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PubMed2006

Estimation of non-linear growth models by linearization: a simulation study using a Gompertz function.

A method based on Taylor series expansion for estimation of location parameters and variance components of non-linear mixed effects models was considered. An attractive property of the method is the opportunity for an easily implemented algorithm. Estimation of non-linear mixed effects models can be done by common methods for linear mixed effects models, and thus existing programs can be used after small modifications. The applicability of this algorithm in animal breeding was studied with simulation using a Gompertz function growth model in pigs. Two growth data sets were analyzed: a full set containing observations from the entire growing period, and a truncated time trajectory set containing animals slaughtered prematurely, which is common in pig breeding. The results from the 50 simulation replicates with full data set indicate that the linearization approach was capable of estimating the original parameters satisfactorily. However, estimation of the parameters related to adult weight becomes unstable in the case of a truncated data set.

Algorithms↗
PubMed1999

Simple model for linear and nonlinear mixing at unstable fluid interfaces in spherical geometry.

A simple model was recently described for predicting linear and nonlinear mixing at an unstable planar fluid interface subjected to an arbitrary time-dependent variable acceleration history [J. D. Ramshaw, Phys. Rev. E 58, 5834 (1998)]. Here we present an analogous model for describing the mixing of two adjacent spherical fluid shells of different density resulting from an arbitrary time-dependent mean interface radius R(t). As in the planar case, the model is based on a heuristic expression for the kinetic energy of the system. This expression is based on that for the kinetic energy of a linearly perturbed interface, but with a dynamically renormalized effective wavelength which becomes proportional to the half-width a(t) of the mixing layer in the nonlinear regime. An equation of motion for s=R(2)a is then derived from Lagrange's equations. This evolution equation properly reduces to Plesset's equation for small perturbations, and to the previous planar model in the limit of very large R. The conservation properties of the model are established, and a suitable numerical scheme which preserves these properties is proposed.

Journal Article↗
PubMed1991

A log-linear modeling framework for selective mixing.

Nonrandom mixing can significantly alter the diffusion path of an infectious disease such as AIDS that requires intimate contact. Recent attempts to model this effect have sought a general framework capable of representing both simple and arbitrarily complicated mixing structures, and of solving the balancing problem in a nonequilibrium multigroup population. Log-linear models are proposed here as a general framework for solving the first problem. This approach offers several additional benefits: The parameters used to govern the mixing have a simple, intuitive interpretation, the framework provides a statistically sound basis for the estimation of these parameters from mixing-matrix data, and the resulting estimates are easily integrated into compartmental models for diffusion. A modified selection model is proposed to solve the second problem of generalizing the selection process to nonequilibrium populations. The distribution of contacts under this model is derived and is found to satisfy the assumptions of statistical inference for log-linear models. Together these techniques provide an integrated and flexible framework for modeling the role of selective mixing in the spread of disease.

Acquired Immunodeficiency Syndrome↗
PubMed2001

Mixed multivariate generalized linear models for assessing lower-limb arterial stenoses.

Experiments and observational studies often involve gathering information on several response variables, enabling us to model their dependence on observable predictor variables. Despite associations between the response variables, they are often analysed separately using general and generalized linear models. This paper investigates applications of multivariate regression analysis to improve the accuracy of predictions and decisions, in the specific context of diagnosing arterial stenoses in human legs. Two basic models are developed for this application, using (i) four binary responses and (ii) a mixture of two binary and two normal responses. The results clearly demonstrate the potential advantages offered by this approach.

Arterial Occlusive Diseases↗
PubMedDate not supplied

[Cyclosporin pharmacokinetic modeling in renal transplant patients].

AIM: To characterize the pharmacokinetic behavior of oral cyclosporin (CsA) in renal transplant patient, based on through blood concentration (C0) value, and to develop and to evaluate a Bayesian method for the individualized adjustment of CsA daily dose (DD). METHODS: Sixty-seven renal allograft recipients (42 men and 25 women) who had been treated with CsA (Sandimmun Neoral) associated with mycophenolate mofetil (2g daily) and prednisone (0,5-1 mg/kg daily) were randomly divided into two groups. Group A (N=48) was used to characterize CsA pharmacokinetic behavior and Group B (N=19) to evaluate Bayesian predictive performance for the model developed. We evaluated different structural models using non linear mixed effects modeling implemented in the NONMEN computer program in order to quantify the relationship between DD and C0. Accuracy and precision were evaluated by the mean standardized prediction error and its standard deviation. RESULTS: The Michaelis-Menten model was found to be optimum for quantifying the relationship between DD and C0. This model includes time-dependent parameters such as the Michaelis-Menten constant (Km) and daily maximum dose (Dmax) as well as first order autoregressive terms DD and C0 included in the structural model in an additive way. In the final model, the Dmax parameter is affected by plasmatic urea values and shows a half-life stabilization time of 90.90 days (95% CI: 52.60 to 250 days). Plasmatic urea values of 50 mg/dL are related to an initial Dmax value of 3 mg/kg daily (95% CI: 1.81 to 4.19 mg/kg daily) which decreases exponentially throughout the post-transplant period until it reaches a constant value of 2.16 mg/kg daily (95% CI: 1.41 to 2.91 mg/kg daily) In the same way, the Km parameter presents a central tendency value of 93.60 ng/mL (95% CI: 28.60 to 158.60 ng/mL) and the half-life necessary for its stabilization is 12.70 days (95% CI: 9.80 to 17.90 days). The residual variability of the model is 8.2%. The mean value of standardized prediction errors for populations and its standard deviation, as well as its confidence intervals of 95%, confirm the appropriate accuracy and precision of both a priori and a posteriori predictions with this model. Also, it reached between 70 and 100% a posteriori sequential predictions with prediction errors below 10%. CONCLUSION: The characterization of the pharmacokinetic behavior of CsA requires us to consider parameters such as Dmax and Km as non lineal functions of time, while the first order autoregressive terms DD and C0 must also be incorporated into the model.

Adult↗
PubMed2006

Pairwise fitting of mixed models for the joint modeling of multivariate longitudinal profiles.

A mixed model is a flexible tool for joint modeling purposes, especially when the gathered data are unbalanced. However, computational problems due to the dimension of the joint covariance matrix of the random effects arise as soon as the number of outcomes and/or the number of used random effects per outcome increases. We propose a pairwise approach in which all possible bivariate models are fitted, and where inference follows from pseudo-likelihood arguments. The approach is applicable for linear, generalized linear, and nonlinear mixed models, or for combinations of these. The methodology will be illustrated for linear mixed models in the analysis of 22-dimensional, highly unbalanced, longitudinal profiles of hearing thresholds.

Auditory Threshold↗
PubMed1997

Mixed models for bivariate response repeated measures data using Gibbs sampling.

Repeated measures data are frequently incomplete, unbalanced and correlated. There has been a great deal of recent interest in mixed effects models for analysing such data. In this paper, we develop bivariate response mixed effects models that are a generalization of linear mixed effects models for a single response variable. We describe their estimation procedures using a Markov chain Monte Carlo method, the Gibbs sampler. We illustrate the methods with analyses of intravenous vitamin D3 administration for secondary hyperparathyroidism in hemodialysis patients. In these data there were two response variables on each individual (PTH and calcium level). This study also suffered from attrition, like many longitudinal studies. While, considering the study design, it was reasonable to assume the drop-out mechanism for the calcium (Ca) level to be 'missing at random', the drop-out mechanism for the PTH level was likely to be non-ignorable. We found that the posterior treatment effects for the PTH level by the single response model were underestimated compared with those obtained by the bivariate response model, while there were little differences in the posterior features for the Ca level under both models.

Calcium↗
PubMed2004

Generic biomass functions for Norway spruce in Central Europe--a meta-analysis approach toward prediction and uncertainty estimation.

To facilitate future carbon and nutrient inventories, we used mixed-effect linear models to develop new generic biomass functions for Norway spruce (Picea abies (L.) Karst.) in Central Europe. We present both the functions and their respective variance-covariance matrices and illustrate their application for biomass prediction and uncertainty estimation for Norway spruce trees ranging widely in size, age, competitive status and site. We collected biomass data for 688 trees sampled in 102 stands by 19 authors. The total number of trees in the "base" model data sets containing the predictor variables diameter at breast height (D), height (H), age (A), site index (SI) and site elevation (HSL) varied according to compartment (roots: n = 114, stem: n = 235, dry branches: n = 207, live branches: n = 429 and needles: n = 551). "Core" data sets with about 40% fewer trees could be extracted containing the additional predictor variables crown length and social class. A set of 43 candidate models representing combinations of lnD, lnH, lnA, SI and HSL, including second-order polynomials and interactions, was established. The categorical variable "author" subsuming mainly methodological differences was included as a random effect in a mixed linear model. The Akaike Information Criterion was used for model selection. The best models for stem, root and branch biomass contained only combinations of D, H and A as predictors. More complex models that included site-related variables resulted for needle biomass. Adding crown length as a predictor for needles, branches and roots reduced both the bias and the confidence interval of predictions substantially. Applying the best models to a test data set of 17 stands ranging in age from 16 to 172 years produced realistic allocation patterns at the tree and stand levels. The 95% confidence intervals (% of mean prediction) were highest for crown compartments (approximately +/- 12%) and lowest for stem biomass (approximately +/- 5%), and within each compartment, they were highest for the youngest and oldest stands, respectively.

Biomass↗
PubMed2002

Application of mixed-effects models for exposure assessment.

The benefits of using linear mixed-effects models for occupational exposure assessment were studied by re-analysing three data sets from two published surveys with repeated exposure measurements. The relative contributions of particular characteristics affecting exposure levels were assessed as in a multiple regression model, while controlling for the correlation between repeated measurements. While one-way ANOVA allows one only to estimate unconditioned variance components, a mixed model enables estimation of between- and within-worker variance components of exposure levels while accounting for the fixed effects of work characteristics. Consequently, we can identify the work characteristics affecting each variance component. Mixed models were applied to the data sets with repeated measurements and auxil iary information on work characteristics. The between-worker variance components were reduced by 35, 66 and 80%, respectively, in the three data sets when work characteristics were taken into account. The within-worker (day-to-day) variability was reduced only in the pig farmer data set, by 25%, when accounting for work activities. In addition, coefficients of work characteristics from the mixed model were compared with coefficients resulting from originally published multiple linear regression models. In the rubber manufacturing data, the coefficients of the mixed model showed similar relative importance, but were generally smaller than the coefficients from regression models. However, in the pig farm data, only the coefficients of work activities were somewhat reduced. The mixed model is a helpful tool for estimating factors affecting exposure and suitable variance components. Identifying the factors in the working environment that affect the between-worker variability facilitates a posteriori grouping of workers into more uniformly exposed groups. Identifying the factors that affect the within-worker variance is helpful for hazard control and in designing efficient sampling schemes with reference to time schedule.

Analysis of Variance↗
PubMed2002

Estimate the time varying brain receptor occupancy in PET imaging experiments using non-linear fixed and mixed effect modeling approach.

Positron-Emission Tomography (PET) is an imaging technology currently used in drug development as a non-invasive measure of drug distribution and interaction with biochemical target system. The level of receptor occupancy achieved by a compound can be estimated by comparing time-activity measurements in an experiment done using tracer alone with the activity measured when the tracer is given following administration of unlabelled compound. The effective use of this surrogate marker as an enabling tool for drug development requires the definition of a model linking the brain receptor occupancy with the fluctuation of plasma concentrations. However, the predictive performance of such a model is strongly related to the precision on the estimate of receptor occupancy evaluated in PET scans collected at different times following drug treatment. Several methods have been proposed for the analysis and the quantification of the ligand-receptor interactions investigated from PET data. The aim of the present study is to evaluate alternative parameter estimation strategies based on the use of non-linear mixed effect models allowing to account for intra and inter-subject variability on the time-activity and for covariates potentially explaining this variability. A comparison of the different modeling approaches is presented using real data. The results of this comparison indicates that the mixed effect approach with a primary model partitioning the variance in term of Inter-Individual Variability (IIV) and Inter-Occasion Variability (IOV) and a second stage model relating the changes on binding potential to the dose of unlabelled drug is definitely the preferred approach.

Animals↗
PubMed2006

A general approach for two-stage analysis of multilevel clustered non-Gaussian data.

In this article, we propose a two-stage approach to modeling multilevel clustered non-Gaussian data with sufficiently large numbers of continuous measures per cluster. Such data are common in biological and medical studies utilizing monitoring or image-processing equipment. We consider a general class of hierarchical models that generalizes the model in the global two-stage (GTS) method for nonlinear mixed effects models by using any square-root-n-consistent and asymptotically normal estimators from stage 1 as pseudodata in the stage 2 model, and by extending the stage 2 model to accommodate random effects from multiple levels of clustering. The second-stage model is a standard linear mixed effects model with normal random effects, but the cluster-specific distributions, conditional on random effects, can be non-Gaussian. This methodology provides a flexible framework for modeling not only a location parameter but also other characteristics of conditional distributions that may be of specific interest. For estimation of the population parameters, we propose a conditional restricted maximum likelihood (CREML) approach and establish the asymptotic properties of the CREML estimators. The proposed general approach is illustrated using quartiles as cluster-specific parameters estimated in the first stage, and applied to the data example from a collagen fibril development study. We demonstrate using simulations that in samples with small numbers of independent clusters, the CREML estimators may perform better than conditional maximum likelihood estimators, which are a direct extension of the estimators from the GTS method.

Animals↗
PubMed2001

Validation of a test for dams carrying foetuses persistently infected with bovine viral-diarrhoea virus based on determination of antibody levels in late pregnancy.

Our objective was to estimate, using a generalised linear mixed-model approach, the sensitivity and specificity of an indirect ELISA when used to identify dams pregnant with persistently bovine viral-diarrhoea virus (BVDV)-infected foetuses. Cows that had been tested for antibodies to BVDV with a positive result during their pregnancy and where the offspring had been tested for both antibody and virus were identified by accessing the Swedish BVD database and the official pedigree records. The resulting data set consisted of 2162 cow-calf pairs in 126 herds, of which 281 included virus-positive calves. The sensitivities and specificities at 12 different decision thresholds (corresponding to optical densities (ODs) between 0.5 and 1.6) were estimated using generalised linear mixed models (binomial error, logit link), in which the gold standard (the BVDV status of the calf) was included as a covariate. In each model, the dependent variable was the dichotomous test result at the decision threshold in question. There was a significant positive interaction between the calf's status and gestational stage in all 12 models--indicating that the sensitivity and specificity at any given decision threshold was improved when the the test was performed later in pregnancy. The test should be applied only when samples have been taken in late gestation--not before the seventh month in pregnancy. If applied during the last months of pregnancy, the point estimate of the sensitivity ranges between 0.94 and 1.0 as the decision threshold is moved from 1.0 and downwards to 0.7. Similarly, the specificity ranges between 0.39 and 0.67 as the decision threshold is moved from 0.8 and upwards to 1.1.

Animals↗
PubMed2004

Evaluation of type I error rates when modeling ordered categorical data in NONMEM.

The development of non-linear mixed pharmacokinetic/pharmacodynamic models for continuous variables is usually guided by graphical assessment of goodness of fit and statistical significance criteria. The latter is usually the likelihood ratio test (LR). When the variable to be modeled is categorical, on the other hand, the available graphical methods are less informative and/or more complicated to use and the modeler needs to rely more heavily on statistical significance assessment in the model development. The aim of this study was to evaluate the type I error rates, obtained from using the LR test, for inclusion of a false parameter in a non-linear mixed effects model for ordered categorical data when modeling with NONMEM. Data with four ordinal categories were simulated from a logistic model. Two nested multinomial models were fitted to the data, the model used for simulation and a model containing one additional parameter. The difference in fit (objective function value) between models was calculated. Three types of models were explored; (i) a model without interindividual variability (IIV) where the addition of a parameter describing IIV was assessed, (ii) a model with IIV where the addition of a drug effect parameter (either categorical or continuous drug exposure measure) was evaluated, and (iii) a model including IIV and drug effect where the inclusion of a random effects parameter on the drug effect was assessed. Alterations were made to the simulation conditions, for example, varying the number of individuals and the size and distribution of the IIV, to explore potential influences on the type I error rate. The estimated type I error rate for inclusion of a false random effect parameter in model (i) and (iii) were, as expected, lower than the nominal. When the additional parameter was a fixed effects parameter describing drug effect (model(II)) the estimated type I error rates were in agreement with the nominal. None of the different simulation conditions tried changed this pattern. Thus, the LR test seems appropriate for judging the statistical significance of fixed effects parameters when modeling categorical data with NONMEM.

Clinical Trials as Topic↗
PubMed2002

Comparisons between exemplar similarity and mixed prototype models using a linearly separable category structure.

Nosofsky and Zaki (2002) found that an exemplar similarity model provided better accounts of individual subject classification and generalization performance than did a mixed prototype model proposed by Smith and Minda (1998; Minda & Smith, 2001). However, these previous tests used a nonlinearly separable category structure. In the present work, the authors extend the previous findings by demonstrating a superiority for the exemplar generalization model over the mixed prototype model in a case involving a linearly separable structure. Because this structure has numerous features that Minda and Smith argued should be conducive to prototype-based processing, the results pose a significant challenge to the mixed prototype view.

Generalization, Psychological↗
PubMed2003

Pharmacoepidemiologic investigation of a clonazepam-valproic acid interaction by mixed effect modeling using routine clinical pharmacokinetic data in Japanese patients.

Non-linear Mixed Effects Modeling (NONMEM) was used to estimate the effects of clonazepam-valproic acid interaction on clearance values using 576 serum levels collected from 317 pediatric and adult epileptic patients (age range, 0.3-32.6 years) during their clinical routine care. Patients received the administration of clonazepam and/or valproic acid. The final model describing clonazepam clearance was CL = 144.0 TBW-0.172 1.14VPA, where CL is total body clearance (mL/kg/h); TBW is total body weight (kg); VPA = 1 for concomitant administration of valproic acid and VPA = zero otherwise. The final model describing valproic acid clearance was CL (mL/kg/h) = 17.2 TBW-0.264 DOSE0.159 0.821CZP 0.896GEN, where DOSE is the daily dose of valproic acid (mg/kg/day); CZP = 1 for concomitant administration of clonazepam and CZP = zero otherwise; GEN = 1 for female and GEN = zero otherwise. Concomitant administration of clonazepam and valproic acid resulted in a 14% increase in clonazepam clearance, and a 17.9% decrease in valproic acid clearance.

Adolescent↗
PubMed2001

Modeling changes in CD4-positive T-lymphocyte counts after the start of highly active antiretroviral therapy and the relation with risk of opportunistic infections: the Aquitaine Cohort, 1996-1997.

After initiation of a treatment for human immunodeficiency virus type 1 infection containing a protease inhibitor, immune restoration associated with increases in CD4-positive (CD4+) T lymphocyte count may be delayed. In a sample of patients who had been prescribed protease inhibitors for the first time, the authors tested to see whether there was a minimal duration of CD4+ cell count increase before the increase had an impact on the occurrence of opportunistic infections. The evolution (difference between time t and baseline) of CD4+ cell count was modeled using a mixed effects linear model. Changes in CD4+ count estimated by this model were then included as time-dependent covariates in a proportional hazards model. Finally, the authors tested for the existence of a CD4+ change x time interaction. The authors used a sample of 553 French patients first prescribed protease inhibitors in 1996 and followed for a median of 16 months. During the first 120 days, there was no association between CD4+ change and the rate of opportunistic infections. After 120 days, each 50-cell/mm3 increase in CD4+ count was associated with a 60% (95% confidence interval: 45, 72) reduction in the incidence of opportunistic infections. These results, based on modeling of CD4+ cell response, at least indirectly reinforce the concept of a delayed but possible immune recovery with the use of protease inhibitors. The findings support the potential for interruption of certain types of prophylaxis against opportunistic infections under reasonable conditions of duration of antiretroviral therapy and sustained CD4+ cell response.

AIDS-Related Opportunistic Infections↗
PubMed2005

Using SAS PROC NLMIXED to fit item response theory models.

Researchers routinely construct tests or questionnaires containing a set of items that measure personality traits, cognitive abilities, political attitudes, and so forth. Typically, responses to these items are scored in discrete categories, such as points on a Likert scale or a choice out of several mutually exclusive alternatives. Item response theory (IRT) explains observed responses to items on a test (questionnaire) by a person's unobserved trait, ability, or attitude. Although applications of IRT modeling have increased considerably because of its utility in developing and assessing measuring instruments, IRT modeling has not been fully integrated into the curriculum of colleges and universities, mainly because existing general purpose statistical packages do not provide built-in routines with which to perform IRT modeling. Recent advances in statistical theory and the incorporation of those advances into general purpose statistical software such as the Statistical Analysis System (SAS) allow researchers to analyze measurement data by using a class of models known as generalized linear mixed effects models (McCulloch & Searle, 2001), which include IRT models as special cases. The purpose of this article is to demonstrate the generality and flexibility of using SAS to estimate IRT model parameters. With real data examples, we illustrate the implementations of a variety of IRT models for dichotomous, polytomous, and nominal responses. Since SAS is widely available in educational institutions, it is hoped that this article will contribute to the spread of IRT modeling in quantitative courses.

Attitude↗
PubMed1998

Automated covariate model building within NONMEM.

PURPOSE: One important task in population pharmacokinetic/pharmacodynamic model building is to identify the relationships between the parameters and demographic factors (covariates). The purpose of this study is to present an automated procedure that accomplishes this. The benefits of the proposed procedure over other commonly used methods are (i) the covariate model is built for all parameters simultaneously, (ii) the covariate model is built within the population modeling program (NONMEM) giving familiar meaning to the significance levels used, (iii) it can appropriately handle covariates that varies over time and (iv) it is not dependent on the quality of the posterior Bayes estimates of the individual parameter values. For situations in which the computer run-times are a limiting factor, a linearization of the non-linear mixed effects model is proposed and evaluated. METHODS: The covariate model is built in a stepwise fashion in which both linear and non-linear relationships between the parameters and covariates are considered. The linearization is basically a linear mixed effects model in which the population predictions and their derivatives with respect to the parameters are fixed from a model without covariates. The stepwise procedure as well as the linearization was evaluated using simulations in which the covariates were taken from a real data set. RESULTS: The covariate models identified agreed well with what could be expected based on the covariates that were actually supported in each of the simulated data sets. The predictive performance of the linearized model was close to that of the non-linearized model. CONCLUSIONS: The proposed procedure identifies covariate models that are close to the model supported by the data set as well as being useful in the prediction of new data. The linearized model performs nearly as well as the non-linearized model.

Mathematical Computing↗