lme4: Mixed-effects modeling with R

Transcription

1 Douglas M. Bates lme4: Mixed-effects modeling with R February 17, 2010 Springer Page: 1 job: lmmwr macro: svmono.cls date/time: 17-Feb-2010/14:23

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4 Contents 1 A Simple, Linear, Mixed-effects Model Mixed-effects Models The Dyestuff and Dyestuff2 Data The Dyestuff Data The Dyestuff2 Data Fitting Linear Mixed Models A Model For the Dyestuff Data A Model For the Dyestuff2 Data Further Assessment of the Fitted Models The Linear Mixed-effects Probability Model Definitions and Results Matrices and Vectors in the Fitted Model Object Assessing the Variability of the Parameter Estimates Confidence Intervals on the Parameters Interpreting the Profile Zeta Plot Profile Pairs Plots Assessing the Random Effects Chapter Summary Exercises Models With Multiple Random-effects Terms A Model With Crossed Random Effects The Penicillin Data A Model For the Penicillin Data A Model With Nested Random Effects The Pastes Data Fitting a Model With Nested Random Effects Parameter Estimates for Model fm Testing H 0 : σ 2 = 0 Versus H a : σ 2 > Assessing the Reduced Model, fm3a A Model With Partially Crossed Random Effects ix

5 x Contents The InstEval Data Structure of L for model fm Chapter Summary Exercises Models Incorporating Covariates Models for the ergostool data Random-effects for both Subject and Type Using Fixed Effects for Type and Random Effects for Subject Covariates Affecting Mathematics Score Gain Rat Brain example Models for Longitudinal Data The sleepstudy Data Characteristics of the sleepstudy Data Plot Mixed-effects Models For the sleepstudy Data A Model With Correlated Random Effects A Model With Uncorrelated Random Effects Generating Z and Λ From Random-effects Terms Comparing Models fm9 and fm Assessing the Precision of the Parameter Estimates Examining the Random Effects and Predictions Chapter Summary Problems Computational Methods for Mixed Models Definitions and Basic Results The Conditional Distribution (U Y = y) Integrating h(u) in the Linear Mixed Model Determining the PLS Solutions, ũ and β θ The Fill-reducing Permutation, P The Value of the Deviance and Profiled Deviance Determining rθ 2 and β θ The REML Criterion Step-by-step Evaluation of the Profiled Deviance Generalizing to Other Forms of Mixed Models Descriptions of the Model Forms Determining the Conditional Mode, ũ Chapter Summary Exercises References Page: x job: lmmwr macro: svmono.cls date/time: 17-Feb-2010/14:23

6 List of Figures 1.1 Yield of dyestuff from 6 batches of an intermediate Simulated data similar in structure to the Dyestuff data Image of the Λ for model fm1ml Image of the random-effects model matrix, Z T, for fm Profile zeta plots of the parameters in model fm1ml Absolute value profile zeta plots of the parameters in model fm1ml Profile zeta plots comparing log(σ), σ and σ 2 in model fm1ml Profile zeta plots comparing log(σ 1 ), σ 1 and σ1 2 in model fm1ml Profile pairs plot for the parameters in model fm % prediction intervals on the random effects in fm1ml, shown as a dotplot % prediction intervals on the random effects in fm1ml versus quantiles of the standard normal distribution Travel time for an ultrasonic wave test on 6 rails Diameter of growth inhibition zone for 6 samples of penicillin Random effects prediction intervals for model fm Image of the random-effects model matrix for fm Images of Λ, Z T Z and L for model fm Profile zeta plot of the parameters in model fm Profile pairs plot of the parameters in model fm Profile pairs plot for model fm2 (log scale) Cross-tabulation image of the batch and sample factors Strength of paste preparations by batch and sample Images of Λ, Z T Z and L for model fm Random effects prediction intervals for model fm Profile zeta plots for the parameters in model fm Profile zeta plots for the parameters in model fm3a Profile pairs plot of the parameters in model fm3a Random effects prediction intervals for model fm xi

7 xii List of Figures 2.16 Image of the sparse Cholesky factor, L, from model fm Effort to arise by subject and stool type Profile zeta plot for the parameters in model fm Profile zeta plot for the parameters in model fm Prediction intervals on the random effects for stool type Profile plot of the parameters in model fm Activation of brain regions in rats Lattice plot of the sleepstudy data Images of Λ, Σ and L for model fm Images of Λ, Σ and L for model fm Images of Z T for models fm8 and fm Profile zeta plots for the parameters in model fm Profile pairs plot for the parameters in model fm Plot of the conditional modes of the random effects for model fm9 (left panel) and the corresponding subject-specific coefficients (right panel) Comparison of within-subject estimates and conditional modes for fm Comparison of predictions from separate fits and fm Prediction intervals on the random effects for model fm Page: xii job: lmmwr macro: svmono.cls date/time: 17-Feb-2010/14:23