Showing papers on "Mixed model published in 2005"

Conditional Akaike information for mixed-effects models

Abstract: SUMMARY This paper focuses on the Akaike information criterion, AIC, for linear mixed-effects models in the analysis of clustered data. We make the distinction between questions regarding the population and questions regarding the particular clusters in the data. We show that the AIC in current use is not appropriate for the focus on clusters, and we propose instead the conditional Akaike information and its corresponding criterion, the conditional AIC, CAIC. The penalty term in CAIC is related to the effective degrees of freedom p for a linear mixed model proposed by Hodges & Sargent (2001); p reflects an intermediate level of complexity between a fixed-effects model with no cluster effect and a corresponding model with fixed cluster effects. The CAIC is defined for both maximum likelihood and residual maximum likelihood estimation. A pharmacokinetics data appli cation is used to illuminate the distinction between the two inference settings, and to illustrate the use of the conditional AIC in model selection.

Mixed Models: Using the General Linear Mixed Model to Analyse Unbalanced Repeated Measures and Longitudinal Data

Abstract: The general linear mixed model provides a useful approach for analysing a wide variety of data structures which practising statisticians often encounter. Two such data structures which can be problematic to analyse are unbalanced repeated measures data and longitudinal data. Owing to recent advances in methods and software, the mixed model analysis is now readily available to data analysts. The model is similar in many respects to ordinary multiple regression, but because it allows correlation between the observations, it requires additional work to specify models and to assess goodness-of-fit. The extra complexity involved is compensated for by the additional flexibility it provides in model fitting. The purpose of this tutorial is to provide readers with a sufficient introduction to the theory to understand the method and a more extensive discussion of model fitting and checking in order to provide guidelines for its use. We provide two detailed case studies, one a clinical trial with repeated measures and dropouts, and one an epidemiological survey with longitudinal follow-up.

Bilinear Mixed Effects Models for Dyadic Data

Abstract: This article discusses the use of a symmetric multiplicative interaction effect to capture certain types of third-order dependence patterns often present in social networks and other dyadic datasets. Such an effect, along with standard linear fixed and random effects, is incorporated into a generalized linear model, and a Markov chain Monte Carlo algorithm is provided for Bayesian estimation and inference. In an example analysis of international relations data, accounting for such patterns improves model fit and predictive performance.

Mixed Models: Modelling Covariance Structure in the Analysis of Repeated Measures Data

Abstract: The term 'repeated measures' refers to data with multiple observations on the same sampling unit. In most cases, the multiple observations are taken over time, but they could be over space. It is usually plausible to assume that observations on the same unit are correlated. Hence, statistical analysis of repeated measures data must address the issue of covariation between measures on the same unit. Until recently, analysis techniques available in computer software only offered the user limited and inadequate choices. One choice was to ignore covariance structure and make invalid assumptions. Another was to avoid the covariance structure issue by analysing transformed data or making adjustments to otherwise inadequate analyses. Ignoring covariance structure may result in erroneous inference, and avoiding it may result in inefficient inference. Recently available mixed model methodology permits the covariance structure to be incorporated into the statistical model. The MIXED procedure of the SAS((R)) System provides a rich selection of covariance structures through the RANDOM and REPEATED statements. Modelling the covariance structure is a major hurdle in the use of PROC MIXED. However, once the covariance structure is modelled, inference about fixed effects proceeds essentially as when using PROC GLM. An example from the pharmaceutical industry is used to illustrate how to choose a covariance structure. The example also illustrates the effects of choice of covariance structure on tests and estimates of fixed effects. In many situations, estimates of linear combinations are invariant with respect to covariance structure, yet standard errors of the estimates may still depend on the covariance structure.

Estimation of value of travel-time savings using mixed logit models

Abstract: In this paper, we discuss some of the issues that arise with the computation of the implied value of travel-time savings in the case of discrete choice models allowing for random taste heterogeneity. We specifically look at the case of models producing a non-zero probability of positive travel-time coefficients, and discuss the consistency of such estimates with theories of rational economic behaviour. We then describe how the presence of unobserved travel-experience attributes or conjoint activities can bias the estimation of the travel-time coefficient, and can lead to false conclusions with regards to the existence of negative valuations of travel-time savings. We note that while it is important not to interpret such estimates as travel-time coefficients per se, it is nevertheless similarly important to allow such effects to manifest themselves; as such, the use of distributions with fixed bounds is inappropriate. On the other hand, the use of unbounded distributions can lead to further problems, as their shape (especially in the case of symmetrical distributions) can falsely imply the presence of positive estimates. We note that a preferable solution is to use bounded distributions where the bounds are estimated from the data during model calibration. This allows for the effects of data impurities or model misspecifications to manifest themselves, while reducing the risk of bias as a result of the shape of the distribution. To conclude, a brief application is conducted to support the theoretical claims made in the paper.

Random effect models for repeated measures of zero-inflated count data:

Abstract: For count responses, the situation of excess zeros (relative to what standard models allow) often occurs in biomedical and sociological applications. Modeling repeated measures of zero-inflated count data presents special challenges. This is because in addition to the problem of extra zeros, the correlation between measurements upon the same subject at different occasions needs to be taken into account. This article discusses random effect models for repeated measurements on this type of response variable. A useful model is the hurdle model with random effects, which separately handles the zero observations and the positive counts. In maximum likelihood model fitting, we consider both a normal distribution and a nonparametric approach for the random effects. A special case of the hurdle model can be used to test for zero inflation. Random effects can also be introduced in a zero-inflated Poisson or negative binomial model, but such a model may encounter fitting problems if there is zero deflation at any s...

Design and analysis of gauge R&R studies : making decisions with confidence intervals in random and mixed ANOVA models

Abstract: Preface 1. Introduction 2. Balanced One-Factor Random Models 3. Balanced Two-Factor Crossed Random Models with Interaction 4. Design of Gauge R&R Experiments 5. Balanced Two-Factor Crossed Random Models with No Interaction 6. Balanced Two-Factor Crossed Mixed Models 7. Unbalanced One- and Two-Factor Models 8. Strategies for Constructing Intervals with ANOVA Models Appendix A. The Analysis of Variance Appendix B. MLS and GCI Methods Appendix C. Tables of F-values Bibliography Index.

Simple fitting of subject‐specific curves for longitudinal data

Abstract: We present a simple semiparametric model for fitting subject-specific curves for longitudinal data. Individual curves are modelled as penalized splines with random coefficients. This model has a mixed model representation, and it is easily implemented in standard statistical software. We conduct an analysis of the long-term effect of radiation therapy on the height of children suffering from acute lymphoblastic leukaemia using penalized splines in the framework of semiparametric mixed effects models. The analysis revealed significant differences between therapies and showed that the growth rate of girls in the study cannot be fully explained by the group-average curve and that individual curves are necessary to reflect the individual response to treatment. We also show how to implement these models in S-PLUS and R in the appendix.

Maximization by Parts in Likelihood Inference

Abstract: This article presents and examines a new algorithm for solving a score equation for the maximum likelihood estimate in certain problems of practical interest. The method circumvents the need to compute second-order derivatives of the full likelihood function. It exploits the structure of certain models that yield a natural decomposition of a very complicated likelihood function. In this decomposition, the first part is a log-likelihood from a simply analyzed model, and the second part is used to update estimates from the first part. Convergence properties of this iterative (fixed-point) algorithm are examined, and asymptotics are derived for estimators obtained using only a finite number of iterations. Illustrative examples considered in the article include multivariate Gaussian copula models, nonnormal random-effects models, generalized linear mixed models, and state-space models. Properties of the algorithm and of estimators are evaluated in simulation studies on a bivariate copula model and a nonnormal...

Fixed and random effects.

Abstract: In performing a multilevel analysis, what is a level? And when should a variable have a random slope? This depends on whether the units in the design should be regarded as being representative of a population and on whether the researcher wishes to draw conclusions primarily about the observed units or primarily about the population. Keywords: fixed effects; random effects; linear model; multilevel analysis; mixed model; population; dummy variables

Statistical models for genotype by environment data: from conventional ANOVA models to eco-physiological QTL models

Abstract: To study the performance of genotypes under different growing conditions, plant breeders evaluate their germplasm in multi-environment trials. These trials produce genotype × environment data. We present statistical models for the analysis of such data that differ in the extent to which additional genetic, physiological, and environmental information is incorporated into the model formulation. The simplest model in our exposition is the additive 2-way analysis of variance model, without genotype × environment interaction, and with parameters whose interpretation depends strongly on the set of included genotypes and environments. The most complicated model is a synthesis of a multiple quantitative trait locus (QTL) model and an eco-physiological model to describe a collection of genotypic response curves. Between those extremes, we discuss linear-bilinear models, whose parameters can only indirectly be related to genetic and physiological information, and factorial regression models that allow direct incorporation of explicit genetic, physiological, and environmental covariables on the levels of the genotypic and environmental factors. Factorial regression models are also very suitable for the modelling of QTL main effects and QTL × environment interaction. Our conclusion is that statistical and physiological models can be fruitfully combined for the study of genotype × environment interaction.

Multilevel linear mixed model for tree diameter increment in stone pine (Pinus pinea): a calibrating approach

Abstract: Diameter increment for stone pine (Pinus pinea L.) is described using a multilevel linear mixed model, where stochastic variability is broken down among period, plot, tree and within-tree components. Covariates acting at tree and stand level, as breast height diameter, density, dominant height or site index are included in the model as fixed effects in order to explain residual random variability. The effect of competition on diameter increment is expressed by including distance independent competition indices. The entrance of regional effects within the model is tested to determine whether a single model is sufficient to explain stone pine diameter increment in Spain, or if, on the contrary, regional models are needed. Diameter increment model can be calibrated by predicting random components using data from past growth measurements taken in a complementary sample of trees. Calibration is carried out by using the best linear unbiased predictor (BLUP) theory. Both the fixed effects model and the calibrated model mean a substantial improvement when compared with the classical approach, widely used in forest management, of assuming constancy in diameter increment for a short projection period.

Analysis of unbalanced data by mixed linear models using the MIXED procedure of the SAS System

Abstract: Unbalanced data are a common problem in plant research based on designed experiments. Such data are often conveniently analysed using linear mixed models. Recent developments in mixed model theory have been implemented in major packages. This paper describes the use of the mixed procedure of the SAS System for the analysis of designed experiments. Special emphasis is given to the specification of options as depending on the assumed mixed model and on the unbalancedness in the data. In addition, we consider a compact representation of multiple comparisons for unbalanced data (letter display). Two small data sets are used to exemplify the methods.

Up hill, down dale: quantitative genetics of curvaceous traits

Abstract: ‘Repeated’ measurements for a trait and individual, taken along some continuous scale such as time, can be thought of as representing points on a curve, where both means and covariances along the trajectory can change, gradually and continually. Such traits are commonly referred to as ‘function-valued’ (FV) traits. This review shows that standard quantitative genetic concepts extend readily to FV traits, with individual statistics, such as estimated breeding values and selection response, replaced by corresponding curves, modelled by respective functions. Covariance functions are introduced as the FV equivalent to matrices of covariances. Considering the class of functions represented by a regression on the continuous covariable, FV traits can be analysed within the linear mixed model framework commonly employed in quantitative genetics, giving rise to the so-called random regression model. Estimation of covariance functions, either indirectly from estimated covariances or directly from the data using restricted maximum likelihood or Bayesian analysis, is considered. It is shown that direct estimation of the leading principal components of covariance functions is feasible and advantageous. Extensions to multi-dimensional analyses are discussed.

Restricted maximum likelihood estimation of genetic principal components and smoothed covariance matrices.

Abstract: Principal component analysis is a widely used 'dimension reduction' technique, albeit generally at a phenotypic level. It is shown that we can estimate genetic principal components directly through a simple reparameterisation of the usual linear, mixed model. This is applicable to any analysis fitting multiple, correlated genetic effects, whether effects for individual traits or sets of random regression coefficients to model trajectories. Depending on the magnitude of genetic correlation, a subset of the principal component generally suffices to capture the bulk of genetic variation. Corresponding estimates of genetic covariance matrices are more parsimonious, have reduced rank and are smoothed, with the number of parameters required to model the dispersion structure reduced from k(k + 1)/2 to m(2k - m + 1)/2 for k effects and m principal components. Estimation of these parameters, the largest eigenvalues and pertaining eigenvectors of the genetic covariance matrix, via restricted maximum likelihood using derivatives of the likelihood, is described. It is shown that reduced rank estimation can reduce computational requirements of multivariate analyses substantially. An application to the analysis of eight traits recorded via live ultrasound scanning of beef cattle is given.

R 2 statistics for mixed models

Abstract: The R statistic, when used in a regression or ANOVA context, is appealing because it summarizes how well the model explains the data in an easy-tounderstand way. R statistics are also useful to gauge the effect of changing a model. Generalizing R to mixed models is not obvious when there are correlated errors, as might occur if data are georeferenced or result from a designed experiment with blocking. Such an R statistic might refer only to the explanation associated with the independent variables, or might capture the explanatory power of the whole model. In the latter case, one might develop an R statistic from Wald or likelihood ratio statistics, but these can yield different numeric results. Example formulas for these generalizations of R are given. Two simulated data sets, one based on a randomized complete block design and the other with spatially correlated observations, demonstrate increases in R as model complexity increases, the result of modeling the covariance structure of the residuals.

Best Linear Unbiased Prediction of Cultivar Effects for Subdivided Target Regions

Abstract: Breeding for local adaptation may be economically viable providing there is sufficient genotype × subregion interaction. If the targeted subregion is part of a larger region covered by a testing network, information from neighboring subregions can be exploited to gain more precise estimates for the targeted subregion. For balanced data, the simplest approach is to use genotypic mean estimates for the whole target region, and this has often been shown to yield better predictions than simple means per subregion. A disadvantage of this approach is that it gives equal weight to all neighboring subregions and the targeted subregion, thus ignoring potential heterogeneity in information content. The objective of the present paper is to propose a method that allows a weighted combination of data from several subregions and to compare that method to other estimators. The proposed method is based on best linear unbiased prediction, which employs a weighted mean of subregion means. It follows from the theory of mixed models that the resulting estimator is optimal under the assumed model, minimizing prediction errors and maximizing the expected gain from selection. Using published variance component estimates, we found the resulting predictions to be superior to other approaches. We also show that the estimator is beneficial when selecting for global adaptation.

One- and Two-Sided Tolerance Intervals for General Balanced Mixed Models and Unbalanced One-Way Random Models

Abstract: In this article we develop procedures for one- and two-sided tolerance intervals for normal general linear models in which there exists a set of independent scaled chi-squared random variables. The proposed procedures are based on the concept of generalized pivotal quantities and are applicable to general mixed models provided that balanced data are available. However, this study focuses on situations involving unbalanced data. Specific attention is given to the unbalanced one-way random model. It is shown that the use of generalized pivotal quantities allows the construction of the tolerance intervals of interest fairly straightforward. Some practical examples are given to illustrate the proposed procedures. Furthermore, detailed statistical simulation studies are conducted to evaluate their performance, showing that the proposed procedures can be recommended for practical use.

A simulation study on tests of hypotheses and confidence intervals for fixed effects in mixed models for blocked experiments with missing data

Abstract: This article considers the analysis of experiments with missing data from various experimental designs frequently used in agricultural research (randomized complete blocks, split plots, strip plots). We investigate the small sample properties of REML-based Wald-type F tests using linear mixed models. Several methods for approximating the denominator degrees of freedom are employed, all of which are available with the MIXED procedure of the SAS System (8.02). The simulation results show that the Kenward-Roger method provides the best control of the Type I error rate and is not inferior to other methods in terms of power.

Multiple imputation under Bayesianly smoothed pattern-mixture models for non-ignorable drop-out

Abstract: Conventional pattern-mixture models can be highly sensitive to model misspecification. In many longitudinal studies, where the nature of the drop-out and the form of the population model are unknown, interval estimates from any single pattern-mixture model may suffer from undercoverage, because uncertainty about model misspecification is not taken into account. In this article, a new class of Bayesian random coefficient pattern-mixture models is developed to address potentially non-ignorable drop-out. Instead of imposing hard equality constraints to overcome inherent inestimability problems in pattern-mixture models, we propose to smooth the polynomial coefficient estimates across patterns using a hierarchical Bayesian model that allows random variation across groups. Using real and simulated data, we show that multiple imputation under a three-level linear mixed-effects model which accommodates a random level due to drop-out groups can be an effective method to deal with non-ignorable drop-out by allowing model uncertainty to be incorporated into the imputation process.

Optimal smoothing in nonparametric mixed-effect models

Abstract: Mixed-effect models are widely used for the analysis of correlated data such as longitudinal data and repeated measures. In this article, we study an approach to the nonparametric estimation of mixed-effect models. We consider models with parametric random effects and flexible fixed effects, and employ the penalized least squares method to estimate the models. The issue to be addressed is the selection of smoothing parameters through the generalized cross-validation method, which is shown to yield optimal smoothing for both real and latent random effects. Simulation studies are conducted to investigate the empirical performance of generalized cross-validation in the context. Real-data examples are presented to demonstrate the applications of the methodology.

Spatial Assessment of Model Errors from Four Regression Techniques

Abstract: Fomst modelers have attempted to account for the spatial autocorrelations among trees in growth and yield models by applying alternative regression techniques such as linear mixed models (LMM), generalized additive models (GAM), and geographicalIy weighted regression (GWR) However, the model errors are commonly assessed using average errors across the entire study area and across tree size classes Little attention has been paid to the spatial heterogeneity of model performance In this study, we used local Moran coefficients to investigate the spatial distributions of the model errors from the four regression models The results indicated that GAM improved model-fitting to the data and provided better predictions for the response variable However, it is nonspatial in nature and, consequently, generated spatial distributions for the model errors similar to the ones from ordinary least-squares (OLS) Furthermore, OLS and GAM yielded more clusters of similar (either positive or negative) model errors, indicating that trees in some subareas were either all underestimated or all overestimated for the response variable In contrast, LMM was able to model the spatial covariance structures in the data and obtain more accurate predictions by accounting for the effects of spatial autocorrelations through the empirical best linear unbiased predictors GWR is a varying-coefficient modeling technique It estimated the model coefficients locally at each tree in the example plot and resulted in more accurate predictions for the response variable Moreover, the spatial distributions of the model errors from LMM and GWR were more desirable, with fewer clusters of dissimilar model errors than the ones derived from OLS and GAM

Influence Diagnostics for Linear Mixed Models

Abstract: Linear mixed models are extremely sensitive to outlying responses and extreme points in the fixed and random effect design spaces. Few diag- nostics are available in standard computing packages. We provide routine diagnostic tools, which are computationally inexpensive. The diagnostics are functions of basic building blocks: studentized residuals, error contrast matrix, and the inverse of the response variable covariance matrix. The ba- sic building blocks are computed only once from the complete data analysis and provide information on the influence of the data on different aspects of the model fit. Numerical examples provide analysts with the complete pictures of the diagnostics. The linear mixed model provides flexibility in fitting models with various combinations of fixed and random effects, and is often used to analyze data in a broad spectrum of areas. It is well known that not all observations in a data set play an equal role in determining estimates, tests and other statistics. Sometimes the character of estimates in the model may be determined by only a few cases while most of the data are essentially ignored. It is important that the data analyst be aware of particular observations that have an unusually large influence on the results of the analysis. Such cases may be assessed as being appropriate and retained in the analysis, may represent inappropriate data and be eliminated from the analysis, may suggest that additional data need to be collected, may suggest the current modeling scheme is inadequate, or may indicate a data reading or data entry error. Regardless of the ultimate assessment of such cases, their identification is necessary before intelligent subject-matter-based decisions can be drawn. In ordinary linear models such model diagnostics are generally available in statistical packages and standard textbooks on applied regression, see for ex- ample, Cook and Weisberg (1982), Chatterjee and Hadi (1986, 1988). Oman

Simultaneous modelling of survival and longitudinal data with an application to repeated quality of life measures.

Abstract: In biomedical studies, interest often focuses on the relationship between patient's characteristics or some risk factors and both quality of life and survival time of subjects under study. In this paper, we propose a simultaneous modelling of both quality of life and survival time using the observed covariates. Moreover, random effects are introduced into the simultaneous models to account for dependence between quality of life and survival time due to unobserved factors. EM algorithms are used to derive the point estimates for the parameters in the proposed model and profile likelihood function is used to estimate their variances. The asymptotic properties are established for our proposed estimators. Finally, simulation studies are conducted to examine the finite-sample properties of the proposed estimators and a liver transplantation data set is analyzed to illustrate our approaches.