Confidence Intervals in Generalized Regression Models

by Uusipaikka; Esa

• ISBN13: 9781420060270
• ISBN10: 1420060279
• eBook ISBN(s): 9781040172810
• Format: Hardcover
• Copyright: 2008-07-25
• Publisher: Chapman & Hall/
• More Book Details

Summary

A Cohesive Approach to Regression ModelsConfidence Intervals in Generalized Regression Modelsintroduces 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 ModelsThe 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 ParameterFunctionsOffering 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.

Table of Contents

List of Tables	p. xiii
List of Figures	p. xvii
Preface	p. xxi
Introduction	p. xxv
Likelihood-Based Statistical Inference	p. 1
Statistical evidence	p. 2
Response and its statistical model	p. 3
Sample space, parameter space, and model function	p. 3
Interest functions	p. 5
Statistical inference	p. 8
Evidential statements	p. 9
Uncertainties of statements	p. 9
Likelihood concepts and law of likelihood	p. 10
Likelihood, score, and observed information functions	p. 10
Law of likelihood and relative likelihood function	p. 15
Likelihood-based methods	p. 17
Likelihood region	p. 19
Uncertainty of likelihood region	p. 20
Profile likelihood-based confidence intervals	p. 22
Profile likelihood function	p. 23
Profile likelihood region and its uncertainty	p. 26
Profile likelihood-based confidence interval	p. 28
Calculation of profile likelihood-based confidence intervals	p. 31
Comparison with the delta method	p. 34
Likelihood ratio tests	p. 37
Model restricted by hypothesis	p. 38
Likelihood of the restricted model	p. 39
General likelihood ratio test statistic (LRT statistic)	p. 41
Likelihood ratio test and its observed significance level	p. 42
Maximum likelihood estimate	p. 45
Maximum likelihood estimate (MLE)	p. 45
Asymptotic distribution of MLE	p. 47
Model selection	p. 47
Bibliographic notes	p. 49
Generalized Regression Model	p. 51
Examples of regression data	p. 51
Definition of generalized regression model	p. 69
Response	p. 70
Distributions of the components of response	p. 70
Regression function and regression parameter	p. 70
Regressors and model matrix (matrices)	p. 71
Example	p. 72
Special cases of GRM	p. 73
Assumptions on parts of GRM	p. 73
Various special GRMs	p. 74
Likelihood inference	p. 75
MLE with iterative reweighted least squares	p. 76
Model checking	p. 78
Bibliographic notes	p. 79
General Linear Model	p. 81
Definition of the general linear model	p. 81
Estimate of regression coefficients	p. 87
Least squares estimate (LSE)	p. 87
Maximum likelihood estimate (MLE)	p. 90
Test of linear hypotheses	p. 92
Confidence regions and intervals	p. 95
Joint confidence regions for finite sets of linear combinations	p. 95
Separate confidence intervals for linear combinations	p. 97
Model checking	p. 100
Bibliographic notes	p. 103
Nonlinear Regression Model	p. 107
Definition of nonlinear regression model	p. 107
Estimate of regression parameters	p. 109
Least squares estimate (LSE) of regression parameters	p. 109
Maximum likelihood estimate (MLE)	p. 112
Approximate distribution of LRT statistic	p. 114
Profile likelihood-basec confidence region	p. 115
Profile likelihood-based confidence interval	p. 115
LRT for a hypothesis on finite set of functions	p. 121
Model checking	p. 123
Bibliographic notes	p. 124
Generalized Linear Model	p. 127
Definition of generalized linear model	p. 127
Distribution, linear predictor, and link function	p. 127
Examples of distributions generating generalized linear models	p. 129
MLE of regression coefficients	p. 136
MLE	p. 136
Newton-Raphson and Fisher-scoring	p. 138
Bibliographic notes	p. 140
Binomial and Logistic Regression Model	p. 141
Data	p. 141
Binomial distribution	p. 144
Link functions	p. 146
Unparametrized link functions	p. 146
Parametrized link functions	p. 149
Likelihood inference	p. 151
Likelihood function of binomial data	p. 151
Estimates of parameters	p. 152
Likelihood ratio statistic or deviance function	p. 154
Distribution of deviance	p. 154
Model checking	p. 156
Logistic regression model	p. 157
Models with other link functions	p. 163
Nonlinear binomial regression model	p. 165
Bibliographic notes	p. 168
Poisson Regression Model	p. 169
Data	p. 169
Poisson distribution	p. 170
Link functions	p. 172
Unparametrized link functions	p. 172
Parametrized link functions	p. 175
Likelihood inference	p. 176
Likelihood function of Poisson data	p. 176
Estimates of parameters	p. 177
Likelihood ratio statistic or deviance function	p. 179
Distribution of deviance	p. 179
Model checking	p. 181
Log-linear model	p. 182
Bibliographic notes	p. 187
Multinomial Regression Model	p. 189
Data	p. 189
Multinomial distribution	p. 191
Likelihood function	p. 191
Logistic multinomial regression model	p. 193
Proportional odds regression model	p. 195
Bibliographic notes	p. 199
Other Generalized Linear Regressions Models	p. 201
Negative binomial regression model	p. 201
Data	p. 201
Negative binomial distribution	p. 203
Likelihood inference	p. 204
Negative binomial logistic regression model	p. 208
Gamma regression model	p. 211
Data	p. 211
Gamma distribution	p. 211
Link function	p. 213
Likelihood inference	p. 214
Model checking	p. 221
Other Generalized Regression Models	p. 225
Weighted general linear model	p. 225
Model	p. 225
Weighted linear regression model as GRM	p. 226
Weighted nonlinear regression model	p. 229
Model	p. 229
Weighted nonlinear regression model as GRM	p. 230
Quality design or Taguchi model	p. 231
Lifetime regression model	p. 237
Cox regression model	p. 240
Bibliographic notes	p. 248
Datasets	p. 251
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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