Applied Mixed Models in Medicine

Helen Brown, Principal Statistician, NHS Scotland, Edinburgh, UK. Robin Prescott, Medical Statistics Unit, University of Edinburgh Medical School, UK.

• Preface to third edition xiiiMixed models notation xviiAbout the Companion Website xix1 Introduction 11.1 The use of mixed models 11.2 Introductory example 31.2.1 Simple model to assess the effects of treatment (Model A) 41.2.2 A model taking patient effects into account (Model B) 61.2.3 Random effects model (Model C) 71.2.4 Estimation (or prediction) of random effects 121.3 A multi-centre hypertension trial 121.3.1 Modelling the data 131.3.2 Including a baseline covariate (Model B) 151.3.3 Modelling centre effects (Model C) 161.3.4 Including centre-by-treatment interaction effects (Model D) 171.3.5 Modelling centre and centre⋅treatment effects as random (Model E) 181.4 Repeated measures data 191.4.1 Covariance pattern models 191.4.2 Random coefficients models 211.5 More about mixed models 221.5.1 What is a mixed model? 221.5.2 Why use mixed models? 231.5.3 Communicating results 241.5.4 Mixed models in medicine 251.5.5 Mixed models in perspective 251.6 Some useful definitions 271.6.1 Containment 281.6.2 Balance 291.6.3 Error strata 312 Normal mixed models 342.1 Model definition 342.1.1 The fixed effects model 352.1.2 The mixed model 372.1.3 The random effects model covariance structure 392.1.4 The random coefficients model covariance structure 422.1.5 The covariance pattern model covariance structure 442.2 Model fitting methods 462.2.1 The likelihood function and approaches to its maximisation 472.2.2 Estimation of fixed effects 502.2.3 Estimation (or prediction) of random effects and coefficients 512.2.4 Estimation of variance parameters 532.3 The Bayesian approach 572.3.1 Introduction 582.3.2 Determining the posterior density 592.3.3 Parameter estimation, probability intervals and p-values 602.3.4 Specifying non-informative prior distributions 622.3.5 Evaluating the posterior distribution 672.4 Practical application and interpretation 692.4.1 Negative variance components 692.4.2 Accuracy of variance parameters 732.4.3 Bias in fixed and random effects standard errors 742.4.4 Significance testing 752.4.5 Confidence intervals 782.4.6 Checking model assumptions 792.4.7 Missing data 812.4.8 Determining whether the simulated posterior distribution has converged 832.5 Example 842.5.1 Analysis models 842.5.2 Results 862.5.3 Discussion of points from Section 2.4 883 Generalised linear mixed models 1133.1 Generalised linear models 1143.1.1 Introduction 1143.1.2 Distributions 1153.1.3 The general form for exponential distributions 1173.1.4 The GLM definition 1183.1.5 Fitting the GLM 1213.1.6 Expressing individual distributions in the general exponential form 1233.1.7 Conditional logistic regression 1253.2 Generalised linear mixed models 1253.2.1 The GLMM definition 1263.2.2 The likelihood and quasi-likelihood functions 1273.2.3 Fitting the GLMM 1293.3 Practical application and interpretation 1333.3.1 Specifying binary data 1333.3.2 Uniform effects categories 1333.3.3 Negative variance components 1343.3.4 Presentation of fixed and random effects estimates 1353.3.5 Accuracy of variance parameters and potential bias 1363.3.6 Bias in fixed and random effects standard errors 1373.3.7 The dispersion parameter 1393.3.8 Significance testing 1403.3.9 Confidence intervals 1413.3.10 Model checking 1423.3.11 Determining whether the simulated posterior distribution has converged 1423.4 Example 1433.4.1 Introduction and models fitted 1433.4.2 Results 1443.4.3 Discussion of points from Section 3.3 1484 Mixed models for categorical data 1684.1 Ordinal logistic regression (fixed effects model) 1684.2 Mixed ordinal logistic regression 1734.2.1 Definition of the mixed ordinal logistic regression model 1734.2.2 Residual variance matrix 1744.2.3 Likelihood and quasi-likelihood functions 1764.2.4 Model fitting methods 1774.3 Mixed models for unordered categorical data 1774.3.1 The G matrix 1804.3.2 The R matrix 1804.3.3 Fitting the model 1804.4 Practical application and interpretation 1804.4.1 Expressing fixed and random effects results 1814.4.2 The proportional odds assumption 1814.4.3 Number of covariance parameters 1814.4.4 Choosing a covariance pattern 1824.4.5 Interpreting covariance parameters 1824.4.6 Checking model assumptions 1824.4.7 The dispersion parameter 1824.4.8 Other points 1834.5 Example 1835 Multi-centre trials and meta-analyses 1975.1 Introduction to multi-centre trials 1975.1.1 What is a multi-centre trial? 1975.1.2 Why use mixed models to analyse multi-centre data? 1985.2 The implications of using different analysis models 1985.2.1 Centre and centre⋅treatment effects fixed 1985.2.2 Centre effects fixed, centre⋅treatment effects omitted 1995.2.3 Centre and centre⋅treatment effects random 2005.2.4 Centre effects random, centre⋅treatment effects omitted 2015.3 Example: a multi-centre trial 2025.4 Practical application and interpretation 2095.4.1 Plausibility of a centre⋅treatment interaction 2095.4.2 Generalisation 2105.4.3 Number of centres 2105.4.4 Centre size 2105.4.5 Negative variance components 2115.4.6 Balance 2115.5 Sample size estimation 2115.5.1 Normal data 2125.5.2 Binary data 2155.5.3 Categorical data 2175.6 Meta-analysis 2175.7 Example: meta-analysis 2185.7.1 Analyses 2195.7.2 Results 2205.7.3 Treatment estimates in individual trials 2216 Repeated measures data 2316.1 Introduction 2316.1.1 Reasons for repeated measurements 2316.1.2 Analysis objectives 2326.1.3 Fixed effects approaches 2326.1.4 Mixed models approaches 2346.2 Covariance pattern models 2346.2.1 Covariance patterns 2356.2.2 Choice of covariance pattern 2396.2.3 Choice of fixed effects 2416.2.4 General points 2426.3 Example: covariance pattern models for normal data 2446.3.1 Analysis models 2456.3.2 Selection of covariance pattern 2456.3.3 Assessing fixed effects 2476.3.4 Model checking 2486.4 Example: covariance pattern models for count data 2546.4.1 Analysis models 2556.4.2 Analysis using a categorical mixed model 2586.5 Random coefficients models 2626.5.1 Introduction 2626.5.2 General points 2646.5.3 Comparisons with fixed effects approaches 2666.6 Examples of random coefficients models 2676.6.1 A linear random coefficients model 2676.6.2 A polynomial random coefficients model 2706.7 Sample size estimation 2866.7.1 Normal data 2866.7.2 Binary data 2886.7.3 Categorical data 2887 Cross-over trials 2897.1 Introduction 2897.2 Advantages of mixed models in cross-over trials 2907.3 The AB/BA cross-over trial 2907.3.1 Example: AB/BA cross-over design 2927.4 Higher order complete block designs 2977.4.1 Inclusion of carry-over effects 2977.4.2 Example: four-period, four-treatment cross-over trial 2987.5 Incomplete block designs 3027.5.1 Example: Three treatment two-period cross-over trial 3037.6 Optimal designs 3057.6.1 Example: Balaam’s design 3057.7 Covariance pattern models 3077.7.1 Structured by period 3087.7.2 Structured by treatment 3087.7.3 Example: four-way cross-over trial 3087.8 Analysis of binary data 3177.9 Analysis of categorical data 3217.10 Use of results from random effects models in trial design 3257.10.1 Example 3257.11 General points 3268 Other applications of mixed models 3298.1 Trials with repeated measurements within visits 3298.1.1 Covariance pattern models 3308.1.2 Example 3358.1.3 Random coefficients models 3418.1.4 Example: random coefficients models 3438.2 Multi-centre trials with repeated measurements 3498.2.1 Example: multi-centre hypertension trial 3498.2.2 Covariance pattern models 3508.3 Multi-centre cross-over trials 3558.4 Hierarchical multi-centre trials and meta-analysis 3568.5 Matched case–control studies 3578.5.1 Example 3588.5.2 Analysis of a quantitative variable 3598.5.3 Check of model assumptions 3608.5.4 Analysis of binary Variables 3628.6 Different variances for treatment groups in a simple between-patient trial 3708.6.1 Example 3718.7 Estimating variance components in an animal physiology trial 3748.7.1 Sample size estimation for a future experiment 3758.8 Inter- and intra-observer variation in foetal scan measurements 3808.9 Components of variation and mean estimates in a cardiology experiment 3818.10 Cluster sample surveys 3848.10.1 Example: cluster sample survey 3848.11 Small area mortality estimates 3868.12 Estimating surgeon performance 3898.13 Event history analysis 3918.13.1 Example 3928.14 A laboratory study using a within-subject 4 × 4 factorial design 3948.15 Bioequivalence studies with replicate cross-over designs 3978.15.1 Example 4028.16 Cluster randomised trials 4118.16.1 Example: A trial to evaluate integrated care pathways for treatment of children with asthma in hospital 4128.16.2 Example: Edinburgh randomised trial of breast screening 4148.17 Analysis of bilateral data 4188.18 Incomplete block designs 4258.18.1 Introduction 4258.18.2 Balanced incomplete block (BIB) designs 4268.18.3 Studies where only comparisons to a particular treatment are of interest 4298.18.4 Designs to produce lower variances for a specific treatment pair 4338.18.5 Design to produce lower variances for more than one treatment pair 4389 Software for fitting mixed models 4529.1 Packages for fitting mixed models 4529.2 PROC MIXED 4549.3 Using SAS to fit mixed models to non-normal data 4769.3.1 PROC GLIMMIX 4779.3.2 PROC GENMOD 4809.4 PROC MCMC 483Glossary 489References 492Index 497SIP Series Titles 513