Random effects model

About: Random effects model is a research topic. Over the lifetime, 8388 publications have been published within this topic receiving 438823 citations. The topic is also known as: random effects & random effect.

An introduction to statistical methods and data analysis

Abstract: PART I: INTRODUCTION 1. WHAT IS STATISTICS? Introduction / Why Study Statistics? / Some Current Applications of Statistics / What Do Statisticians Do? / Quality and Process Improvement / A Note to the Student / Summary / Supplementary Exercises PART II: COLLECTING THE DATA 2. USING SURVEYS AND SCIENTIFIC STUDIES TO COLLECT DATA Introduction / Surveys / Scientific Studies / Observational Studies / Data Management: Preparing Data for Summarization and Analysis / Summary PART III: SUMMARIZING DATA 3. DATA DESCRIPTION Introduction / Describing Data on a Single Variable: Graphical Methods / Describing Data on a Single Variable: Measures of Central Tendency / Describing Data on a Single Variable: Measures of Variability / The Box Plot / Summarizing Data from More Than One Variable / Calculators, Computers, and Software Systems / Summary / Key Formulas / Supplementary Exercises PART IV: TOOLS AND CONCEPTS 4. PROBABILITY AND PROBABILITY DISTRIBUTIONS How Probability Can Be Used in Making Inferences / Finding the Probability of an Event / Basic Event Relations and Probability Laws / Conditional Probability and Independence / Bayes's Formula / Variables: Discrete and Continuous / Probability Distributions for Discrete Random Variables / A Useful Discrete Random Variable: The Binomial / Probability Distributions for Continuous Random Variables / A Useful Continuous Random Variable: The Normal Distribution / Random Sampling / Sampling Distributions / Normal Approximation to the Binomial / Summary / Key Formulas / Supplementary Exercises PART V: ANALYZING DATA: CENTRAL VALUES, VARIANCES, AND PROPORTIONS 5. INFERENCES ON A POPULATION CENTRAL VALUE Introduction and Case Study / Estimation of / Choosing the Sample Size for Estimating / A Statistical Test for / Choosing the Sample Size for Testing / The Level of Significance of a Statistical Test / Inferences about for Normal Population, s Unknown / Inferences about the Population Median / Summary / Key Formulas / Supplementary Exercises 6. COMPARING TWO POPULATION CENTRAL VALUES Introduction and Case Study / Inferences about 1 - 2: Independent Samples / A Nonparametric Alternative: The Wilcoxon Rank Sum Test / Inferences about 1 - 2: Paired Data / A Nonparametric Alternative: The Wilcoxon Signed-Rank Test / Choosing Sample Sizes for Inferences about 1 - 2 / Summary / Key Formulas / Supplementary Exercises 7. INFERENCES ABOUT POPULATION VARIANCES Introduction and Case Study / Estimation and Tests for a Population Variance / Estimation and Tests for Comparing Two Population Variances / Tests for Comparing k > 2 Population Variances / Summary / Key Formulas / Supplementary Exercises 8. INFERENCES ABOUT POPULATION CENTRAL VALUES Introduction and Case Study / A Statistical Test About More Than Two Population Variances / Checking on the Assumptions / Alternative When Assumptions are Violated: Transformations / A Nonparametric Alternative: The Kruskal-Wallis Test / Summary / Key Formulas / Supplementary Exercises 9. MULTIPLE COMPARISONS Introduction and Case Study / Planned Comparisons Among Treatments: Linear Contrasts / Which Error Rate Is Controlled / Multiple Comparisons with the Best Treatment / Comparison of Treatments to a Control / Pairwise Comparison on All Treatments / Summary / Key Formulas / Supplementary Exercises 10. CATEGORICAL DATA Introduction and Case Study / Inferences about a Population Proportion p / Comparing Two Population Proportions p1 - p2 / Probability Distributions for Discrete Random Variables / The Multinomial Experiment and Chi-Square Goodness-of-Fit Test / The Chi-Square Test of Homogeneity of Proportions / The Chi-Square Test of Independence of Two Nominal Level Variables / Fisher's Exact Test, a Permutation Test / Measures of Association / Combining Sets of Contingency Tables / Summary / Key Formulas / Supplementary Exercises PART VI: ANALYZING DATA: REGRESSION METHODS, MODEL BUILDING 11. SIMPLE LINEAR REGRESSION AND CORRELATION Linear Regression and the Method of Least Squares / Transformations to Linearize Data / Correlation / A Look Ahead: Multiple Regression / Summary of Key Formulas. Supplementary Exercises. 12. INFERENCES RELATED TO LINEAR REGRESSION AND CORRELATION Introduction and Case Study / Diagnostics for Detecting Violations of Model Conditions / Inferences about the Intercept and Slope of the Regression Line / Inferences about the Population Mean for a Specified Value of the Explanatory Variable / Predictions and Prediction Intervals / Examining Lack of Fit in the Model / The Inverse Regression Problem (Calibration): Predicting Values for x for a Specified Value of y / Summary / Key Formulas / Supplementary Exercises 13. MULTIPLE REGRESSION AND THE GENERAL LINEAR MODEL Introduction and Case Study / The General Linear Model / Least Squares Estimates of Parameters in the General Linear Model / Inferences about the Parameters in the General Linear Model / Inferences about the Population Mean and Predictions from the General Linear Model / Comparing the Slope of Several Regression Lines / Logistic Regression / Matrix Formulation of the General Linear Model / Summary / Key Formulas / Supplementary Exercises 14. BUILDING REGRESSION MODELS WITH DIAGNOSTICS Introduction and Case Study / Selecting the Variables (Step 1) / Model Formulation (Step 2) / Checking Model Conditions (Step 3) / Summary / Key Formulas / Supplementary Exercises PART VII: ANALYZING DATA: DESIGN OF EXPERIMENTS AND ANOVA 15. DESIGN CONCEPTS FOR EXPERIMENTS AND STUDIES Experiments, Treatments, Experimental Units, Blocking, Randomization, and Measurement Units / How Many Replications? / Studies for Comparing Means versus Studies for Comparing Variances / Summary / Key Formulas / Supplementary Exercises 16. ANALYSIS OF VARIANCE FOR STANDARD DESIGNS Introduction and Case Study / Completely Randomized Design with Single Factor / Randomized Block Design / Latin Square Design / Factorial Experiments in a Completely Randomized Design / The Estimation of Treatment Differences and Planned Comparisons in the Treatment Means / Checking Model Conditions / Alternative Analyses: Transformation and Friedman's Rank-Based Test / Summary / Key Formulas / Supplementary Exercises 17. ANALYSIS OF COVARIANCE Introduction and Case Study / A Completely Randomized Design with One Covariate / The Extrapolation Problem / Multiple Covariates and More Complicated Designs / Summary / Key Formulas / Supplementary Exercises 18. ANALYSIS OF VARIANCE FOR SOME UNBALANCED DESIGNS Introduction and Case Study / A Randomized Block Design with One or More Missing Observations / A Latin Square Design with Missing Data / Incomplete Block Designs / Summary / Key Formulas / Supplementary Exercises 19. ANALYSIS OF VARIANCE FOR SOME FIXED EFFECTS, RANDOM EFFECTS, AND MIXED EFFECTS MODELS Introduction and Case Study / A One-Factor Experiment with Random Treatment Effects / Extensions of Random-Effects Models / A Mixed Model: Experiments with Both Fixed and Random Treatment Effects / Models with Nested Factors / Rules for Obtaining Expected Mean Squares / Summary / Key Formulas / Supplementary Exercises 20. SPLIT-PLOT DESIGNS AND EXPERIMENTS WITH REPEATED MEASURES Introduction and Case Study / Split-Plot Designs / Single-Factor Experiments with Repeated Measures / Two-Factor Experiments with Repeated Measures on One of the Factors / Crossover Design / Summary / Key Formulas / Supplementary Exercises PART VIII: COMMUNICATING AND DOCUMENTING THE RESULTS OF A STUDY OR EXPERIMENT 21. COMMUNICATING AND DOCUMENTING THE RESULTS OF A STUDY OR EXPERIMENT Introduction / The Difficulty of Good Communication / Communication Hurdles: Graphical Distortions / Communication Hurdles: Biased Samples / Communication Hurdles: Sample Size / The Statistical Report / Documentation and Storage of Results / Summary / Supplementary Exercises

Approximate inference in generalized linear mixed models

Abstract: Statistical approaches to overdispersion, correlated errors, shrinkage estimation, and smoothing of regression relationships may be encompassed within the framework of the generalized linear mixed model (GLMM). Given an unobserved vector of random effects, observations are assumed to be conditionally independent with means that depend on the linear predictor through a specified link function and conditional variances that are specified by a variance function, known prior weights and a scale factor. The random effects are assumed to be normally distributed with mean zero and dispersion matrix depending on unknown variance components. For problems involving time series, spatial aggregation and smoothing, the dispersion may be specified in terms of a rank deficient inverse covariance matrix. Approximation of the marginal quasi-likelihood using Laplace's method leads eventually to estimating equations based on penalized quasilikelihood or PQL for the mean parameters and pseudo-likelihood for the variances. Im...

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

Multilevel and Longitudinal Modeling Using Stata

Abstract: Preface LINEAR VARIANCE-COMPONENTS MODELS Introduction How reliable are expiratory flow measurements? The variance-components model Modeling the Mini Wright measurements Estimation methods Assigning values to the random intercepts Summary and further reading Exercises LINEAR RANDOM-INTERCEPT MODELS Introduction Are tax preparers useful? The longitudinal data structure Panel data and correlated residuals The random-intercept model Different kinds of effects in panel models Endogeneity and between-taxpayer effects Residual diagnostics Summary and further reading Exercises LINEAR RANDOM-COEFFICIENT AND GROWTH-CURVE MODELS Introduction How effective are different schools? Separate linear regressions for each school The random-coefficient model How do children grow? Growth-curve modeling Two-stage model formulation Prediction of trajectories for individual children Complex level-1 variation or heteroskedasticity Summary and further reading Exercises DICHOTOMOUS OR BINARY RESPONSES Models for dichotomous responses Which treatment is best for toenail infection? The longitudinal data structure Population-averaged or marginal probabilities Random-intercept logistic regression Subject-specific vs. population-averaged relationships Maximum likelihood estimation using adaptive quadrature Empirical Bayes (EB) predictions Other approaches to clustered dichotomous data Summary and further reading Exercises ORDINAL RESPONSES Introduction Cumulative models for ordinal responses Are antipsychotic drugs effective for patients with schizophrenia? Longitudinal data structure and graphs A proportional-odds model A random-intercept proportional-odds model A random-coefficient proportional-odds model Marginal and patient-specific probabilities Do experts differ in their grading of student essays? A random-intercept model with grader bias Including grader-specific measurement error variances Including grader-specific thresholds Summary and further reading Exercises COUNTS Introduction Types of counts Poisson model for counts Did the German health-care reform reduce the number of doctor visits? Longitudinal data structure Poisson regression ignoring overdispersion and clustering Poisson regression with overdispersion but ignoring clustering Random-intercept Poisson regression Random-coefficient Poisson regression Other approaches to clustered counts Which Scottish countries have a high risk of lip cancer? Standardized mortality ratios Random-intercept Poisson regression Nonparametric maximum likelihood estimation Summary and further reading Exercises HIGHER LEVEL MODELS AND NESTED RANDOM EFFECTS Introduction Which method is best for measuring expiratory flow? Two-level variance-components models Three-level variance-components models Did the Guatemalan immunization campaign work? A three-level logistic random-intercept model Summary and further reading Exercises CROSSED RANDOM EFFECTS Introduction How does investment depend on expected profit and capital stock? A two-way error-components model How much do primary and secondary schools affect attainment at age 16? An additive crossed random-effects model Including a random interaction A trick requiring fewer random effects Summary and further reading Exercises APPENDIX A: Syntax for gllamm, eq, and gllapred APPENDIX B: Syntax for gllamm APPENDIX C: Syntax for gllapred APPENDIX D: Syntax for gllasim References Author Index Subject Index

A basic introduction to fixed‐effect and random‐effects models for meta‐analysis

Abstract: There are two popular statistical models for meta-analysis, the fixed-effect model and the random-effects model. The fact that these two models employ similar sets of formulas to compute statistics, and sometimes yield similar estimates for the various parameters, may lead people to believe that the models are interchangeable. In fact, though, the models represent fundamentally different assumptions about the data. The selection of the appropriate model is important to ensure that the various statistics are estimated correctly. Additionally, and more fundamentally, the model serves to place the analysis in context. It provides a framework for the goals of the analysis as well as for the interpretation of the statistics. In this paper we explain the key assumptions of each model, and then outline the differences between the models. We conclude with a discussion of factors to consider when choosing between the two models. Copyright © 2010 John Wiley & Sons, Ltd.