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PubMed2002

Marginally specified generalized linear mixed models: a robust approach.

Longitudinal data modeling is complicated by the necessity to deal appropriately with the correlation between observations made on the same individual. Building on an earlier nonrobust version proposed by Heagerty (1999, Biometrics 55, 688-698), our robust marginally specified generalized linear mixed model (ROBMS-GLMM) provides an effective method for dealing with such data. This model is one of the first to allow both population-averaged and individual-specific inference. As well, it adopts the flexibility and interpretability of generalized linear mixed models for introducing dependence but builds a regression structure for the marginal mean, allowing valid application with time-dependent (exogenous) and time-independent covariates. These new estimators are obtained as solutions of a robustified likelihood equation involving Huber's least favorable distribution and a collection of weights. Huber's least favorable distribution produces estimates that are resistant to certain deviations from the random effects distributional assumptions. Innovative weighting strategies enable the ROBMS-GLMM to perform well when faced with outlying observations both in the response and covariates. We illustrate the methodology with an analysis of a prospective longitudinal study of laryngoscopic endotracheal intubation, a skill that numerous health-care professionals are expected to acquire. The principal goal of our research is to achieve robust inference in longitudinal analyses.

Biometry↗

PubMed2003

Random effects selection in linear mixed models.

We address the important practical problem of how to select the random effects component in a linear mixed model. A hierarchical Bayesian model is used to identify any random effect with zero variance. The proposed approach reparameterizes the mixed model so that functions of the covariance parameters of the random effects distribution are incorporated as regression coefficients on standard normal latent variables. We allow random effects to effectively drop out of the model by choosing mixture priors with point mass at zero for the random effects variances. Due to the reparameterization, the model enjoys a conditionally linear structure that facilitates the use of normal conjugate priors. We demonstrate that posterior computation can proceed via a simple and efficient Markov chain Monte Carlo algorithm. The methods are illustrated using simulated data and real data from a study relating prenatal exposure to polychlorinated biphenyls and psychomotor development of children.

Algorithms↗

PubMed2006

QTL methodology for response curves on the basis of non-linear mixed models, with an illustration to senescence in potato.

The improvement of quantitative traits in plant breeding will in general benefit from a better understanding of the genetic basis underlying their development. In this paper, a QTL mapping strategy is presented for modelling the development of phenotypic traits over time. Traditionally, crop growth models are used to study development. We propose an integration of crop growth models and QTL models within the framework of non-linear mixed models. We illustrate our approach with a QTL model for leaf senescence in a diploid potato cross. Assuming a logistic progression of senescence in time, two curve parameters are modelled, slope and inflection point, as a function of QTLs. The final QTL model for our example data contained four QTLs, of which two affected the position of the inflection point, one the senescence progression-rate, and a final one both inflection point and rate.

Nonlinear Dynamics↗

PubMed2001

Linear mixed models with flexible distributions of random effects for longitudinal data.

Normality of random effects is a routine assumption for the linear mixed model, but it may be unrealistic, obscuring important features of among-individual variation. We relax this assumption by approximating the random effects density by the seminonparameteric (SNP) representation of Gallant and Nychka (1987, Econometrics 55, 363-390), which includes normality as a special case and provides flexibility in capturing a broad range of nonnormal behavior, controlled by a user-chosen tuning parameter. An advantage is that the marginal likelihood may be expressed in closed form, so inference may be carried out using standard optimization techniques. We demonstrate that standard information criteria may be used to choose the tuning parameter and detect departures from normality, and we illustrate the approach via simulation and using longitudinal data from the Framingham study.

Biometry↗

PubMedDate not supplied

Multivariate linear mixed models for multiple outcomes.

We propose a multivariate linear mixed (MLMM) for the analysis of multiple outcomes, which generalizes the latent variable model of Sammel and Ryan. The proposed model assumes a flexible correlation structure among the multiple outcomes, and allows a global test of the impact of exposure across outcomes. In contrast to the Sammel-Ryan model, the MLMM separates the mean and correlation parameters so that the mean estimation will remain reasonably robust even if the correlation is misspecified. The model is applied to birth defects data, where continuous data on the size of infants who were exposed to anticonvulsant medications in utero are compared to controls.

Abnormalities, Drug-Induced↗

PubMed2002

A Monte Carlo EM algorithm for generalized linear mixed models with flexible random effects distribution.

A popular way to represent clustered binary, count, or other data is via the generalized linear mixed model framework, which accommodates correlation through incorporation of random effects. A standard assumption is that the random effects follow a parametric family such as the normal distribution; however, this may be unrealistic or too restrictive to represent the data. We relax this assumption and require only that the distribution of random effects belong to a class of 'smooth' densities and approximate the density by the seminonparametric (SNP) approach of Gallant and Nychka (1987). This representation allows the density to be skewed, multi-modal, fat- or thin-tailed relative to the normal and includes the normal as a special case. Because an efficient algorithm to sample from an SNP density is available, we propose a Monte Carlo EM algorithm using a rejection sampling scheme to estimate the fixed parameters of the linear predictor, variance components and the SNP density. The approach is illustrated by application to a data set and via simulation.

Journal Article↗

PubMed2004

Applying linear mixed models to estimate reliability in clinical trial data with repeated measurements.

Repeated measures are exploited to study reliability in the context of psychiatric health sciences. It is shown how test-retest reliability can be derived using linear mixed models when the scale is continuous or quasi-continuous. The advantage of this approach is that the full modeling power of mixed models can be used. Repeated measures with a different mean structure can be used to usefully study reliability, correction for covariate effects is possible, and a complicated variance-covariance structure between measurements is allowed. In case the variance structure reduces to a random intercept (compound symmetry), classical methods are recovered. With more complex variance structures (e.g., including random slopes of time and/or serial correlation), time-dependent reliability functions are obtained. The methodology is motivated by and applied to data from five double-blind randomized clinical trials comparing the effects of risperidone to conventional antipsychotic agents for the treatment of chronic schizophrenia. Model assumptions are investigated through residual plots and by investigating the effect of influential observations.

Analysis of Variance↗

PubMed2004

A comparison of the general linear mixed model and repeated measures ANOVA using a dataset with multiple missing data points.

Longitudinal methods are the methods of choice for researchers who view their phenomena of interest as dynamic. Although statistical methods have remained largely fixed in a linear view of biology and behavior, more recent methods, such as the general linear mixed model (mixed model), can be used to analyze dynamic phenomena that are often of interest to nurses. Two strengths of the mixed model are (1) the ability to accommodate missing data points often encountered in longitudinal datasets and (2) the ability to model nonlinear, individual characteristics. The purpose of this article is to demonstrate the advantages of using the mixed model for analyzing nonlinear, longitudinal datasets with multiple missing data points by comparing the mixed model to the widely used repeated measures ANOVA using an experimental set of data. The decision-making steps in analyzing the data using both the mixed model and the repeated measures ANOVA are described.

Analysis of Variance↗

PubMed2000

Use of linear mixed models to detect irregular fetal growth.

Ultrasonographic fetal biometry data can be used to estimate fetal weights repeatedly over time. Different types of reference bands have been presented to judge such data. But relevant aspects of an individual growth curve are not taken in to account. How can individual growth curves be used efficiently to detect irregular fetal growth? Linear mixed model analysis of a sample of 167 uncomplicated pregnancies with 304 ultrasonic measurements revealed that a random intercept model is sufficient and heterogeneity in slopes is negligible. Hence irregularity of a sequence of weights can be captured by three quantities' residual level, trend and mean squared error' for which reference ranges are derived from the fitted model. After standardization they are independently normal and chi-square distributed in the reference population. Regularity of fetal growth may then be reported in terms of level and trend either in the form of Z-scores or as percentile indices. A sample of 97 pregnancies diagnosed as cases of chronic placental insufficiency postnatally is analysed to validate the ability to detected abnormal growth. Only the mean level and slope revealed to be able to detect IUGR. MSE was not discriminative. The sensitivity of the method depends on the number and placement of observation times, which opens the possibility of a model-based cost-effectiveness analysis. The derived scores make efficient use of available information and are easily implementable into any computer based monitoring device.

Decision Making, Computer-Assisted↗

PubMed1999

Solving large mixed linear models using preconditioned conjugate gradient iteration.

Continuous evaluation of dairy cattle with a random regression test-day model requires a fast solving method and algorithm. A new computing technique feasible in Jacobi and conjugate gradient based iterative methods using iteration on data is presented. In the new computing technique, the calculations in multiplication of a vector by a matrix were recorded to three steps instead of the commonly used two steps. The three-step method was implemented in a general mixed linear model program that used preconditioned conjugate gradient iteration. Performance of this program in comparison to other general solving programs was assessed via estimation of breeding values using univariate, multivariate, and random regression test-day models. Central processing unit time per iteration with the new three-step technique was, at best, one-third that needed with the old technique. Performance was best with the test-day model, which was the largest and most complex model used. The new program did well in comparison to other general software. Programs keeping the mixed model equations in random access memory required at least 20 and 435% more time to solve the univariate and multivariate animal models, respectively. Computations of the second best iteration on data took approximately three and five times longer for the animal and test-day models, respectively, than did the new program. Good performance was due to fast computing time per iteration and quick convergence to the final solutions. Use of preconditioned conjugate gradient based methods in solving large breeding value problems is supported by our findings.

Algorithms↗

PubMed1999

Genetic variance components analysis for binary phenotypes using generalized linear mixed models (GLMMs) and Gibbs sampling.

The common complex diseases such as asthma are an important focus of genetic research, and studies based on large numbers of simple pedigrees ascertained from population-based sampling frames are becoming commonplace. Many of the genetic and environmental factors causing these diseases are unknown and there is often a strong residual covariance between relatives even after all known determinants are taken into account. This must be modelled correctly whether scientific interest is focused on fixed effects, as in an association analysis, or on the covariances themselves. Analysis is straightforward for multivariate Normal phenotypes, but difficulties arise with other types of trait. Generalized linear mixed models (GLMMs) offer a potentially unifying approach to analysis for many classes of phenotype including multivariate Normal traits, binary traits, and censored survival times. Markov Chain Monte Carlo methods, including Gibbs sampling, provide a convenient framework within which such models may be fitted. In this paper, Bayesian inference Using Gibbs Sampling (a generic Gibbs sampler; BUGS) is used to fit GLMMs for multivariate Normal and binary phenotypes in nuclear families. BUGS is easy to use and readily available. We motivate a suitable model structure for Normal phenotypes and show how the model extends to binary traits. We discuss parameter interpretation and statistical inference and show how to circumvent a number of important theoretical and practical problems that we encountered. Using simulated data we show that model parameters seem consistent and appear unbiased in smaller data sets. We illustrate our methods using data from an ongoing cohort study.

Binomial Distribution↗

PubMed2000

Variance components analysis for pedigree-based censored survival data using generalized linear mixed models (GLMMs) and Gibbs sampling in BUGS.

Complex human diseases are an increasingly important focus of genetic research. Many of the determinants of these diseases are unknown and there is often a strong residual covariance between relatives even when all known genetic and environmental factors have been taken into account. This must be modeled correctly whether scientific interest is focused on fixed effects, as in an association analysis, or on the covariance structure itself. Analysis is straightforward for multivariate normally distributed traits, but difficulties arise with other types of trait. Generalized linear mixed models (GLMMs) offer a potentially unifying approach to analysis for many classes of phenotype including right censored survival times. This includes age-at-onset and age-at-death data and a variety of other censored traits. Markov chain Monte Carlo (MCMC) methods, including Gibbs sampling, provide a convenient framework within which such GLMMs may be fitted. In this paper, we use BUGS ("Bayesian inference using Gibbs sampling": a readily available, generic Gibbs sampler) to fit GLMMs for right-censored survival times in nuclear and extended families. We discuss parameter interpretation and statistical inference, and show how to circumvent a number of important theoretical and practical problems. Using simulated data, we show that model parameters are consistent. We further illustrate our methods using data from an ongoing cohort study. Finally, we propose that the random effects associated with a genetic component of variance (e.g., sigma(2)(A)) in a GLMM may be regarded as an adjusted "phenotype" and used as input to a conventional model-based or model-free linkage analysis. This provides a simple way to conduct a linkage analysis for a trait reflected in a right-censored survival time while comprehensively adjusting for observed confounders at the level of the individual and latent environmental effects shared across families.

Bayes Theorem↗

PubMed2007

The statistical performance of an MCF-7 cell culture assay evaluated using generalized linear mixed models and a score test.

Biological assays often utilize experimental designs where observations are replicated at multiple levels, and where each level represents a separate component of the assay's overall variance. Statistical analysis of such data usually ignores these design effects, whereas more sophisticated methods would improve the statistical power of assays. This report evaluates the statistical performance of an in vitro MCF-7 cell proliferation assay (E-SCREEN) by identifying the optimal generalized linear mixed model (GLMM) that accurately represents the assay's experimental design and variance components. Our statistical assessment found that 17beta-oestradiol cell culture assay data were best modelled with a GLMM configured with a reciprocal link function, a gamma error distribution, and three sources of design variation: plate-to-plate; well-to-well, and the interaction between plate-to-plate variation and dose. The gamma-distributed random error of the assay was estimated to have a coefficient of variation (COV) = 3.2 per cent, and a variance component score test described by X. Lin found that each of the three variance components were statistically significant. The optimal GLMM also confirmed the estrogenicity of five weakly oestrogenic polychlorinated biphenyls (PCBs 17, 49, 66, 74, and 128). Based on information criteria, the optimal gamma GLMM consistently out-performed equivalent naive normal and log-normal linear models, both with and without random effects terms. Because the gamma GLMM was by far the best model on conceptual and empirical grounds, and requires only trivially more effort to use, we encourage its use and suggest that naive models be avoided when possible.

Biological Assay↗

PubMed2001

Use of generalized linear mixed models in the spatial analysis of small-area malaria incidence rates in Kwazulu Natal, South Africa.

Spatial statistical analysis of 1994-1995 small-area malaria incidence rates in the population of the northernmost districts of KwaZulu Natal, South Africa, was undertaken to identify factors that might explain very strong heterogeneity in the rates. In this paper, the authors describe a method of adjusting the regression analysis results for strong spatial correlation in the rates by using generalized linear mixed models and variograms. The results of the spatially adjusted, multiple regression analysis showed that malaria incidence was significantly positively associated with higher winter rainfall and a higher average maximum temperature and was significantly negatively associated with increasing distance from water bodies. The statistical model was used to produce a map of predicted malaria incidence in the area, taking into account local variation from the model prediction if this variation was supported by the data. The predictor variables showed that even small differences in climate can have very marked effects on the intensity of malaria transmission, even in areas subject to malaria control for many years. The results of this study have important implications for malaria control programs in the area.

Climate↗

PubMed2003

Estimating correlation by using a general linear mixed model: evaluation of the relationship between the concentration of HIV-1 RNA in blood and semen.

Estimating the correlation coefficient between two outcome variables is one of the most important aspects of epidemiological and clinical research. A simple Pearson's correlation coefficient method is usually employed when there are complete independent data points for both outcome variables. However, researchers often deal with correlated observations in a longitudinal setting with missing values where a simple Pearson's correlation coefficient method cannot be used. General linear mixed models (GLMM) techniques were used to estimate correlation coefficients in a longitudinal data set with missing values. A random regression mixed model with unstructured covariance matrix was employed to estimate correlation coefficients between concentrations of HIV-1 RNA in blood and seminal plasma. The effects of CD4 count and antiretroviral therapy were also examined. We used data sets from three different centres (650 samples from 238 patients) where blood and seminal plasma HIV-1 RNA concentrations were collected from patients; 137 samples from 90 different patients without antiviral therapy and 513 samples from 148 patients receiving therapy were considered for analysis. We found no significant correlation between blood and semen HIV-1 RNA concentration in the absence of antiviral therapy. However, a moderate correlation between blood and semen HIV-1 RNA was observed among subjects with lower CD4 counts receiving therapy. Our findings confirm and extend the idea that the concentrations of HIV-1 in semen often differ from the HIV-1 concentration in blood. Antiretroviral therapy administered to subjects with low CD4 counts result in sufficient concomitant reduction of HIV-1 in blood and semen so as to improve the correlation between these compartments. These results have important implications for studies related to the sexual transmission of HIV, and development of HIV prevention strategies.

Anti-HIV Agents↗

PubMed1998

Use of generalized linear mixed models in analyzing mutant frequency data from the transgenic mouse assay.

The transgenic mouse assay is now widely used for the study of mutagenesis in diverse rodent tissues and to test chemicals for genotoxic potential. This kind of assay generally involves nested observations at several levels of sampling, e.g., animals, packaging reactions, and plates. Due to the common origin, the mutant frequency (MF) in tissues from the same animal are likely to be positively correlated, inducing extra variation relative to the common binomial variation. In this article, a generalized linear mixed model is used to analyze the overdispersed binomial data on mutant frequency from the transgenic mouse assay, with a random effect for each level of the sampling hierarchy. This is a comprehensive framework within which different sources of variation in the data can be evaluated in nested factorial experiments and treatment effects can be assessed simultaneously. It avoids the current practice of repeated testing for excess binomial variability at each level of the sampling hierarchy and aggregating data up the levels, but fits the data with one single model. Parameters associated with the fixed effects, particularly dose, and the variance components for the random effects (e.g., animals, packages, and plates) can be estimated and tested for significance. Data previously reported in the literature involving the lacl gene from the Big Blue mouse are used to illustrate the proposed method.

Animals↗

PubMed2004

A generalized linear mixed models approach for detecting incident clusters of disease in small areas, with an application to biological terrorism.

Since the intentional dissemination of anthrax through the US postal system in the fall of 2001, there has been increased interest in surveillance for detection of biological terrorism. More generally, this could be described as the detection of incident disease clusters. In addition, the advent of affordable and quick geocoding allows for surveillance on a finer spatial scale than has been possible in the past. Surveillance for incident clusters of disease in both time and space is a relatively undeveloped arena of statistical methodology. Surveillance for bioterrorism detection, in particular, raises unique issues with methodological relevance. For example, the bioterrorism agents of greatest concern cause initial symptoms that may be difficult to distinguish from those of naturally occurring disease. In this paper, the authors propose a general approach to evaluating whether observed counts in relatively small areas are larger than would be expected on the basis of a history of naturally occurring disease. They implement the approach using generalized linear mixed models. The approach is illustrated using data on health-care visits (1996-1999) from a large Massachusetts managed care organization/multispecialty practice group in the context of syndromic surveillance for anthrax. The authors argue that there is great value in using the geographic data.

Bioterrorism↗

PubMed1999

Generalised linear mixed models analysis of risk factors for contamination of Danish broiler flocks with Salmonella typhimurium.

We present a retrospective observational study of risk factors associated with the occurrence of Salmonella typhimurium (ST) in Danish broiler flocks. The study is based on recordings from 1994 in the ante-mortem database maintained by the Danish Poultry Council. The epidemiological units are the broiler flocks (about 4000 flocks) which are clustered within producers. Broiler flocks with ST-infected parent stocks show increased risk of salmonella infection, and also the hatchery affects the salmonella status significantly. Among the rearing factors, only the use of medicine as well as the time of rearing, and the sampling method are significant. Epidemiological control would seem most efficient on starting at the top levels of the production hierarchy from which a major part of the ST contamination is derived. A secondary purpose of the study is to evaluate different statistical approaches and software for the analysis of a moderately-sized data set of veterinary origin. We compare the results from five analyses of the generalised linear mixed model (GLMM) type. The first observation is that the results agree reasonably well and lead to similar conclusions. A closer look reveals certain patterns of bias and estimation accuracy that correspond well with theoretical findings and practical experience reported in the statistical literature.

Animal Husbandry↗