Showing papers on "Random effects model published in 2011"

Multilevel analysis : an introduction to basic and advanced multilevel modeling

Abstract: Preface second edition Preface to first edition Introduction Multilevel analysis Probability models This book Prerequisites Notation Multilevel Theories, Multi-Stage Sampling and Multilevel Models Dependence as a nuisance Dependence as an interesting phenomenon Macro-level, micro-level, and cross-level relations Glommary Statistical Treatment of Clustered Data Aggregation Disaggregation The intraclass correlation Within-group and between group variance Testing for group differences Design effects in two-stage samples Reliability of aggregated variables Within-and between group relations Regressions Correlations Estimation of within-and between-group correlations Combination of within-group evidence Glommary The Random Intercept Model Terminology and notation A regression model: fixed effects only Variable intercepts: fixed or random parameters? When to use random coefficient models Definition of the random intercept model More explanatory variables Within-and between-group regressions Parameter estimation 'Estimating' random group effects: posterior means Posterior confidence intervals Three-level random intercept models Glommary The Hierarchical Linear Model Random slopes Heteroscedasticity Do not force ?01 to be 0! Interpretation of random slope variances Explanation of random intercepts and slopes Cross-level interaction effects A general formulation of fixed and random parts Specification of random slope models Centering variables with random slopes? Estimation Three or more levels Glommary Testing and Model Specification Tests for fixed parameters Multiparameter tests for fixed effects Deviance tests More powerful tests for variance parameters Other tests for parameters in the random part Confidence intervals for parameters in the random part Model specification Working upward from level one Joint consideration of level-one and level-two variables Concluding remarks on model specification Glommary How Much Does the Model Explain? Explained variance Negative values of R2? Definition of the proportion of explained variance in two-level models Explained variance in three-level models Explained variance in models with random slopes Components of variance Random intercept models Random slope models Glommary Heteroscedasticity Heteroscedasticity at level one Linear variance functions Quadratic variance functions Heteroscedasticity at level two Glommary Missing Data General issues for missing data Implications for design Missing values of the dependent variable Full maximum likelihood Imputation The imputation method Putting together the multiple results Multiple imputations by chained equations Choice of the imputation model Glommary Assumptions of the Hierarchical Linear Model Assumptions of the hierarchical linear model Following the logic of the hierarchical linear model Include contextual effects Check whether variables have random effects Explained variance Specification of the fixed part Specification of the random part Testing for heteroscedasticity What to do in case of heteroscedasticity Inspection of level-one residuals Residuals at level two Influence of level-two units More general distributional assumptions Glommary Designing Multilevel Studies Some introductory notes on power Estimating a population mean Measurement of subjects Estimating association between variables Cross-level interaction effects Allocating treatment to groups or individuals Exploring the variance structure The intraclass correlation Variance parameters Glommary Other Methods and Models Bayesian inference Sandwich estimators for standard errors Latent class models Glommary Imperfect Hierarchies A two-level model with a crossed random factor Crossed random effects in three-level models Multiple membership models Multiple membership multiple classification models Glommary Survey Weights Model-based and design-based inference Descriptive and analytic use of surveys Two kinds of weights Choosing between model-based and design-based analysis Inclusion probabilities and two-level weights Exploring the informativeness of the sampling design Example: Metacognitive strategies as measured in the PISA study Sampling design Model-based analysis of data divided into parts Inclusion of weights in the model How to assign weights in multilevel models Appendix Matrix expressions for the single-level estimators Glommary Longitudinal Data Fixed occasions The compound symmetry models Random slopes The fully multivariate model Multivariate regression analysis Explained variance Variable occasion designs Populations of curves Random functions Explaining the functions 2741524 Changing covariates Autocorrelated residuals Glommary Multivariate Multilevel Models Why analyze multiple dependent variables simultaneously? The multivariate random intercept model Multivariate random slope models Glommary Discrete Dependent Variables Hierarchical generalized linear models Introduction to multilevel logistic regression Heterogeneous proportions The logit function: Log-odds The empty model The random intercept model Estimation Aggregation Further topics on multilevel logistic regression Random slope model Representation as a threshold model Residual intraclass correlation coefficient Explained variance Consequences of adding effects to the model Ordered categorical variables Multilevel event history analysis Multilevel Poisson regression Glommary Software Special software for multilevel modeling HLM MLwiN The MIXOR suite and SuperMix Modules in general-purpose software packages SAS procedures VARCOMP, MIXED, GLIMMIX, and NLMIXED R Stata SPSS, commands VARCOMP and MIXED Other multilevel software PinT Optimal Design MLPowSim Mplus Latent Gold REALCOM WinBUGS References Index

Robust Inference with Multi-way Clustering

Abstract: In this article we propose a variance estimator for the OLS estimator as well as for nonlinear estimators such as logit, probit, and GMM. This variance estimator enables cluster-robust inference when there is two-way or multiway clustering that is nonnested. The variance estimator extends the standard cluster-robust variance estimator or sandwich estimator for one-way clustering (e.g., Liang and Zeger 1986; Arellano 1987) and relies on similar relatively weak distributional assumptions. Our method is easily implemented in statistical packages, such as Stata and SAS, that already offer cluster-robust standard errors when there is one-way clustering. The method is demonstrated by a Monte Carlo analysis for a two-way random effects model; a Monte Carlo analysis of a placebo law that extends the state–year effects example of Bertrand, Duflo, and Mullainathan (2004) to two dimensions; and by application to studies in the empirical literature where two-way clustering is present.

The arcsine is asinine: the analysis of proportions in ecology

Abstract: The arcsine square root transformation has long been standard procedure when analyzing proportional data in ecology, with applications in data sets containing binomial and non-binomial response variables. Here, we argue that the arcsine transform should not be used in either circumstance. For binomial data, logistic regression has greater interpretability and higher power than analyses of transformed data. However, it is important to check the data for additional unexplained variation, i.e., overdispersion, and to account for it via the inclusion of random effects in the model if found. For non-binomial data, the arcsine transform is undesirable on the grounds of interpretability, and because it can produce nonsensical predictions. The logit transformation is proposed as an alternative approach to address these issues. Examples are presented in both cases to illustrate these advantages, comparing various methods of analyzing proportions including untransformed, arcsine- and logit-transformed linear models and logistic regression (with or without random effects). Simulations demonstrate that logistic regression usually provides a gain in power over other methods.

Interpretation of random effects meta-analyses.

Abstract: Summary estimates of treatment effect from random effects meta-analysis give only the average effect across all studies. Inclusion of prediction intervals, which estimate the likely effect in an individual setting, could make it easier to apply the results to clinical practice

Prediction-Corrected Visual Predictive Checks for Diagnosing Nonlinear Mixed-Effects Models

Abstract: Informative diagnostic tools are vital to the development of useful mixed-effects models. The Visual Predictive Check (VPC) is a popular tool for evaluating the performance of population PK and PKPD models. Ideally, a VPC will diagnose both the fixed and random effects in a mixed-effects model. In many cases, this can be done by comparing different percentiles of the observed data to percentiles of simulated data, generally grouped together within bins of an independent variable. However, the diagnostic value of a VPC can be hampered by binning across a large variability in dose and/or influential covariates. VPCs can also be misleading if applied to data following adaptive designs such as dose adjustments. The prediction-corrected VPC (pcVPC) offers a solution to these problems while retaining the visual interpretation of the traditional VPC. In a pcVPC, the variability coming from binning across independent variables is removed by normalizing the observed and simulated dependent variable based on the typical population prediction for the median independent variable in the bin. The principal benefit with the pcVPC has been explored by application to both simulated and real examples of PK and PKPD models. The investigated examples demonstrate that pcVPCs have an enhanced ability to diagnose model misspecification especially with respect to random effects models in a range of situations. The pcVPC was in contrast to traditional VPCs shown to be readily applicable to data from studies with a priori and/or a posteriori dose adaptations.

What Forty Years of Research Says About the Impact of Technology on Learning: A Second-Order Meta-Analysis and Validation Study

Abstract: This research study employs a second-order meta-analysis procedure to summarize 40 years of research activity addressing the question, does computer technology use affect student achievement in formal face-to-face classrooms as compared to classrooms that do not use technology? A study-level meta-analytic validation was also conducted for purposes of comparison. An extensive literature search and a systematic review process resulted in the inclusion of 25 meta-analyses with minimal overlap in primary literature, encompassing 1,055 primary studies. The random effects mean effect size of 0.35 was significantly different from zero. The distribution was heterogeneous under the fixed effects model. To validate the second-order meta-analysis, 574 individual independent effect sizes were extracted from 13 out of the 25 meta-analyses. The mean effect size was 0.33 under the random effects model, and the distribution was heterogeneous. Insights about the state of the field, implications for technology use, and pro...

NICE DSU Technical Support Document 2: A Generalised Linear Modelling Framework for Pairwise and Network Meta-Analysis of Randomised Controlled Trials

Abstract: This paper sets out a generalised linear model (GLM) framework for the synthesis of data from randomised controlled trials (RCTs). We describe a common model taking the form of a linear regression for both fixed and random effects synthesis, that can be implemented with Normal, Binomial, Poisson, and Multinomial data. The familiar logistic model for meta- analysis with Binomial data is a GLM with a logit link function, which is appropriate for probability outcomes. The same linear regression framework can be applied to continuous outcomes, rate models, competing risks, or ordered category outcomes, by using other link functions, such as identity, log, complementary log-log, and probit link functions. The common core model for the linear predictor can be applied to pair-wise meta-analysis, indirect comparisons, synthesis of multi-arm trials, and mixed treatment comparisons, also known as network meta-analysis, without distinction.We take a Bayesian approach to estimation and provide WinBUGS program code for a Bayesian analysis using Markov chain Monte Carlo (MCMC) simulation. An advantage of this approach is that it is straightforward to extend to shared parameter models where different RCTs report outcomes in different formats but from a common underlying model. Use of the GLM framework allows us to present a unified account of how models can be compared using the Deviance Information Criterion (DIC), and how goodness of fit can be assessed using the residual deviance. WinBUGS code for model critique is provided. Our approach is illustrated through a range of worked examples for the commonly encountered evidence formats, including shared parameter models.We give suggestions on computational issues that sometimes arise in MCMC evidence synthesis, and comment briefly on alternative software.

New Directions in Spatial Econometrics

Abstract: 1 New Directions in Spatial Econometrics: Introduction.- 1.1 Introduction.- 1.2 Spatial Effects in Regression Models.- 1.2.1 Specification of Spatial Dependence.- 1.2.2 Spatial Data and Model Transformations.- 1.3 Spatial Effects in Limited Dependent Variable Models.- 1.4 Heterogeneity and Dependence in Space-Time Models.- 1.5 Future Directions.- References.- I-A: Spatial Effects in Linear Regression Models Specification of Spatial Dependence.- 2 Small Sample Properties of Tests for Spatial Dependence in Regression Models: Some Further Results.- 2.1 Introduction.- 2.2 Tests for Spatial Dependence.- 2.2.1 Null and Alternative Hypotheses.- 2.2.2 Tests for Spatial Error Dependence.- 2.2.3 Tests for Spatial Lag Dependence.- 2.3 Experimental Design.- 2.4 Results of Monte Carlo Experiments.- 2.4.1 Empirical Size of the Tests.- 2.4.2 Power of Tests Against First Order Spatial Error Dependence.- 2.4.3 Power of Tests Against Spatial Autoregressive Lag Dependence.- 2.4.4 Power of Tests Against Second Order Spatial Error Dependence.- 2.4.5 Power of Tests Against a SARMA (1,1) Process.- 2.5 Conclusions.- Acknowledgements.- References.- Appendix 1: Tables.- 3 Spatial Correlation: A Suggested Alternative to the Autoregressive Model.- 3.1 Introduction.- 3.2 The Spatial AR Model of Autocorrelation.- 3.3 The Singularity of (I - pM).- 3.3.1 Theoretical Issues.- 3.3.2 Independent Corroborative Evidence.- 3.4 The Parameter Space.- 3.5 A Suggested Variation of the Spatial AR Model.- 3.5.1 The Suggested Model.- 3.5.2 Some Limiting Correlations.- 3.5.3 A Generalization.- 3.6 Suggestions for Further Work.- Acknowledgements.- References.- Appendix 1: Spatial Weighting Matrices.- 4 Spatial Autoregressive Error Components in Travel Flow Models: An Application to Aggregate Mode Choice.- 4.1 Introduction.- 4.2 The First-Order Spatially Autoregressive Error Components Formulation.- 4.3 Estimation Issues.- 4.4 Empirical Example.- 4.4.1 An Illustration Based on Synthetic Data.- 4.5 Conclusions.- References.- I-B: Spatial Effects in Linear Regression Models Spatial Data and Model Transformations.- 5 The Impacts of Misspecified Spatial Interaction in Linear Regression Models.- 5.1 Introduction.- 5.2 Aggregation and the Identification of Spatial Interaction.- 5.3 Experimental Design.- 5.3.1 Sample Size.- 5.3.2 Spatial Interaction Structures.- 5.3.3 Spatial Models and Parameter Space.- 5.3.4 Test Statistics and Estimators.- 5.3.5 Forms of Misspecification.- 5.4 Empirical Results.- 5.4.1 Size of Tests Under the Null.- 5.4.2 Power of Tests.- 5.4.3 Misspecification Effects on the Power of Tests for Spatial Dependence.- 5.4.4 Sensitivity of Parameter Estimation to Specification of Weight Matrix.- 5.4.5 Impact of Misspecification of Weight Matrix on Estimation.- 5.5 General Inferences References.- 6 Computation of Box-Cox Transform Parameters: A New Method and its Application to Spatial Econometrics.- 6.1 Introduction.- 6.2 The Elasticity Method: Further Elaboration.- 6.2.1 Linearization Bias.- 6.2.2 Discretization Bias.- 6.2.3 Specification Bias.- 6.3 The One Exogenous Variable Test.- 6.4 An Application to Spatial Econometrics.- 6.5 The Multiple Exogenous Variable Computation.- 6.6 Conclusions.- References.- 7 Data Problems in Spatial Econometric Modeling.- 7.1 Introduction.- 7.2 Data for Spatial Econometric Analysis.- 7.3 Data Problems in Spatial Econometrics.- 7.4 Methodologies for Handling Data Problems.- 7.4.1 Influential Cases in the Standard Regression Model.- 7.4.2 Influential Cases in a Spatial Regression Model.- 7.4.3 An Example.- 7.5 Implementing Methodologies.- References.- 8 Spatial Filtering in a Regression Framework: Examples Using Data on Urban Crime, Regional Inequality, and Government Expenditures.- 8.1 Introduction.- 8.2 Rationale for a Spatial Filter.- 8.3 The Gi Statistic.- 8.4 The Filtering Procedure.- 8.5 Filtering Variables: Three Examples.- 8.5.1 Example 1: Urban Crime.- 8.5.2 Example 2: Regional Inequality.- 8.5.3 Example 3: Government Expenditures.- >8.6 Conclusions.- >Acknowledgments.- References.- II: Spatial Effects in Limited Dependent Variable Models.- 9 Spatial Effects in Probit Models: A Monte Carlo Investigation.- 9.1 Introduction.- 9.2 Sources of Heteroscedasticity.- 9.3 Heteroscedastic Probit.- 9.4 Monte Carlo Design.- 9.5 Tests.- 9.6 Monte Carlo Results.- 9.7 Conclusions.- References.- Appendix 1: Monte Carlo Results.- Appendix 2: Heteroscedastic Probit Computer Programs.- Appendix 3: Monte Carlo Computer Programs.- 10 Estimating Logit Models with Spatial Dependence.- 10.1 Introduction.- 10.1.1 Model.- 10.2 Simulation Example.- 10.3 Conclusions.- >References.- Appendix 1: Gauss Program for Finding ML Estimates.- Appendix 2: Gauss Program to Estimate Asymptotic Variances of ML Estimates.- 11 Utility Variability within Aggregate Spatial Units and its Relevance to Discrete Models of Destination Choice.- 11.1 Introduction.- 11.2 Theoretical Background.- 11.3 Estimation of the Maximum Utility Model.- 11.4 Model Specifications and Simulations.- 11.4.1 Specification Issues.- 11.4.2 Description of Simulation Method.- 11.4.3 Results.- 11.5 Conclusions.- Acknowledgement.- References.- III: Heterogeneity and Dependence in Space-Time Models.- 12 The General Linear Model and Spatial Autoregressive Models.- 12.1 Introduction.- 12.2 The GLM.- 12.3 Data Preprocessing.- 12.3.1 Analysis of the 1964 Benchmark Data.- 12.3.2 Evaluation of Missing USDA Values Estimation.- >12.4 Implementation of the Spatial Statistical GLM.- 12.4.1 Preliminary Spatial Analysis of Milk Yields: AR Trend Surface GLMs.- 12.4.2 AR GLM Models for the Repeated Measures Case.- 12.4.3 A Spatially Adjusted Canonical Correlation Analysis of the Milk Production Data.- 12.5 Conclusions.- >References.- >Appendix 1: SAS Computer Code to Compute the Popular Spatial Autocorrelation Indices.- Appendix 2: SAS Code for Estimating Missing Values in the 1969 Data Set.- Appendix 3: SAS Code for 1969 USDA Data Analysis.- 13 Econometric Models and Spatial Parametric Instability: Relevant Concepts and an Instability Index.- 13.1 Introduction.- 13.2 The Expansion Method.- 13.3 Parametric Instability.- 13.3.1 Example.- 13.4 Conclusions.- 13.4.1 Instability Measures: Scope.- 13.4.2 Instability Measures: Significance.- References.- 14 Bayesian Hierarchical Forecasts for Dynamic Systems: Case Study on Backcasting School District Income Tax Revenues.- 14.1 Introduction.- 14.2 Literature Review.- 14.3 The C-MSKF Model: Time Series Prediction with Spatial Adjustments.- 14.3.1 Multi-State Kaiman Filter.- 14.3.2 Spatial Adjustment via Hierarchical Random Effects Model.- 14.3.3 CIHM Method.- 14.3.4 C-MSKF.- 14.4 Case Study and Observational Setting.- 14.4.1 Data.- 14.4.2 Treatments.- 14.5 Results.- >14.6 Conclusions.- >References.- Appendix 1: Poolbayes Program.- 15 A Multiprocess Mixture Model to Estimate Space-Time Dimensions of Weekly Pricing of Certificates of Deposit.- 15.1 Introduction.- 15.2 A Dynamic Targeting Model of CD Rate-Setting Behavior.- 15.2.1 The Model.- 15.2.2 The Decision Rule.- 15.3 The Spatial Econometric Model.- 15.3.1 Spatial Time-Varying Parameters.- 15.3.2 Parameter Estimation.- 15.3.3 Testing Hypotheses with the Model.- 15.4 Implementing the Model.- 15.4.1 The Data.- 15.4.2 Prior Information.- 15.4.3 Empirical Results.- 15.5 Conclusions.- Acknowledgements.- References.- Appendix 1: FORTRAN Program for the Spatial Mixture.- Author Index.- Contributors.

A Random Effects Branch-Site Model for Detecting Episodic Diversifying Selection

Abstract: Adaptive evolution frequently occurs in episodic bursts, localized to a few sites in a gene, and to a small number of lineages in a phylogenetic tree. A popular class of “branch-site” evolutionary models provides a statistical framework to search for evidence of such episodic selection. For computational tractability, current branch-site models unrealistically assume that all branches in the tree can be partitioned a priori into two rigid classes—“foreground” branches that are allowed to undergo diversifying selective bursts and “background” branches that are negatively selected or neutral. We demonstrate that this assumption leads to unacceptably high rates of false positives or false negatives when the evolutionary process along background branches strongly deviates from modeling assumptions. To address this problem, we extend Felsenstein's pruning algorithm to allow efficient likelihood computations for models in which variation over branches (and not just sites) is described in the random effects likelihood framework. This enables us to model the process at every branch-site combination as a mixture of three Markov substitution models—our model treats the selective class of every branch at a particular site as an unobserved state that is chosen independently of that at any other branch. When benchmarked on a previously published set of simulated sequences, our method consistently matched or outperformed existing branch-site tests in terms of power and error rates. Using three empirical data sets, previously analyzed for episodic selection, we discuss how modeling assumptions can influence inference in practical situations.

Quantifying, displaying and accounting for heterogeneity in the meta-analysis of RCTs using standard and generalised Q statistics.

Abstract: Clinical researchers have often preferred to use a fixed effects model for the primary interpretation of a meta-analysis. Heterogeneity is usually assessed via the well known Q and I 2 statistics, along with the random effects estimate they imply. In recent years, alternative methods for quantifying heterogeneity have been proposed, that are based on a 'generalised' Q statistic. We review 18 IPD meta-analyses of RCTs into treatments for cancer, in order to quantify the amount of heterogeneity present and also to discuss practical methods for explaining heterogeneity. Differing results were obtained when the standard Q and I 2 statistics were used to test for the presence of heterogeneity. The two meta-analyses with the largest amount of heterogeneity were investigated further, and on inspection the straightforward application of a random effects model was not deemed appropriate. Compared to the standard Q statistic, the generalised Q statistic provided a more accurate platform for estimating the amount of heterogeneity in the 18 meta-analyses. Explaining heterogeneity via the pre-specification of trial subgroups, graphical diagnostic tools and sensitivity analyses produced a more desirable outcome than an automatic application of the random effects model. Generalised Q statistic methods for quantifying and adjusting for heterogeneity should be incorporated as standard into statistical software. Software is provided to help achieve this aim.

Fixed effects and variance components estimation in three-level meta-analysis.

Abstract: Meta-analytic methods have been widely applied to education, medicine, and the social sciences. Much of meta-analytic data are hierarchically structured because effect size estimates are nested within studies, and in turn, studies can be nested within level-3 units such as laboratories or investigators, and so forth. Thus, multilevel models are a natural framework for analyzing meta-analytic data. This paper discusses the application of a Fisher scoring method in two-level and three-level meta-analysis that takes into account random variation at the second and third levels. The usefulness of the model is demonstrated using data that provide information about school calendar types. sas proc mixed and hlm can be used to compute the estimates of fixed effects and variance components. Copyright © 2011 John Wiley & Sons, Ltd.

On the Analysis of Combined Experiments

Abstract: The replication of experiments over multiple environments such as locations and years is a common practice in field research. A major reason for the practice is to estimate the effects of treatments over a variety of environments. Environments are frequently classed as random effects in the model for statistical analysis, while treatments are almost always classed as fixed effects. Where environments are random and treatments are fixed, it is not always necessary to include all possible interactions between treatments and environments as random effects in the model. The rationale for decisions about the inclusion or exclusion of fixed by random effects in a mixed model is presented. Where the effects of treatments over broad populations of environments are to be estimated, it is often most appropriate to include only those fixed by random effects that reference experimental units.

Multivariate Random-effects Meta-regression: Updates to Mvmeta:

Abstract: An extension of mvmeta, my program for multivariate random-effects meta-analysis, is described The extension handles meta-regression Estima- tion methods available are restricted maximum likelihood, maximum likelihood, method of moments, and fixed effects The program also allows a wider range of models (Riley’s overall correlation model and structured between-studies covari- ance); better estimation (using Mata for speed and correctly allowing for missing data); and new postestimation facilities (I-squared, standard errors and confidence intervals for between-studies standard deviations and correlations, and identifi- cation of the best intervention) The program is illustrated using a multiple- treatments meta-analysis Copyright 2011 by StataCorp LP

Measuring individual differences in reaction norms in field and experimental studies: a power analysis of random regression models

Abstract: Summary 1. Interest in measuring individual variation in reaction norms using mixed-effects and, more specifically, random regression models have grown apace in the last few years within evolution and ecology. However, these are data hungry methods, and little effort to date has been put into understanding how much and what kind of data we need to collect in order to apply these models usefully and reliably. 2. We conducted simulations to address three central questions. First, what is the best sampling strategy to collect sufficient data to test for individual variation using random regression models? Second, on occasions when precision is difficult to assess, can we be confident that a failure to detect significant variance in plasticity using random regression represents a biological reality rather than a lack of statistical power? Finally, does the common practice of censoring individuals with one or few repeated measures improve or reduce power to estimate individual variation in random regressions? 3. We have also developed a series of easy-to-use functions in the ‘pamm’ statistical package for R, which is freely available, that will allow researchers to conduct similar power analyses tailored more specifically to their own data. 4. Our results reveal potentially useful rules of thumb: large data sets (N > 200) are needed to evaluate the variance of individual-specific slopes; a number of individuals ⁄ number of observations per individual ratio of approximately 0AE5 consistently yielded the highest power to detect random effects; individuals with one or few observations should not generally be censored as this reduces power to detect variance in plasticity. 5. We discuss the wider implications of these simulations and remaining challenges and suggest a new way to standardize results that would better facilitate the comparison of findings across empirical studies.

Bayesian Ideas and Data Analysis : An Introduction for Scientists and Statisticians

Abstract: Prologue Probability of a Defective: Binomial Data Brass Alloy Zinc Content: Normal Data Armadillo Hunting: Poisson Data Abortion in Dairy Cattle: Survival Data Ache Hunting with Age Trends Lung Cancer Treatment: Log-Normal Regression Survival with Random Effects: Ache Hunting Fundamental Ideas I Simple Probability Computations Science, Priors, and Prediction Statistical Models Posterior Analysis Commonly Used Distributions Integration versus Simulation Introduction WinBUGS I: Getting Started Method of Composition Monte Carlo Integration Posterior Computations in R Fundamental Ideas II Statistical Testing Exchangeability Likelihood Functions Sufficient Statistics Analysis Using Predictive Distributions Flat Priors Jeffreys' Priors Bayes Factors Other Model Selection Criteria Normal Approximations to Posteriors Bayesian Consistency and Inconsistency Hierarchical Models Some Final Comments on Likelihoods Identifiability and Noninformative Data Comparing Populations Inference for Proportions Inference for Normal Populations Inference for Rates Sample Size Determination Illustrations: Foundry Data Medfly Data Radiological Contrast Data Reyes Syndrome Data Corrosion Data Diasorin Data Ache Hunting Data Breast Cancer Data Simulations Generating Random Samples Traditional Monte Carlo Methods Basics of Markov Chain Theory Markov Chain Monte Carlo Basic Concepts of Regression Introduction Data Notation and Format Predictive Models: An Overview Modeling with Linear Structures Illustration: FEV Data Binomial Regression The Sampling Model Binomial Regression Analysis Model Checking Prior Distributions Mixed Models Illustrations: Space Shuttle Data Trauma Data Onychomycosis Fungis Data Cow Abortion Data Linear Regression The Sampling Model Reference Priors Conjugate Priors Independence Priors ANOVA Model Diagnostics Model Selection Nonlinear Regression Illustrations: FEV Data Bank Salary Data Diasorin Data Coleman Report Data Dugong Growth Data Correlated Data Introduction Mixed Models Multivariate Normal Models Multivariate Normal Regression Posterior Sampling and Missing Data Illustrations: Interleukin Data Sleeping Dog Data Meta-Analysis Data Dental Data Count Data Poisson Regression Over-Dispersion and Mixtures of Poissons Longitudinal Data Illustrations: Ache Hunting Data Textile Faults Data Coronary Heart Disease Data Foot and Mouth Disease Data Time to Event Data Introduction One-Sample Models Two-Sample Data Plotting Survival and Hazard Functions Illustrations: Leukemia Cancer Data Breast Cancer Data Time to Event Regression Accelerated Failure Time Models Proportional Hazards Modeling Survival with Random Effects Illustrations: Leukemia Cancer Data Larynx Cancer Data Cow Abortion Data Kidney Transplant Data Lung Cancer Data Ache Hunting Data Binary Diagnostic Tests Basic Ideas One Test, One Population Two Tests, Two Populations Prevalence Distributions Illustrations: Coronary Artery Disease Paratuberculosis Data Nucleospora Salmonis Data Ovine Progressive Pnemonia Data Nonparametric Models Flexible Density Shapes Flexible Regression Functions Proportional Hazards Modeling Illustrations: Galaxy Data ELISA Data for Johnes Disease Fungus Data Test Engine Data Lung Cancer Data Appendix A: Matrices and Vectors Appendix B: Probability Appendix C: Getting Started in R References

Hierarchical diffusion models for two-choice response times.

Abstract: Two-choice response times are a common type of data, and much research has been devoted to the development of process models for such data. However, the practical application of these models is notoriously complicated, and flexible methods are largely nonexistent. We combine a popular model for choice response times—the Wiener diffusion process—with techniques from psychometrics in order to construct a hierarchical diffusion model. Chief among these techniques is the application of random effects, with which we allow for unexplained variability among participants, items, or other experimental units. These techniques lead to a modeling framework that is highly flexible and easy to work with. Among the many novel models this statistical framework provides are a multilevel diffusion model, regression diffusion models, and a large family of explanatory diffusion models. We provide examples and the necessary computer code.

Regression for Categorical Data

Abstract: 1. Introduction 2. Binary regression: the logit model 3. Generalized linear models 4. Modeling of binary data 5. Alternative binary regression models 6. Regularization and variable selection for parametric models 7. Regression analysis of count data 8. Multinomial response models 9. Ordinal response models 10. Semi- and nonparametric generalized regression 11. Tree-based methods 12. The analysis of contingency tables: log-linear and graphical models 13. Multivariate response models 14. Random effects models 15. Prediction and classification Appendix A. Distributions Appendix B. Some basic tools Appendix C. Constrained estimation Appendix D. Kullback-Leibler distance and information-based criteria of model fit Appendix E. Numerical integration and tools for random effects modeling.

Sequential methods for random‐effects meta‐analysis

Abstract: Although meta-analyses are typically viewed as retrospective activities, they are increasingly being applied prospectively to provide up-to-date evidence on specific research questions. When meta-analyses are updated account should be taken of the possibility of false-positive findings due to repeated significance tests. We discuss the use of sequential methods for meta-analyses that incorporate random effects to allow for heterogeneity across studies. We propose a method that uses an approximate semi-Bayes procedure to update evidence on the among-study variance, starting with an informative prior distribution that might be based on findings from previous meta-analyses. We compare our methods with other approaches, including the traditional method of cumulative meta-analysis, in a simulation study and observe that it has Type I and Type II error rates close to the nominal level. We illustrate the method using an example in the treatment of bleeding peptic ulcers. Copyright © 2010 John Wiley & Sons, Ltd.

Global testing under sparse alternatives: ANOVA, multiple comparisons and the higher criticism

Abstract: Testing for the significance of a subset of regression coefficients in a linear model, a staple of statistical analysis, goes back at least to the work of Fisher who introduced the analysis of variance (ANOVA). We study this problem under the assumption that the coefficient vector is sparse, a common situation in modern high-dimensional settings. Suppose we have p covariates and that under the alternative, the response only depends upon the order of p^(1−α) of those, 0 ≤ α ≤ 1. Under moderate sparsity levels, that is, 0 ≤ α ≤ 1/2, we show that ANOVA is essentially optimal under some conditions on the design. This is no longer the case under strong sparsity constraints, that is, α > 1/2. In such settings, a multiple comparison procedure is often preferred and we establish its optimality when α ≥ 3/4. However, these two very popular methods are suboptimal, and sometimes powerless, under moderately strong sparsity where 1/2 1/2. This optimality property is true for a variety of designs, including the classical (balanced) multi-way designs and more modern “p > n” designs arising in genetics and signal processing. In addition to the standard fixed effects model, we establish similar results for a random effects model where the nonzero coefficients of the regression vector are normally distributed.

Misspecifying the Shape of a Random Effects Distribution: Why Getting It Wrong May Not Matter

Abstract: Statistical models that include random effects are commonly used to analyze longitudinal and correlated data, often with strong and parametric assumptions about the random effects distribution. There is marked disagreement in the literature as to whether such parametric assumptions are important or innocuous. In the context of generalized linear mixed models used to analyze clustered or longitudinal data, we examine the impact of random effects distribution misspecification on a variety of inferences, including prediction, inference about covariate effects, prediction of random effects and estimation of random effects variances. We describe examples, theoretical calculations and simulations to elucidate situations in which the specification is and is not important. A key conclusion is the large degree of robustness of maximum likelihood for a wide variety of commonly encountered situations.

How many studies do you need? A primer on statistical power for meta-analysis

Abstract: In this article, the authors outline methods for using fixed and random effects power analysis in the context of meta-analysis. Like statistical power analysis for primary studies, power analysis for meta-analysis can be done either prospectively or retrospectively and requires assumptions about parameters that are unknown. The authors provide some suggestions for thinking about these parameters, in particular for the random effects variance component. The authors also show how the typically uninformative retrospective power analysis can be made more informative. The authors then discuss the value of confidence intervals, show how they could be used in addition to or instead of retrospective power analysis, and also demonstrate that confidence intervals can convey information more effectively in some situations than power analyses alone. Finally, the authors take up the question “How many studies do you need to do a meta-analysis?” and show that, given the need for a conclusion, the answer is “two studies...

Fixed and Random Effects Selection in Mixed Effects Models

Abstract: Summary We consider selecting both fixed and random effects in a general class of mixed effects models using maximum penalized likelihood (MPL) estimation along with the smoothly clipped absolute deviation (SCAD) and adaptive least absolute shrinkage and selection operator (ALASSO) penalty functions. The MPL estimates are shown to possess consistency and sparsity properties and asymptotic normality. A model selection criterion, called the ICQ statistic, is proposed for selecting the penalty parameters (Ibrahim, Zhu, and Tang, 2008, Journal of the American Statistical Association 103, 1648–1658). The variable selection procedure based on ICQ is shown to consistently select important fixed and random effects. The methodology is very general and can be applied to numerous situations involving random effects, including generalized linear mixed models. Simulation studies and a real data set from a Yale infant growth study are used to illustrate the proposed methodology.

Joint modelling of longitudinal and time-to-event data with application to predicting abdominal aortic aneurysm growth and rupture

Abstract: Shared random effects joint models are becoming increasingly popular for investigating the relationship between longitudinal and time-to-event data. Although appealing, such complex models are computationally intensive, and quick, approximate methods may provide a reasonable alternative. In this paper, we first compare the shared random effects model with two approximate approaches: a naive proportional hazards model with time-dependent covariate and a two-stage joint model, which uses plug-in estimates of the fitted values from a longitudinal analysis as covariates in a survival model. We show that the approximate approaches should be avoided since they can severely underestimate any association between the current underlying longitudinal value and the event hazard. We present classical and Bayesian implementations of the shared random effects model and highlight the advantages of the latter for making predictions. We then apply the models described to a study of abdominal aortic aneurysms (AAA) to investigate the association between AAA diameter and the hazard of AAA rupture. Out-of-sample predictions of future AAA growth and hazard of rupture are derived from Bayesian posterior predictive distributions, which are easily calculated within an MCMC framework. Finally, using a multivariate survival sub-model we show that underlying diameter rather than the rate of growth is the most important predictor of AAA rupture.

The Hausman test in a Cliff and Ord panel model

Abstract: Summary This paper studies the random effects model and the fixed effects model for spatial panel data. The model includes a Cliff and Ord type spatial lag of the dependent variable as well as a spatially lagged one-way error component structure, accounting for both heterogeneity and spatial correlation across units. We discuss instrumental variable estimation under both the fixed and the random effects specifications and propose a spatial Hausman test which compares these two models accounting for spatial autocorrelation in the disturbances. We derive the large sample properties of our estimation procedures and show that the test statistic is asymptotically chi-square distributed. A small Monte Carlo study demonstrates that this test works well even in small panels.

Dealing with substantial heterogeneity in Cochrane reviews. Cross-sectional study

Abstract: Dealing with heterogeneity in meta-analyses is often tricky, and there is only limited advice for authors on what to do. We investigated how authors addressed different degrees of heterogeneity, in particular whether they used a fixed effect model, which assumes that all the included studies are estimating the same true effect, or a random effects model where this is not assumed. We sampled randomly 60 Cochrane reviews from 2008, which presented a result in its first meta-analysis with substantial heterogeneity (I2 greater than 50%, i.e. more than 50% of the variation is due to heterogeneity rather than chance). We extracted information on choice of statistical model, how the authors had handled the heterogeneity, and assessed the methodological quality of the reviews in relation to this. The distribution of heterogeneity was rather uniform in the whole I2 interval, 50-100%. A fixed effect model was used in 33 reviews (55%), but there was no correlation between I2 and choice of model (P = 0.79). We considered that 20 reviews (33%), 16 of which had used a fixed effect model, had major problems. The most common problems were: use of a fixed effect model and lack of rationale for choice of that model, lack of comment on even severe heterogeneity and of reservations and explanations of its likely causes. The problematic reviews had significantly fewer included trials than other reviews (4.3 vs. 8.0, P = 0.024). The problems became less pronounced with time, as those reviews that were most recently updated more often used a random effects model. One-third of Cochrane reviews with substantial heterogeneity had major problems in relation to their handling of heterogeneity. More attention is needed to this issue, as the problems we identified can be essential for the conclusions of the reviews.

Fitting Multilevel Models With Ordinal Outcomes: Performance of Alternative Specifications and Methods of Estimation

Abstract: Previous research has compared methods of estimation for multilevel models fit to binary data but there are reasons to believe that the results will not always generalize to the ordinal case. This paper thus evaluates (a) whether and when fitting multilevel linear models to ordinal outcome data is justified and (b) which estimator to employ when instead fitting multilevel cumulative logit models to ordinal data, Maximum Likelihood (ML) or Penalized Quasi-Likelihood (PQL). ML and PQL are compared across variations in sample size, magnitude of variance components, number of outcome categories, and distribution shape. Fitting a multilevel linear model to ordinal outcomes is shown to be inferior in virtually all circumstances. PQL performance improves markedly with the number of ordinal categories, regardless of distribution shape. In contrast to binary data, PQL often performs as well as ML when used with ordinal data. Further, the performance of PQL is typically superior to ML when the data includes a small to moderate number of clusters (i.e., ≤ 50 clusters).

Network meta-analysis of survival data with fractional polynomials

Abstract: Pairwise meta-analysis, indirect treatment comparisons and network meta-analysis for aggregate level survival data are often based on the reported hazard ratio, which relies on the proportional hazards assumption. This assumption is implausible when hazard functions intersect, and can have a huge impact on decisions based on comparisons of expected survival, such as cost-effectiveness analysis. As an alternative to network meta-analysis of survival data in which the treatment effect is represented by the constant hazard ratio, a multi-dimensional treatment effect approach is presented. With fractional polynomials the hazard functions of interventions compared in a randomized controlled trial are modeled, and the difference between the parameters of these fractional polynomials within a trial are synthesized (and indirectly compared) across studies. The proposed models are illustrated with an analysis of survival data in non-small-cell lung cancer. Fixed and random effects first and second order fractional polynomials were evaluated. (Network) meta-analysis of survival data with models where the treatment effect is represented with several parameters using fractional polynomials can be more closely fitted to the available data than meta-analysis based on the constant hazard ratio.