Design of Experiments for Generalized Linear Models

Cover......Page 1
Half Title......Page 2
Series Page......Page 3
Title Page......Page 4
Copyright Page......Page 5
Dedication......Page 6
Contents......Page 8
Preface......Page 12
 1.1 Introduction......Page 16
 1.2 Mathematics of the general linear model......Page 22
 1.3 Towards the generalized linear model......Page 24
 1.4 Generalized linear models......Page 27
 1.5 Estimating the values of the model parameters......Page 31
 2.1 Introduction......Page 36
  2.2.2 A function of more than one variable......Page 37
 2.3 Restrictions on independent variables......Page 43
  2.4.1 The function constrOptim......Page 47
  2.4.2 Additional features of constrOptim......Page 51
  2.4.3 Constrained optimisation without constrOptim......Page 52
  2.4.4 Notes and examples......Page 55
  2.4.5 Using the function optim......Page 56
  2.4.6 Initial values for optimisations......Page 58
 2.5 Numerical integration......Page 59
 2.6 Conclusion......Page 62
  3.3.1 Exact and approximate designs......Page 64
  3.3.2 Constructing an exact design from an approximate design......Page 65
  3.3.3 Constructing an exact design directly......Page 66
  3.4.1 Thinking about criteria for selection......Page 67
  3.4.2 The standardised variance......Page 75
  3.5.1 Theory......Page 77
  3.5.2 Example: a simple logistic regression......Page 78
 3.6 Difficulties caused by small samples......Page 81
  3.7.1 Number of support points......Page 83
  3.7.3 A-optimality......Page 85
  3.7.4 D-optimality......Page 86
  3.7.5 Ds-optimality......Page 87
  3.7.6 E-optimality......Page 89
 3.8 Example......Page 90
  3.8.1 Using constrOptim......Page 92
  3.8.2 Using optim......Page 95
 3.9 The general equivalence theorem......Page 97
 3.10 Where next?......Page 102
 4.2 Notation......Page 104
  4.3.1 The logit link function......Page 105
  4.3.2 The probit link function......Page 106
  4.3.3 The complementary log-log link function......Page 107
  4.3.4 Comparing the three link functions......Page 108
  4.4.1 The logit link......Page 109
  4.4.2 The probit link......Page 110
  4.4.3 The complementary log-log link......Page 113
  4.5.2 The logit link......Page 119
  4.5.3 The probit link......Page 135
  4.5.5 Comparison of designs for different links......Page 136
 4.6 Designs for m > 2 explanatory variables......Page 137
 4.7 Obtaining an exact design......Page 143
  4.8.1 Maximum penalised likelihood estimation......Page 147
  4.8.2 IMSE-optimality......Page 152
  4.8.3 Speeding up the calculations......Page 157
  4.8.4 Extending the previous work on IMSE-optimality......Page 158
 4.9 DS-optimality......Page 160
 4.10 Uncertainty over aspects of the model......Page 162
 5.2 Modelling the Poisson distribution......Page 164
  5.3.2 The model η = β0 + β1x1 + ··· + βmxm......Page 166
  5.3.3 When Theorem 5.1 cannot be used......Page 171
 5.4 Small values of the total sample size, N......Page 174
  6.2.1 Modelling data from a multinomial distribution......Page 186
  6.2.2 Estimating the parameter vector β......Page 189
  6.2.3 Some designs......Page 191
 6.4 No specified distribution......Page 198
 7.1 Introduction......Page 206
  7.2.1 Finding the D-optimal design......Page 207
  7.2.2 Next steps......Page 212
  7.2.3 Updating the probability distribution for β1, ···, βh......Page 213
  7.3.1 Finding the D-optimal design......Page 216
  7.3.2 Using results from a previous experiment......Page 224
 7.4 Exact Bayesian design......Page 227
 7.5 Final comments......Page 232
Bibliography......Page 234
Index......Page 238