Applied Mixed Models in Medicine

by Helen Brown (University of Edinburgh, UK); Robin Prescott (University of Edinburgh, UK)

• ISBN13: 9780471965541
• ISBN10: 0471965545
• Format: Hardcover
• Copyright: 2000-01-01
• Publisher: WILEY

Summary

Provides a clear understanding of the application of mixed models, and describes the benefits to be gained from their use as well as the practical implications. Mixed models is becoming a popular method of statistical analysis used for analysing medical data, particularly in the pharmaceutical industry. This method often gives improvements over conventional fixed effect models, especially when data are unbalanced. Presently there is no other book covering the application of mixed models to clinical data, making this book essential reading for those involved in this subject. Features include: * Takes a balanced view of mixed models by discussing some of the problems in their use and indicates where more conventional fixed effect models might be preferred. * Easily accessible to practitioners in any areas where mixed models are used, including medical statisticians and economists * Illustrated with numerous medical examples which clearly demonstrate the application of the theory * Extensive coverage of the underlying theory * Devotes a complete chapter to the use of software procedures and macros to fit mixed models. This title is aimed at medical, applied and bio-statisticians, along with teachers and students of advanced statistics courses in mixed models. The book is also suitable for medical scientists who need to understand the techniques used and the assumuptions which underpin their use.

Table of Contents

Preface	xiii
Series Preface	xvii
Mixed Models Notation	xix
Introduction	1	(32)
The Use of Mixed Models	1	(2)
Introductory Example	3	(9)
Simple model to assess the effects of treatment (Model A)	3	(3)
A model taking patient effects into account (Model B)	6	(1)
Random effects model (Model C)	6	(5)
Estimation (or prediction) of random effects	11	(1)
A Multi-Centre Hypertension Trial	12	(6)
Modelling the data	14	(1)
Including a baseline covariate (Model B)	14	(2)
Modelling centre effects (Model C)	16	(1)
Including centre-by-treatment interaction effects (Model D)	16	(1)
Modelling centre and centre treatment effects as random (Model E)	17	(1)
Repeated Measures Data	18	(3)
Covariance pattern models	19	(1)
Random coefficients models	20	(1)
More About Mixed Models	21	(5)
What is a mixed model?	22	(1)
Why use mixed models?	23	(1)
Communicating results	24	(1)
Mixed models in medicine	25	(1)
Mixed models in perspective	25	(1)
Some Useful Definitions	26	(7)
Containment	27	(1)
Balance	28	(2)
Error strata	30	(3)
Normal Mixed Models	33	(70)
Model Definition	33	(11)
The fixed effects model	33	(3)
The mixed model	36	(2)
The random effects model covariance structure	38	(2)
The random coefficients model covariance structure	40	(2)
The covariance pattern model covariance structure	42	(2)
Model Fitting Methods	44	(11)
The likelihood function and approaches to its maximisation	45	(3)
Estimation of fixed effects	48	(1)
Estimation (or prediction) of random effects and coefficients	49	(2)
Estimating variance parameters	51	(4)
The Bayesian Approach	55	(13)
Introduction	56	(1)
Determining the posterior density	57	(1)
Parameter estimation, probability intervals and p-values	58	(2)
Specifying non-informative prior distributions	60	(4)
Evaluating the posterior distribution	64	(4)
Practical Application and Interpretation	68	(11)
Negative variance components	69	(4)
Accuracy of variance parameters	73	(1)
Bias in fixed and random effects standard errors	73	(1)
Significance testing	74	(3)
Confidence intervals	77	(1)
Model checking	77	(2)
Missing data	79	(1)
Example	79	(24)
Analysis models	80	(2)
Results	82	(1)
Discussion of points from Section 2.4	83	(20)
Generalised Linear Mixed Models (GLMMs)	103	(46)
Generalised Linear Models (GLMs)	104	(12)
Introduction	104	(1)
Distributions	105	(2)
The general form for exponential distributions	107	(1)
The GLM definition	108	(2)
Interpreting results from GLMs	110	(2)
Fitting the GLM	112	(2)
Expressing individual distributions in the general exponential form	114	(2)
Conditional logistic regression	116	(1)
Generalised Linear Mixed Models (GLMMs)	116	(10)
The GLMM definition	116	(2)
The likelihood and quasi-likelihood functions	118	(2)
Fitting the GLMM	120	(4)
Some flaws with GLMMs	124	(2)
Reparameterising random effects models as covariance pattern models	126	(1)
Practical Application and Interpretation	126	(8)
Specifying binary data	127	(1)
Difficulties with fitting random effects (and random coefficients) models	127	(1)
Accuracy of variance parameters	128	(1)
Bias in fixed and random effects standard errors	129	(1)
Negative variance components	129	(1)
Uniform fixed effect categories	130	(1)
Uniform random effect categories	130	(1)
The dispersion parameter	130	(2)
Significance testing	132	(1)
Confidence intervals	133	(1)
Model checking	133	(1)
Example	134	(15)
Introduction and models fitted	134	(2)
Results	136	(3)
Discussion of points from Section 3.3	139	(10)
Mixed Models for Categorical Data	149	(22)
Ordinal Logistic Regression (Fixed Effects Model)	149	(2)
Mixed Ordinal Logistic Regression	151	(8)
Definition of the mixed ordinal logistic regression model	151	(3)
Residual variance matrix, R	154	(3)
Reparameterising random effects models as covariance pattern models	157	(1)
Likelihood and quasi-likelihood functions	158	(1)
Model fitting methods	158	(1)
Mixed Models for Unordered Categorical Data	159	(3)
The G matrix	162	(1)
The R matrix	162	(1)
Fitting the model	162	(1)
Practical Application and Interpretation	162	(2)
The proportional odds assumption	163	(1)
Number of covariance parameters	163	(1)
Choosing a covariance pattern	163	(1)
Interpreting covariance parameters	163	(1)
Fixed and random effects estimates	164	(1)
Checking model assumptions	164	(1)
The dispersion parameter	164	(1)
Other points	164	(1)
Example	164	(7)
Multi-Centre Trials and Meta-Analyses	171	(28)
Introduction to Multi-Centre Trials	171	(1)
What is a multi-centre trial?	171	(1)
Why use mixed models to analyse multi-centre data?	171	(1)
The Implications of Using Different Analysis Models	172	(4)
Centre and centre-treatment effects fixed	172	(1)
Centre effects fixed, centre-treatment effects omitted	173	(1)
Centre and centre-treatment effects random	174	(1)
Centre effects random, centre-treatment effects omitted	175	(1)
Example: A Multi-Centre Trial	176	(5)
Practical Application and Interpretation	181	(2)
Plausibility of a centre-treatment interaction	181	(1)
Generalisation	182	(1)
Number of centres	182	(1)
Centre size	182	(1)
Negative variance components	183	(1)
Balance	183	(1)
Sample Size Estimation	183	(6)
Normal data	184	(3)
Non-normal data	187	(2)
Meta-Analysis	189	(1)
Example: Meta-Analysis	190	(9)
Analyses	190	(1)
Results	191	(1)
Treatment estimates in individual trials	192	(7)
Repeated Measures Data	199	(62)
Introduction	199	(3)
Reasons for repeated measurements	199	(1)
Analysis objectives	200	(1)
Fixed effects approaches	200	(2)
Mixed model approaches	202	(1)
Covariance Pattern Models	202	(9)
Covariance patterns	203	(4)
Choice of covariance pattern	207	(2)
Choice of fixed effects	209	(1)
General points	210	(1)
Example: Covariance Pattern Models for Normal Data	211	(15)
Analysis models	212	(1)
Selection of covariance pattern	212	(2)
Assessing fixed effects	214	(2)
Model checking	216	(10)
Example: Covariance Pattern Models for Count Data	226	(9)
GLMM analysis models	227	(3)
Analysis using a categorical mixed model	230	(5)
Random Coefficients Models	235	(4)
Introduction	235	(2)
General points	237	(1)
Comparisons with fixed effects approaches	238	(1)
Examples of Random Coefficients Models	239	(17)
A linear random coefficients model	239	(2)
A polynomial random coefficients model	241	(15)
Sample Size Estimation	256	(5)
Normal data	256	(2)
Non-normal data	258	(1)
Categorical data	259	(2)
Cross-Over Trials	261	(34)
Introduction	261	(1)
Advantages of Mixed Models in Cross-Over Trials	261	(1)
The AB/BA Cross-Over Trial	262	(7)
Example: AB/BA cross-over design	264	(5)
Higher Order Complete Block Designs	269	(3)
Inclusion of carry-over effects	269	(1)
Example: Four-period, four-treatment cross-over trial	269	(3)
Incomplete Block Designs	272	(4)
The three-treatment, two-period design (Koch's design)	273	(1)
Example: Two-period cross-over trial	274	(2)
Optimal Designs	276	(2)
Example: Balaam's design	276	(2)
Covariance Pattern Models	278	(8)
Structured by period	278	(1)
Structured by treatment	279	(1)
Example: Four-way cross-over trial	279	(7)
Analysis of Binary Data	286	(4)
Analysis of Categorical Data	290	(1)
Use of Results from Random Effects Models in Trial Design	291	(2)
Example	292	(1)
General Points	293	(2)
Other Applications of Mixed Models	295	(64)
Trials with Repeated Measurements Within Visits	295	(17)
Covariance pattern models	296	(4)
Example	300	(6)
Random coefficients models	306	(2)
Example: Random coefficients models	308	(4)
Multi-Centre Trials with Repeated Measures	312	(6)
Example: Multi-centre hypertension trial	313	(1)
Covariance pattern models	314	(4)
Multi-Centre Cross-Over Trials	318	(1)
Hierarchical Multi-Centre Trials and Meta-Analysis	319	(1)
Matched Case-Control Studies	320	(13)
Example	321	(1)
Analysis of a quantitative variable	321	(2)
Check of model assumptions	323	(2)
Analysis of binary variables	325	(8)
Different Variances for Treatment Groups in a Simple Between-Patient Trial	333	(3)
Example	334	(2)
Estimating Variance Components in an Animal Physiology Trial	336	(6)
Sample size estimation for a future experiment	337	(5)
Inter- and Intra-Observer Variation in Foetal Scan Measurements	342	(2)
Components of Variation and Mean Estimates in a Cardiology Experiment	344	(2)
Cluster Sample Surveys	346	(2)
Example: Cluster sample survey	346	(2)
Small Area Mortality Estimates	348	(5)
Estimating Surgeon Performance	353	(2)
Event History Analysis	355	(4)
Example	355	(4)
Software for Fitting Mixed Models	359	(24)
Packages for Fitting Mixed Models	359	(1)
Basic Use of PROC MIXED	360	(19)
Syntax	360	(2)
PROC MIXED statement options	362	(17)
Basic Use of PROC GENMOD and the GLIMMIX Macro	379	(4)
PROC GENMOD	379	(2)
The GLIMMIX macro	381	(2)
Glossary	383	(4)
References	387	(4)
Contacts	391	(2)
Index	393