2. Normal mixed models
    2.1 Model definition
       2.1.1 The fixed effects model
       2.1.2 The mixed model
       2.1.3 Random effects model covariance structure
       2.1.4 Random coefficients model covariance structure
       2.1.5 Covariance pattern model covariance structure
   2.2   Model fitting methods
       2.2.1 Mixed model methods
       2.2.2 Estimating fixed effects
       2.2.3 Estimating random effects
       2.2.4 Estimating variance parameters
       2.2.5 Comparison of methods
   2.3 The Bayesian approach
                    2.3.1 Introduction
       2.3.2 Determining the posterior density
       2.3.3 Parameter estimation, probability intervals
                and p-values
       2.3.4 Specifying non-informative prior distributions
       2.3.5   Evaluating the posterior distribution
   2.4 Practical application and interpretation
     2.4.1 Negative variance components
      2.4.2 Accuracy of variance parameters
      2.4.3 Bias in fixed and random effect standard errors
      2.4.4 Significance testing
      2.4.5 Confidence intervals
      2.4.6 Model checking
      2.4.7 Missing data
   2.5 Example
      2.5.1 Analysis models
      2.5.2 Results
      2.5.3 Discussion of points from Section 2.4

3. Generalised linear mixed models (GLMs)
    3.1 Generalised linear models (GLMs)
       3.1.1 Introduction
       3.1.2 Distributions
       3.1.3 General form for exponential distributions
       3.1.4 GLM definition
       3.1.5 Interpreting results from GLMs
       3.1.6 Fitting the GLM
       3.1.7 Expressing individual distributions in the
                  general exponential form
       3.1.8 Conditional logistic regression
    3.2 Generalised linear mixed models (GLMMs)
       3.2.1 GLMM definition
       3.2.2 Likelihood and quasi-likelihood functions
       3.2.3 Fitting the GLMM
          3.2.3.1 Pseudo-likelihood
          3.2.3.2 Generalised estimating equations
          3.2.3.3 Marginal quasi-likelihood
          3.2.3.4 Bayesian methods
    3.2.4 Some flaws with GLMMs
3.3 Practical application and interpretation
   3.3.1 Specifying binary Data
   3.3.2 Difficulties with fitting random effects
   (and random coefficients) models
   3.3.3 Accuracy of variance parameters
   3.3.4 Bias in fixed and random effects standard errors
   3.3.5 Negative variance components
   3.3.6 Uniform fixed effect categories
   3.3.7 Uniform random effect categories
   3.3.8 The dispersion parameter
   3.3.9 Significance testing
   3.3.10 Confidence intervals
   3.3.11 Checking model assumptions
3.4 Example
   3.4.1 Introduction and models fitted
   3.4.2 Results
   3.4.3 Discussion of points from Section 3.3

4. Mixed models for categorical data
   4.1 Ordinal logistic regression (fixed effects model)
   4.2 Mixed ordinal logistic regression
   4.2.1 Definition of mixed ordinal logistic regression model
   4.2.2 Residual variance matrix, R
   4.2.3 Reparameterising random effects models as covariance pattern models
   4.2.4 Likelihood and quasi-likelihood functions
   4.2.5 Model fitting methods
   4.3 Mixed models for unordered categorical data
   4.4 Practical application and interpretation
         4.4.1 Proportional odds assumption
         4.4.2 Number of covariance parameters
         4.4.3 Choosing a covariance pattern
         4.4.4 Interpreting covariance parameters
         4.4.5 Fixed and random effects estimates
         4.4.6 Checking model assumptions
         4.4.7 Dispersion parameter
         4.4.8 Other points
    4.5 Example

5. Multicentre trials and meta analyses
   5.1 Introduction to multicentre trials
    5.2 Implications of using different analysis models
    5.3 Example: multicentre trial
   5.4 Practical application and interpretation
      5.4.1 Plausibility of a centre.treatment interaction
      5.4.2 Generalisation
      5.4.3 Number of centres
      5.4.4 Centre size
      5.4.5 Negative variance components
      5.4.6 Balance
      5.5 Sample size estimation
      5.6 Meta analysis
      5.7 Example: meta analysis

6. Repeated measures data
    6.1 Introduction
    6.2 Covariance Pattern models
          6.2.1 Covariance patterns
          6.2.2 Choice of covariance pattern
          6.2.3 Choice of fixed effects
          6.2.4 General points
   6.3 Example: covariance pattern models for normal data
      6.3.1 Analysis models
      6.3.2 Selection of covariance pattern
      6.3.3 Assessing fixed effects
      6.3.4 Model checking
   6.4 Example: covariance pattern models for count data
   6.5 Random coefficients models
       6.5.1 Introduction
       6.5.2 General points
        6.5.3 Comparisons with fixed effects approaches
   6.6 Examples: random coefficients models
       6.6.1 A linear random coefficients model
       6.6.2 A polynomial random coefficients model
    6.7 Sample size estimation

7. Cross-over trials
   7.1 Introduction
   7.2 Advantages of mixed models in cross-over trials
   7.3 The AB/BA cross-over trial
   7.4 Higher order complete block designs
   7.5 Incomplete block designs
   7.6 Optimal designs
   7.7 Covariance pattern models
   7.8 Analysis of binary data
   7.9 Analysis of categorical data
   7.10 Use of results from random effects models in trial design
   7.11 General points

8. Other applications of mixed models
    8.1 Trials with repeated measurements within visits
        8.1.1 Covariance pattern models
        8.1.2 Example: covariance pattern models
        8.1.3 Random coefficients models
        8.1.4 Example: random coefficients models
    8.2 Multicentre trials with repeated measures
    8.3 Multicentre cross-over trials
    8.4 Hierarchical multicentre trials and meta data
    8.5 Matched case-control studies
    8.6 Different variances for treatment groups in a
          simple between patient trial
    8.7 Estimating variance components in an animal
            physiology trial
    8.8 Inter and intra observer variation in foetal scan
          measurements
    8.9 Components of variation and mean estimates in
           cardiology experiment
    8.10 Cluster sample surveys
    8.11 Small area mortality estimates
    8.12 Estimating surgeon performance
    8.13 Event history analysis

9. Software for mixed models
    9.1 Packages for fitting mixed models
    9.2 Basic use of PROC MIXED
            Basic syntax
            Simple example
            PROC MIXED statement
            MODEL statement
            RANDOM statement
            LSMEANS statement
            ESTIMATE statement
            CONTRAST statement
            REPEATED statement
            PRIOR statement
            PARMS statement
            MAKE statement
            ID statement
            WEIGHT statement
    9.3 Basic use of PROC GENMOD and GLIMMIX macro
         PROC GENMOD
        GLIMMIX macro

References

Glossary

Mixed models notation

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