Goodness-of-fit in generalized nonlinear mixed-effects models

In recent years, generalized linear and nonlinear mixed-effects models have proved to be powerful tools for the analysis of unbalanced longitudinal data. To date, much of the work has focused on various methods for estimating and comparing the parameters of mixed-effects models. Very little work has been done in the area of model selection and goodness-of-fit, particularly with respect to the assumed variance-covariance structure. In this paper, we present a goodness-of-fit statistic which can be used in a manner similar to the R2 criterion in linear regression for assessing the adequacy of an assumed mean and variance-covariance structure. In addition, we introduce an approximate pseudo-likelihood ratio test for testing the adequacy of the hypothesized convariance structure. These methods are illustrated and compared to the usual normal theory likelihood methods (Akaike's information criterion and the likelihood ratio test) using three examples. Simulation results indicate the pseudo-likelihood ratio test compares favorably with the standard normal theory likelihood ratio test, but both procedures are sensitive to departures from normality.     

Goodness-of-Fit Tests and Model Diagnostics for Negative Binomial Regression of RNA Sequencing Data

This work is about assessing model adequacy for negative binomial (NB) regression, particularly (1) assessing the adequacy of the NB assumption, and (2) assessing the appropriateness of models for NB dispersion parameters. Tools for the first are appropriate for NB regression generally; those for the second are primarily intended for RNA sequencing (RNA-Seq) data analysis. The typically small number of biological samples and large number of genes in RNA-Seq analysis motivate us to address the trade-offs between robustness and statistical power using NB regression models. One widely-used power-saving strategy, for example, is to assume some commonalities of NB dispersion parameters across genes via simple models relating them to mean expression rates, and many such models have been proposed. As RNA-Seq analysis is becoming ever more popular, it is appropriate to make more thorough investigations into power and robustness of the resulting methods, and into practical tools for model assessment. In this article, we propose simulation-based statistical tests and diagnostic graphics to address model adequacy. We provide simulated and real data examples to illustrate that our proposed methods are effective for detecting the misspecification of the NB mean-variance relationship as well as judging the adequacy of fit of several NB dispersion models.     

On the modeling and interpretation of oxygen uptake kinetics from ramp work rate tests

Ramp work rate tests have been used to estimate aerobic parameters in exercise stress testing. Previous studies have suggested an assumption of a linear dynamic system for O2 uptake kinetics. The implication is that model parameters estimated from ramp tests should be similar to those estimated from other dynamic tests. In nine healthy subjects, we found that model parameters used to characterize O2 consumption ramp data were not consistent with those used to characterize step data, when the comparison was made on a subject-to-subject basis. Furthermore the ramp data model parameter values were highly dependent (P less than 0.0001) on the ramp slope. A linear dynamic system interpretation of the ramp data model does not appear to be appropriate, suggesting that caution is needed in the interpretation of ramp data aerobic parameters. The data may be better described by nonlinear or higher order function. Ramp exercise testing is not suitable for assessing dynamic control properties of the cardiorespiratory response to exercise.     

Statistical analysis of nonlinear parameter estimation for Monod biodegradation kinetics using bivariate data

A nonlinear regression technique for estimating the Monod parameters describing biodegradation kinetics is presented and analyzed. Two model data sets were taken from a study of aerobic biodegradation of the polycyclic aromatic hydrocarbons (PAHs), naphthalene and 2-methylnaphthalene, as the growth-limiting substrates, where substrate and biomass concentrations were measured with time. For each PAH, the parameters estimated were: q(max), the maximum substrate utilization rate per unit biomass; K(S), the half-saturation coefficient; and Y, the stoichiometric yield coefficient. Estimating parameters when measurements have been made for two variables with different error structures requires a technique more rigorous than least squares regression. An optimization function is derived from the maximumlikelihood equation assuming an unknown, nondiagonal covariance matrix for the measured variables. Because the derivation is based on an assumption of normally distributed errors in the observations, the error structures of the regression variables were examined. Through residual analysis, the errors in the substrate concentration data were found to be distributed log-normally, demonstrating a need for log transformation of this variable. The covariance between ln C and X was found to be small but significantly nonzero at the 67% confidence level for NPH and at the 94% confidence level for 2MN. The nonlinear parameter estimation yielded unique values for q(max), K(S), and Y for naphthalene. Thus, despite the low concentrations of this sparingly soluble compound, the data contained sufficient information for parameter estimation. For 2-methylnaphthalene, the values of q(max) and K(S) could not be estimated uniquely; however, q(max)/K(S) was estimated. To assess the value of including the relatively imprecise biomass concentration data, the results from the bivariate method were compared with a univariate method using only the substrate concentration data. The results demonstrated that the bivariate data yielded a better confidence in the estimates and provided additional information about the model fit and model adequacy. The combination of the value of the bivariate data set and their nonzero covariance justifies the need for maximum likelihood estimation over the simpler nonlinear least squares regression.     

A latent variable approach to study gene-environment interactions in the presence of multiple correlated exposures

Many existing cohort studies initially designed to investigate disease risk as a function of environmental exposures have collected genomic data in recent years with the objective of testing for gene-environment interaction (G × E) effects. In environmental epidemiology, interest in G × E arises primarily after a significant effect of the environmental exposure has been documented. Cohort studies often collect rich exposure data; as a result, assessing G × E effects in the presence of multiple exposure markers further increases the burden of multiple testing, an issue already present in both genetic and environment health studies. Latent variable (LV) models have been used in environmental epidemiology to reduce dimensionality of the exposure data, gain power by reducing multiplicity issues via condensing exposure data, and avoid collinearity problems due to presence of multiple correlated exposures. We extend the LV framework to characterize gene-environment interaction in presence of multiple correlated exposures and genotype categories. Further, similar to what has been done in case-control G × E studies, we use the assumption of gene-environment (G-E) independence to boost the power of tests for interaction. The consequences of making this assumption, or the issue of how to explicitly model G-E association has not been previously investigated in LV models. We postulate a hierarchy of assumptions about the LV model regarding the different forms of G-E dependence and show that making such assumptions may influence inferential results on the G, E, and G × E parameters. We implement a class of shrinkage estimators to data adaptively trade-off between the most restrictive to most flexible form of G-E dependence assumption and note that such class of compromise estimators can serve as a benchmark of model adequacy in LV models. We demonstrate the methods with an example from the Early Life Exposures in Mexico City to Neuro-Toxicants Study of lead exposure, iron metabolism genes, and birth weight.     

Empirical problems of the hierarchical likelihood ratio test for model selection

Advocates of maximum likelihood (ML) approaches to phylogenetics commonly cite as one of their primary advantages the use of objective statistical criteria for model selection. Currently, a particular implementation of the likelihood ratio test (LRT) is the most commonly used model-selection criterion in phylogenetics. This approach requires the choice of a starting point and a parameter addition (or removal) sequence that can affect all ML inferences (i.e., topology, model, and all evolutionary parameters). Here, several alternative starting points and parameter sequences are tested in empirical data sets to assess their influence on model selection and optimal topology. In the studied data sets, varying model-selection protocols leads to selection of different models that, in some cases, lead to different ML trees. Given the sensitivity of the LRT, some possible solutions to model selection (within the hypothesis testing approach) are outlined, and alternative model-selection criteria are discussed. Some of the suggested alternatives seem to lack these problems, although their behavior and adequacy for phylogenetics needs to be further explored.     

Finding identifiable parameter combinations in nonlinear ODE models and the rational reparameterization of their input-output equations

When examining the structural identifiability properties of dynamic system models, some parameters can take on an infinite number of values and yet yield identical input-output data. These parameters and the model are then said to be unidentifiable. Finding identifiable combinations of parameters with which to reparameterize the model provides a means for quantitatively analyzing the model and computing solutions in terms of the combinations. In this paper, we revisit and explore the properties of an algorithm for finding identifiable parameter combinations using Gröbner Bases and prove useful theoretical properties of these parameter combinations. We prove a set of M algebraically independent identifiable parameter combinations can be found using this algorithm and that there exists a unique rational reparameterization of the input-output equations over these parameter combinations. We also demonstrate application of the procedure to a nonlinear biomodel.     

Generalized Binomial Regression Model

The generalized negative binomial distribution has been found useful in fitting over-dispersed as well as under-dispersed count data. We define and study the generalized binomial regression model which is used to predict a count response variable affected by one or more explanatory variables. The methods of maximum likelihood and moments are given for estimating the model parameters. Approximate tests for the adequacy of the model are considered. The generalized binomial regression model has been applied to two observed data sets to which binomial regression model was applied earlier.     

Parameter inference for biochemical systems that undergo a Hopf bifurcation

The increasingly widespread use of parametric mathematical models to describe biological systems means that the ability to infer model parameters is of great importance. In this study, we consider parameter inferability in nonlinear ordinary differential equation models that undergo a bifurcation, focusing on a simple but generic biochemical reaction model. We systematically investigate the shape of the likelihood function for the model's parameters, analyzing the changes that occur as the model undergoes a Hopf bifurcation. We demonstrate that there exists an intrinsic link between inference and the parameters’ impact on the modeled system's dynamical stability, which we hope will motivate further research in this area.     

Hierarchical multi-scale occupancy estimation for monitoring wildlife populations

Occupancy estimation is an effective analytic framework, but requires repeated surveys of a sample unit to estimate the probability of detection. Detection rates can be estimated from spatially replicated rather than temporally replicated surveys, but this may violate the closure assumption and result in biased estimates of occupancy. We present a new application of a multi-scale occupancy model that permits the simultaneous use of presence–absence data collected at 2 spatial scales and uses a removal design to estimate the probability of detection. Occupancy at the small scale corresponds to local territory occupancy, whereas occupancy at the large scale corresponds to regional occupancy of the sample units. Small-scale occupancy also corresponds to a spatial availability or coverage parameter where a species may be unavailable for sampling at a fraction of the survey stations. We applied the multi-scale occupancy model to a hierarchical sample design for 2 bird species in the Black Hills National Forest: brown creeper (Certhia americana) and lark sparrow (Chondestes grammacus). Our application of the multi-scale occupancy model is particularly well suited for hierarchical sample designs, such as spatially replicated survey stations within sample units that are typical of avian monitoring programs. The model appropriately accounts for the non-independence of the spatially replicated survey stations, addresses the closure assumption for the spatially replicated survey stations, and is useful for decomposing the observation process into detection and availability parameters. This analytic approach is likely to be useful for monitoring at local and regional scales, modeling multi-scale habitat relationships, and estimating population state variables for rare species of conservation concern. © 2011 The Wildlife Society.     

Parameter estimation from experimental laboratory data of HSV-1 by using alternative regression method

In this paper, an estimation of model parameters is performed by using the Alternative Regression (AR) approach on an experimental data set of Herpes Simplex Virus type-1 (HSV-1) infection with innate immune response. Throughout the specified course of time, the measurements of monocytes, neutrophils, and viral load were obtained from the corneas of infected mice. C57BL/6 (B6) mice were used at Oakland University, Department of Biological Sciences, and the outcome measurements were divided into training and testing data sets. The HSV-1 nonlinear dynamic model is proposed based on the observed data patterns and biological system information. The simulation results of the proposed model showed that they consistently fit the experimental data set. In addition, the sensitivity test and model validation diagnostics are considered to determine the most significant key parameters that affect the dynamics of the HSV-1 system.     

Analysis of longitudinal count data with serial correlation

We propose a state space model for analyzing equally or unequally spaced longitudinal count data with serial correlation. With a log link function, the mean of the Poisson response variable is a nonlinear function of the fixed and random effects. The random effects are assumed to be generated from a Gaussian first order autoregression (AR(1)). In this case, the mean of the observations has a log normal distribution. We use a combination of linear and nonlinear methods to take advantage of the Gaussian process embedded in a nonlinear function. The state space model uses a modified Kalman filter recursion to estimate the mean and variance of the AR(1) random error given the previous observations. The marginal likelihood is approximated by numerically integrating out the AR(1) random error. Simulation studies with different sets of parameters show that the state space model performs well. The model is applied to Epileptic Seizure data and Primary Care Visits Data. Missing and unequally spaced observations are handled naturally with this model.     

Linking hyperelastic theoretical models and experimental data of vaginal tissue through histological data

Mechanical characterization of living tissues and computer-based simulations related to medical issues, has become increasingly important to improve diagnostic processes and treatments evaluation. This work proposes a link between the mechanical testing and the material model predictions through histological data of vaginal tissue. Histological data was used to link tensile testing experiments with material-dependent parameters; the approach was adequate to capture the nonlinear response of ovine vaginal tissue over a large strain range.The experimental data obtained on a previous study, has two main components: tensile testing and histological analysis of the ovine vaginal tissue. Uniaxial tensile test data and histological data were collected from three sheep groups: virgins, pregnant and parous. The distal part of vaginal wall was selected since it is prone to tears induced by vaginal delivery.The HGO (Holzapfel-Gasser-Ogden) model parameters were fitted using a stochastic approach, namely the Simple Genetic Algorithm (SGA). The SGA was able to fit the experimental data successfully (R² > 0.986). The dimensionless coefficient ξ, was highly correlated with histological data. The ratio was seen to increase linearly with increasing collagen content.Coefficient ξ brings a new way of interpreting and understanding experimental data; it connects the nonlinear mechanical behaviour (tensile test) with tissue’s morphology (histology). It can be used as an ‘inverse’ (approximate) method to estimate the mechanical properties without direct experimental measurements, through basic histology.In this context, the proposed methodology appears very promising in estimating the response of the tissue via histological information.     

Polytomies and phylogenetically independent contrasts: examination of the bounded degrees of freedom approach

We examined the effect of soft polytomies on the performance (Type I error rate and bias) of Felsenstein's (1985; Am. Nat. 125:1-15) method of phylogenetically independent contrasts for estimating a bivariate correlation. We specifically tested the adequacy of bounding degrees of freedom, as suggested by Purvis and Garland (1993; Syst. Biol. 42:569-575). We simulated bivariate character evolution under Brownian motion (assumed by independent contrasts) and eight other models on five phylogenetic trees. For non-Brownian motion simulations, the adequacy of branch-length standardization was checked with a simple diagnostic (Garland et al., 1992; Syst. Biol. 41:18-32), and transformations were applied as indicated. Surprisingly, soft polytomies tended to have negligible effects on Type I error rates when models other than Brownian motion were used. Overall, and irrespective of evolutionary model, degrees of freedom were appropriately bounded for hypothesis testing, and unbiased estimates of the correlation coefficient were obtained. Our results, along with those of previous simulation studies, suggest that independent contrasts can reliably be applied to real data, even with phylogenetic uncertainty.     

A new method of identifying a systems based on the minimal quadratic discrepancy criterion for biophysics problems

A wide range of biophysical systems are described by nonlinear dynamic models mathematically presented as a set of ordinary differential equations in the Cauchy explicit form: [formula: see text] Fij(X1(t),..,XN(t),t), (i = 1,...,N, j = 1,...,M), where Fij (X1(t), ..., XN(t), t) is a set of basis functions satisfying the Lipschitz condition. We investigate the problem of evaluation of model constants aij (the system identification) using experimental data about the time dependence of the dynamic parameters of the system Xi(t). A new method of system identification for the class of similar nonlinear dynamic models is proposed. It is shown that the problem of identifying an initial nonlinear model can be reduced to the solution of a system of linear equations for the matrix of the dynamic model constants [aj]i. It is proposed to determine the set of dynamic model constants aij using the criterion of minimal quadratic discrepancy for the time dependence of the set of dynamic parameters Xi(t). An important special case of the nonlinear model, the quadratic model, is considered. Test problems of identification using this method are presented for two nonlinear systems: the Van der Pol type multiparametric nonlinear oscillator and the strange attractor of Ressler, a widely known example of dynamic systems showing the stochastic behavior.     

System for dynamic measurements of membrane capacitance in intact epithelial monolayers.

Dynamic measurements of exocytosis have been difficult to perform in intact epithelial monolayers. We have designed a system that estimates with +/-1% accuracy (99% confidence) the total membrane capacitance of monolayers represented by a lumped model. This impedance measurement and analysis system operates through a conventional transepithelial electrophysiology clamp, performing all signal measurements as frequently as every 5 s. Total membrane capacitance (the series combination of apical and basolateral membranes) is the inverse of one of three unique coefficients that describe the monolayer impedance. These coefficients are estimated using a weighted, nonlinear, least-squares algorithm. Using the estimated coefficients, solution ranges for individual membrane parameters are calculated, frequently providing results within +/-20% of true values without additional electrophysiological measurements. We determined the measurement system specifications and statistical significance of estimated parameters using 1) analytical testing with circuit simulation software and equation-generated data; 2) a system noise analysis combined with Monte Carlo simulations; and 3) analog model circuits for calibration of the electronic system and to check equation-generated results. Finally, the time course of capacitance changes associated with purinergically stimulated mucin exocytosis are quantified in monolayers of the colonic goblet cell-like cell line HT29-CI.16E.     

Nonlinear mixed effects models for repeated measures data

We propose a general, nonlinear mixed effects model for repeated measures data and define estimators for its parameters. The proposed estimators are a natural combination of least squares estimators for nonlinear fixed effects models and maximum likelihood (or restricted maximum likelihood) estimators for linear mixed effects models. We implement Newton-Raphson estimation using previously developed computational methods for nonlinear fixed effects models and for linear mixed effects models. Two examples are presented and the connections between this work and recent work on generalized linear mixed effects models are discussed.     

REPLICATED SIMILARITY TESTING USING PAIRED COMPARISON METHOD

The present article discusses the replicated similarity testing using a paired comparison method. The “two one‐sided tests” (TOST) combined with a beta‐binomial model are proposed for this testing. Simulation studies provide evidence that the proposed testing is valid and reliable.