing autocorrelation in ALVMs is not as important as specifying a suitably rich random effects structure.

dOFV distributions: a new diagnostic for the adequacy of parameter uncertainty in nonlinear mixed-effects models applied to the bootstrap.

Knowledge of the uncertainty in model parameters is essential for decision-making in drug development. Contrarily to other aspects of nonlinear mixed effects models (NLMEM), scrutiny towards assumptions around parameter uncertainty is low, and no diagnostic exists to judge whether the estimated uncertainty is appropriate. This work aims at introducing a diagnostic capable of assessing the appropriateness of a given parameter uncertainty distribution. The new diagnostic was applied to case bootstrap examples in order to investigate for which dataset sizes case bootstrap is appropriate for NLMEM. The proposed diagnostic is a plot comparing the distribution of differences in objective function values (dOFV) of the proposed uncertainty distribution to a theoretical Chi square distribution with degrees of freedom equal to the number of estimated model parameters. The uncertainty distribution was deemed appropriate if its dOFV distribution was overlaid with or below the theoretical distribution. The diagnostic was applied to the bootstrap of two real data and two simulated data examples, featuring pharmacokinetic and pharmacodynamic models and datasets of 20-200 individuals with between 2 and 5 observations on average per individual. In the real data examples, the diagnostic indicated that case bootstrap was unsuitable for NLMEM analyses with around 70 individuals. A measure of parameter-specific "effective" sample size was proposed as a potentially better indicator of bootstrap adequacy than overall sample size. In the simulation examples, bootstrap confidence intervals were shown to underestimate inter-individual variability at low sample sizes. The proposed diagnostic proved a relevant tool for assessing the appropriateness of a given parameter uncertainty distribution and as such it should be routinely used.

Improving the estimation of parameter uncertainty distributions in nonlinear mixed effects models using sampling importance resampling.

Taking parameter uncertainty into account is key to make drug development decisions such as testing whether trial endpoints meet defined criteria. Currently used methods for assessing parameter uncertainty in NLMEM have limitations, and there is a lack of diagnostics for when these limitations occur. In this work, a method based on sampling importance resampling (SIR) is proposed, which has the advantage of being free of distributional assumptions and does not require repeated parameter estimation. To perform SIR, a high number of parameter vectors are simulated from a given proposal uncertainty distribution. Their likelihood given the true uncertainty is then approximated by the ratio between the likelihood of the data given each vector and the likelihood of each vector given the proposal distribution, called the importance ratio. Non-parametric uncertainty distributions are obtained by resampling parameter vectors according to probabilities proportional to their importance ratios. Two simulation examples and three real data examples were used to define how SIR should be performed with NLMEM and to investigate the performance of the method. The simulation examples showed that SIR was able to recover the true parameter uncertainty. The real data examples showed that parameter 95 % confidence intervals (CI) obtained with SIR, the covariance matrix, bootstrap and log-likelihood profiling were generally in agreement when 95 % CI were symmetric. For parameters showing asymmetric 95 % CI, SIR 95 % CI provided a close agreement with log-likelihood profiling but often differed from bootstrap 95 % CI which had been shown to be suboptimal for the chosen examples. This work also provides guidance towards the SIR workflow, i.e.,which proposal distribution to choose and how many parameter vectors to sample when performing SIR, using diagnostics developed for this purpose. SIR is a promising approach for assessing parameter uncertainty as it is applicable in many situations where other methods for assessing parameter uncertainty fail, such as in the presence of small datasets, highly nonlinear models or meta-analysis.

A Note on Recurring Misconceptions When Fitting Nonlinear Mixed Models.

Nonlinear mixed-effects (NLME) models are used when analyzing continuous repeated measures data taken on each of a number of individuals where the focus is on characteristics of complex, nonlinear individual change. Challenges with fitting NLME models and interpreting analytic results have been well documented in the statistical literature. However, parameter estimates as well as fitted functions from NLME analyses in recent articles have been misinterpreted, suggesting the need for clarification of these issues before these misconceptions become fact. These misconceptions arise from the choice of popular estimation algorithms, namely, the first-order linearization method (FO) and Gaussian-Hermite quadrature (GHQ) methods, and how these choices necessarily lead to population-average (PA) or subject-specific (SS) interpretations of model parameters, respectively. These estimation approaches also affect the fitted function for the typical individual, the lack-of-fit of individuals' predicted trajectories, and vice versa.