Confidence intervals in generalized regression models

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Summary

A Cohesive Approach to Regression Models

Confidence Intervals in Generalized Regression Models introduces a unified representation--the generalized regression model (GRM)--of various types of regression models. It also uses a likelihood-based approach for performing statistical inference from statistical evidence consisting of data and its statistical model.

Provides a Large Collection of Models

The book encompasses a number of different regression models, from very simple to more complex ones. It covers the general linear model (GLM), nonlinear regression model, generalized linear model (GLIM), logistic regression model, Poisson regression model, multinomial regression model, and Cox regression model. The author also explains methods of constructing confidence regions, profile likelihood-based confidence intervals, and likelihood ratio tests.

Uses Statistical Inference Package to Make Inferences on Real-Valued Parameter Functions

Offering software that helps with statistical analyses, this book focuses on producing statistical inferences for data modeled by GRMs. It contains numerical and graphical results while providing the code online.

Author Notes

Uusipaikka, Esa

List of Tables	p. xiii
List of Figures	p. xvii
Preface	p. xxi
Introduction	p. xxv
1 Likelihood-Based Statistical Inference	p. 1
1.1 Statistical evidence	p. 2
1.1.1 Response and its statistical model	p. 3
1.1.2 Sample space, parameter space, and model function	p. 3
1.1.3 Interest functions	p. 5
1.2 Statistical inference	p. 8
1.2.1 Evidential statements	p. 9
1.2.2 Uncertainties of statements	p. 9
1.3 Likelihood concepts and law of likelihood	p. 10
1.3.1 Likelihood, score, and observed information functions	p. 10
1.3.2 Law of likelihood and relative likelihood function	p. 15
1.4 Likelihood-based methods	p. 17
1.4.1 Likelihood region	p. 19
1.4.2 Uncertainty of likelihood region	p. 20
1.5 Profile likelihood-based confidence intervals	p. 22
1.5.1 Profile likelihood function	p. 23
1.5.2 Profile likelihood region and its uncertainty	p. 26
1.5.3 Profile likelihood-based confidence interval	p. 28
1.5.4 Calculation of profile likelihood-based confidence intervals	p. 31
1.5.5 Comparison with the delta method	p. 34
1.6 Likelihood ratio tests	p. 37
1.6.1 Model restricted by hypothesis	p. 38
1.6.2 Likelihood of the restricted model	p. 39
1.6.3 General likelihood ratio test statistic (LRT statistic)	p. 41
1.6.4 Likelihood ratio test and its observed significance level	p. 42
1.7 Maximum likelihood estimate	p. 45
1.7.1 Maximum likelihood estimate (MLE)	p. 45
1.7.2 Asymptotic distribution of MLE	p. 47
1.8 Model selection	p. 47
1.9 Bibliographic notes	p. 49
2 Generalized Regression Model	p. 51
2.1 Examples of regression data	p. 51
2.2 Definition of generalized regression model	p. 69
2.2.1 Response	p. 70
2.2.2 Distributions of the components of response	p. 70
2.2.3 Regression function and regression parameter	p. 70
2.2.4 Regressors and model matrix (matrices)	p. 71
2.2.5 Example	p. 72
2.3 Special cases of GRM	p. 73
2.3.1 Assumptions on parts of GRM	p. 73
2.3.2 Various special GRMs	p. 74
2.4 Likelihood inference	p. 75
2.5 MLE with iterative reweighted least squares	p. 76
2.6 Model checking	p. 78
2.7 Bibliographic notes	p. 79
3 General Linear Model	p. 81
3.1 Definition of the general linear model	p. 81
3.2 Estimate of regression coefficients	p. 87
3.2.1 Least squares estimate (LSE)	p. 87
3.2.2 Maximum likelihood estimate (MLE)	p. 90
3.3 Test of linear hypotheses	p. 92
3.4 Confidence regions and intervals	p. 95
3.4.1 Joint confidence regions for finite sets of linear combinations	p. 95
3.4.2 Separate confidence intervals for linear combinations	p. 97
3.5 Model checking	p. 100
3.6 Bibliographic notes	p. 103
4 Nonlinear Regression Model	p. 107
4.1 Definition of nonlinear regression model	p. 107
4.2 Estimate of regression parameters	p. 109
4.2.1 Least squares estimate (LSE) of regression parameters	p. 109
4.2.2 Maximum likelihood estimate (MLE)	p. 112
4.3 Approximate distribution of LRT statistic	p. 114
4.4 Profile likelihood-basec confidence region	p. 115
4.5 Profile likelihood-based confidence interval	p. 115
4.6 LRT for a hypothesis on finite set of functions	p. 121
4.7 Model checking	p. 123
4.8 Bibliographic notes	p. 124
5 Generalized Linear Model	p. 127
5.1 Definition of generalized linear model	p. 127
5.1.1 Distribution, linear predictor, and link function	p. 127
5.1.2 Examples of distributions generating generalized linear models	p. 129
5.2 MLE of regression coefficients	p. 136
5.2.1 MLE	p. 136
5.2.2 Newton-Raphson and Fisher-scoring	p. 138
5.3 Bibliographic notes	p. 140
6 Binomial and Logistic Regression Model	p. 141
6.1 Data	p. 141
6.2 Binomial distribution	p. 144
6.3 Link functions	p. 146
6.3.1 Unparametrized link functions	p. 146
6.3.2 Parametrized link functions	p. 149
6.4 Likelihood inference	p. 151
6.4.1 Likelihood function of binomial data	p. 151
6.4.2 Estimates of parameters	p. 152
6.4.3 Likelihood ratio statistic or deviance function	p. 154
6.4.4 Distribution of deviance	p. 154
6.4.5 Model checking	p. 156
6.5 Logistic regression model	p. 157
6.6 Models with other link functions	p. 163
6.7 Nonlinear binomial regression model	p. 165
6.8 Bibliographic notes	p. 168
7 Poisson Regression Model	p. 169
7.1 Data	p. 169
7.2 Poisson distribution	p. 170
7.3 Link functions	p. 172
7.3.1 Unparametrized link functions	p. 172
7.3.2 Parametrized link functions	p. 175
7.4 Likelihood inference	p. 176
7.4.1 Likelihood function of Poisson data	p. 176
7.4.2 Estimates of parameters	p. 177
7.4.3 Likelihood ratio statistic or deviance function	p. 179
7.4.4 Distribution of deviance	p. 179
7.4.5 Model checking	p. 181
7.5 Log-linear model	p. 182
7.6 Bibliographic notes	p. 187
8 Multinomial Regression Model	p. 189
8.1 Data	p. 189
8.2 Multinomial distribution	p. 191
8.3 Likelihood function	p. 191
8.4 Logistic multinomial regression model	p. 193
8.5 Proportional odds regression model	p. 195
8.6 Bibliographic notes	p. 199
9 Other Generalized Linear Regressions Models	p. 201
9.1 Negative binomial regression model	p. 201
9.1.1 Data	p. 201
9.1.2 Negative binomial distribution	p. 203
9.1.3 Likelihood inference	p. 204
9.1.4 Negative binomial logistic regression model	p. 208
9.2 Gamma regression model	p. 211
9.2.1 Data	p. 211
9.2.2 Gamma distribution	p. 211
9.2.3 Link function	p. 213
9.2.4 Likelihood inference	p. 214
9.2.5 Model checking	p. 221
10 Other Generalized Regression Models	p. 225
10.1 Weighted general linear model	p. 225
10.1.1 Model	p. 225
10.1.2 Weighted linear regression model as GRM	p. 226
10.2 Weighted nonlinear regression model	p. 229
10.2.1 Model	p. 229
10.2.2 Weighted nonlinear regression model as GRM	p. 230
10.3 Quality design or Taguchi model	p. 231
10.4 Lifetime regression model	p. 237
10.5 Cox regression model	p. 240
10.6 Bibliographic notes	p. 248
A Datasets	p. 251
B Notation Used for Statistical Models	p. 271
References	p. 277
Data Index	p. 283
Author Index	p. 285
Subject Index	p. 287

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Author Notes

Table of Contents