Formulating Mixed Models for Experiments, Including Longitudinal Experiments

Mixed models have become important in analyzing the results of experiments, particularly those that require more complicated models (e.g., those that involve longitudinal data). This article describes a method for deriving the terms in a mixed model. Our approach extends an earlier method by Brien and Bailey to explicitly identify terms for which autocorrelation and smooth trend arising from longitudinal observations need to be incorporated in the model. At the same time we retain the principle that the model used should include, at least, all the terms that are justified by the randomization. This is done by dividing the factors into sets, called tiers, based on the randomization and determining the crossing and nesting relationships between factors. The method is applied to formulate mixed models for a wide range of examples. We also describe the mixed model analysis of data from a three-phase experiment to investigate the effect of time of refinement on Eucalyptus pulp from four different sources. Cubic smoothing splines are used to describe differences in the trend over time and unstructured covariance matrices between times are found to be necessary.

Mitochondrial sequence data and Dipsacales phylogeny: Mixed models, partitioned Bayesian analyses, and model selection

Phylogenetic analyses of chloroplast DNA sequences, morphology, and combined data have provided consistent support for many of the major branches within the angiosperm, clade Dipsacales. Here we use sequences from three mitochondrial loci to test the existing broad scale phylogeny and in an attempt to resolve several relationships that have remained uncertain. Parsimony, maximum likelihood, and Bayesian analyses of a combined mitochondrial data set recover trees broadly consistent with previous studies, although resolution and support are lower than in the largest chloroplast analyses. Combining chloroplast and mitochondrial data results in a generally well-resolved and very strongly supported topology but the previously recognized problem areas remain. To investigate why these relationships have been difficult to resolve we conducted a series of experiments using different data partitions and heterogeneous substitution models. Usually more complex modeling schemes are favored regardless of the partitions recognized but model choice had little effect on topology or support values. In contrast there are consistent but weakly supported differences in the topologies recovered from coding and non-coding matrices. These conflicts directly correspond to relationships that were poorly resolved in analyses of the full combined chloroplast-mitochondrial data set. We suggest incongruent signal has contributed to our inability to confidently resolve these problem areas. (c) 2007 Elsevier Inc. All rights reserved.

Robust linear mixed models with skew-normal independent distributions from a Bayesian perspective

Linear mixed models were developed to handle clustered data and have been a topic of increasing interest in statistics for the past 50 years. Generally. the normality (or symmetry) of the random effects is a common assumption in linear mixed models but it may, sometimes, be unrealistic, obscuring important features of among-subjects variation. In this article, we utilize skew-normal/independent distributions as a tool for robust modeling of linear mixed models under a Bayesian paradigm. The skew-normal/independent distributions is an attractive class of asymmetric heavy-tailed distributions that includes the skew-normal distribution, skew-t, skew-slash and the skew-contaminated normal distributions as special cases, providing an appealing robust alternative to the routine use of symmetric distributions in this type of models. The methods developed are illustrated using a real data set from Framingham cholesterol study. (C) 2009 Elsevier B.V. All rights reserved.

Influence diagnostics in nonlinear mixed-effects elliptical models

In this work we propose and analyze nonlinear elliptical models for longitudinal data, which represent an alternative to gaussian models in the cases of heavy tails, for instance. The elliptical distributions may help to control the influence of the observations in the parameter estimates by naturally attributing different weights for each case. We consider random effects to introduce the within-group correlation and work with the marginal model without requiring numerical integration. An iterative algorithm to obtain maximum likelihood estimates for the parameters is presented, as well as diagnostic results based on residual distances and local influence [Cook, D., 1986. Assessment of local influence. journal of the Royal Statistical Society - Series B 48 (2), 133-169; Cook D., 1987. Influence assessment. journal of Applied Statistics 14 (2),117-131; Escobar, L.A., Meeker, W.Q., 1992, Assessing influence in regression analysis with censored data, Biometrics 48, 507-528]. As numerical illustration, we apply the obtained results to a kinetics longitudinal data set presented in [Vonesh, E.F., Carter, R.L., 1992. Mixed-effects nonlinear regression for unbalanced repeated measures. Biometrics 48, 1-17], which was analyzed under the assumption of normality. (C) 2009 Elsevier B.V. All rights reserved.

Sampling, WLS, and Mixed Models

Mixed models may be defined with or without reference to sampling, and can be used to predict realized random effects, as when estimating the latent values of study subjects measured with response error. When the model is specified without reference to sampling, a simple mixed model includes two random variables, one stemming from an exchangeable distribution of latent values of study subjects and the other, from the study subjects` response error distributions. Positive probabilities are assigned to both potentially realizable responses and artificial responses that are not potentially realizable, resulting in artificial latent values. In contrast, finite population mixed models represent the two-stage process of sampling subjects and measuring their responses, where positive probabilities are only assigned to potentially realizable responses. A comparison of the estimators over the same potentially realizable responses indicates that the optimal linear mixed model estimator (the usual best linear unbiased predictor, BLUP) is often (but not always) more accurate than the comparable finite population mixed model estimator (the FPMM BLUP). We examine a simple example and provide the basis for a broader discussion of the role of conditioning, sampling, and model assumptions in developing inference.

Leverage analysis for linear mixed models

We consider a generalized leverage matrix useful for the identification of influential units and observations in linear mixed models and show how a decomposition of this matrix may be employed to identify high leverage points for both the marginal fitted values and the random effect component of the conditional fitted values. We illustrate the different uses of the two components of the decomposition with a simulated example as well as with a real data set.

Likelihood ratio tests for variance components in linear mixed models

Although the asymptotic distributions of the likelihood ratio for testing hypotheses of null variance components in linear mixed models derived by Stram and Lee [1994. Variance components testing in longitudinal mixed effects model. Biometrics 50, 1171-1177] are valid, their proof is based on the work of Self and Liang [1987. Asymptotic properties of maximum likelihood estimators and likelihood tests under nonstandard conditions. J. Amer. Statist. Assoc. 82, 605-610] which requires identically distributed random variables, an assumption not always valid in longitudinal data problems. We use the less restrictive results of Vu and Zhou [1997. Generalization of likelihood ratio tests under nonstandard conditions. Ann. Statist. 25, 897-916] to prove that the proposed mixture of chi-squared distributions is the actual asymptotic distribution of such likelihood ratios used as test statistics for null variance components in models with one or two random effects. We also consider a limited simulation study to evaluate the appropriateness of the asymptotic distribution of such likelihood ratios in moderately sized samples. (C) 2008 Elsevier B.V. All rights reserved.

Local Influence Analysis for Skew-Normal Linear Mixed Models

In this article, we consider local influence analysis for the skew-normal linear mixed model (SN-LMM). As the observed data log-likelihood associated with the SN-LMM is intractable, Cook`s well-known approach cannot be applied to obtain measures of local influence. Instead, we develop local influence measures following the approach of Zhu and Lee (2001). This approach is based on the use of an EM-type algorithm and is measurement invariant under reparametrizations. Four specific perturbation schemes are discussed. Results obtained for a simulated data set and a real data set are reported, illustrating the usefulness of the proposed methodology.

A note on influence diagnostics in nonlinear mixed-effects elliptical models

This paper provides general matrix formulas for computing the score function, the (expected and observed) Fisher information and the A matrices (required for the assessment of local influence) for a quite general model which includes the one proposed by Russo et al. (2009). Additionally, we also present an expression for the generalized leverage on fixed and random effects. The matrix formulation has notational advantages, since despite the complexity of the postulated model, all general formulas are compact, clear and have nice forms. (C) 2010 Elsevier B.V. All rights reserved.

Detection of catechol using mixed Langmuir-Blodgett films of a phospholipid and phthalocyanines as voltammetric sensors

The combination of metallic phthalocyanines (MPcs) and biomolecules has been explored in the literature either as mimetic systems to investigate molecular interactions or as supporting layers to immobilize biomolecules. Here, Langmuir-Blodgett (LB) films containing the phospholipid dimyristoyl phosphatidic acid (DMPA) mixed either with iron phthalocyanine (FePc) or with lutetium bisphthalocyanine (LuPc(2)) were applied as ITO modified-electrodes in the detection of catechol using cyclic voltammetry. The mixed Langmuir films of FePc + DMPA and LuPc(2) + DMPA displayed surface-pressure isotherms with no evidence of molecular-level interactions. The Fourier Transform Infrared (FTIR) spectra of the multilayer LB films confirmed the lack of interaction between the components. The DMPA and the FePc molecules were found to be oriented perpendicularly to the substrate, while LuPc(2) molecules were randomly organized. The phospholipid matrix induced a remarkable electrocatalytic effect on the phthalocyanines; as a result the mixed LB films deposited on ITO could be used to detect catechol with detection limits of 4.30 x 10(-7) and 3.34 x 10(-7) M for FePc + DMPA and LuPc(2) + DMPA, respectively. Results from kinetics experiments revealed that ion diffusion dominated the response of the modified electrodes. The sensitivity was comparable to that of other non-enzymatic sensors, which is sufficient to detect catechol in the food industry. The higher stability of the electrochemical response of the LB films and the ability to control the molecular architecture are promising for further studies with incorporation of biomolecules.

Coupled reliability and boundary element model for probabilistic fatigue life assessment in mixed mode crack propagation

Fatigue and crack propagation are phenomena affected by high uncertainties, where deterministic methods fail to predict accurately the structural life. The present work aims at coupling reliability analysis with boundary element method. The latter has been recognized as an accurate and efficient numerical technique to deal with mixed mode propagation, which is very interesting for reliability analysis. The coupled procedure allows us to consider uncertainties during the crack growth process. In addition, it computes the probability of fatigue failure for complex structural geometry and loading. Two coupling procedures are considered: direct coupling of reliability and mechanical solvers and indirect coupling by the response surface method. Numerical applications show the performance of the proposed models in lifetime assessment under uncertainties, where the direct method has shown faster convergence than response surface method. (C) 2010 Elsevier Ltd. All rights reserved.

Log-modified Weibull regression models with censored data: Sensitivity and residual analysis

This paper proposes a regression model considering the modified Weibull distribution. This distribution can be used to model bathtub-shaped failure rate functions. Assuming censored data, we consider maximum likelihood and Jackknife estimators for the parameters of the model. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and we also present some ways to perform global influence. Besides, for different parameter settings, sample sizes and censoring percentages, various simulations are performed and the empirical distribution of the modified deviance residual is displayed and compared with the standard normal distribution. These studies suggest that the residual analysis usually performed in normal linear regression models can be straightforwardly extended for a martingale-type residual in log-modified Weibull regression models with censored data. Finally, we analyze a real data set under log-modified Weibull regression models. A diagnostic analysis and a model checking based on the modified deviance residual are performed to select appropriate models. (c) 2008 Elsevier B.V. All rights reserved.

On estimation and influence diagnostics for zero-inflated negative binomial regression models

The zero-inflated negative binomial model is used to account for overdispersion detected in data that are initially analyzed under the zero-Inflated Poisson model A frequentist analysis a jackknife estimator and a non-parametric bootstrap for parameter estimation of zero-inflated negative binomial regression models are considered In addition an EM-type algorithm is developed for performing maximum likelihood estimation Then the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and some ways to perform global influence analysis are derived In order to study departures from the error assumption as well as the presence of outliers residual analysis based on the standardized Pearson residuals is discussed The relevance of the approach is illustrated with a real data set where It is shown that zero-inflated negative binomial regression models seems to fit the data better than the Poisson counterpart (C) 2010 Elsevier B V All rights reserved

The constrained compartmentalized knapsack problem: mathematical models and solution methods

The constrained compartmentalized knapsack problem can be seen as an extension of the constrained knapsack problem. However, the items are grouped into different classes so that the overall knapsack has to be divided into compartments, and each compartment is loaded with items from the same class. Moreover, building a compartment incurs a fixed cost and a fixed loss of the capacity in the original knapsack, and the compartments are lower and upper bounded. The objective is to maximize the total value of the items loaded in the overall knapsack minus the cost of the compartments. This problem has been formulated as an integer non-linear program, and in this paper, we reformulate the non-linear model as an integer linear master problem with a large number of variables. Some heuristics based on the solution of the restricted master problem are investigated. A new and more compact integer linear model is also presented, which can be solved by a branch-and-bound commercial solver that found most of the optimal solutions for the constrained compartmentalized knapsack problem. On the other hand, heuristics provide good solutions with low computational effort. (C) 2011 Elsevier BM. All rights reserved.

Bayesian nonlinear regression models with scale mixtures of skew-normal distributions: Estimation and case influence diagnostics

The purpose of this paper is to develop a Bayesian analysis for nonlinear regression models under scale mixtures of skew-normal distributions. This novel class of models provides a useful generalization of the symmetrical nonlinear regression models since the error distributions cover both skewness and heavy-tailed distributions such as the skew-t, skew-slash and the skew-contaminated normal distributions. The main advantage of these class of distributions is that they have a nice hierarchical representation that allows the implementation of Markov chain Monte Carlo (MCMC) methods to simulate samples from the joint posterior distribution. In order to examine the robust aspects of this flexible class, against outlying and influential observations, we present a Bayesian case deletion influence diagnostics based on the Kullback-Leibler divergence. Further, some discussions on the model selection criteria are given. The newly developed procedures are illustrated considering two simulations study, and a real data previously analyzed under normal and skew-normal nonlinear regression models. (C) 2010 Elsevier B.V. All rights reserved.