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书籍名称:Generalized Linear Models and Extensions, Fourth Edition
出版社:Stata Press
作者: James W. Hardin and Joseph M. Hilbe
出版时间:2018-07-01
语种: 英语
页数: 598
印刷日期:2018-07-12
开本: 胶版纸
纸张:598 I S B N: 978-1-59718-225-6
装订: 平装

简介

Generalized linear models (GLMs) extend linear regression to models with a non-Gaussian or even discrete response. GLM theory is predicated on the exponential family of distributions—a class so rich that it includes the commonly used logit, probit, and Poisson models. Although one can fit these models in Stata by using specialized commands (for example, logit for logit models), fitting them as GLMs with Stata’s glm command offers some advantages. For example, model diagnostics may be calculated and interpreted similarly regardless of the assumed distribution. This text thoroughly covers GLMs, both theoretically and computationally, with an emphasis on Stata. The theory consists of showing how the various GLMs are special cases of the exponential family, showing general properties of this family of distributions, and showing the derivation of maximum likelihood (ML) estimators and standard errors. Hardin and Hilbe show how iteratively reweighted least squares, another method of parameter estimation, is a consequence of ML estimation using Fisher scoring. The authors also discuss different methods of estimating standard errors, including robust methods, robust methods with clustering, Newey–West, outer product of the gradient, bootstrap, and jackknife. The thorough coverage of model diagnostics includes measures of influence such as Cook’s distance, several forms of residuals, the Akaike and Bayesian information criteria, and various R2-type measures of explained variability. After presenting general theory, Hardin and Hilbe then break down each distribution. Each distribution has its own chapter that explains the computational details of applying the general theory to that particular distribution. Pseudocode plays a valuable role here because it lets the authors describe computational algorithms relatively simply. Devoting an entire chapter to each distribution (or family, in GLM terms) also allows for the inclusion of real-data examples showing how Stata fits such models, as well as the presentation of certain diagnostics and analytical strategies that are unique to that family. The chapters on binary data and on count (Poisson) data are excellent in this regard. Hardin and Hilbe give ample attention to the problems of overdispersion and zero inflation in count-data models. The final part of the text concerns extensions of GLMs. First, the authors cover multinomial responses, both ordered and unordered. Although multinomial responses are not strictly a part of GLM, the theory is similar in that one can think of a multinomial response as an extension of a binary response. The examples presented in these chapters often use the authors’ own Stata programs, augmenting official Stata’s capabilities. Second, GLMs may be extended to clustered data through generalized estimating equations (GEEs), and one chapter covers GEE theory and examples. GLMs may also be extended by programming one’s own family and link functions for use with Stata’s official glm command, and the authors detail this process. Finally, the authors describe extensions for multivariate models and Bayesian analysis. The fourth edition includes two new chapters. The first introduces bivariate and multivariate models for binary and count outcomes. The second covers Bayesian analysis and demonstrates how to use the bayes: prefix and the bayesmh command to fit Bayesian models for many of the GLMs that were discussed in previous chapters. Additionally, the authors added discussions of the generalized negative binomial models of Waring and Famoye. New explanations of working with heaped data, clustered data, and bias-corrected GLMs are included. The new edition also incorporates more examples of creating synthetic data for models such as Poisson, negative binomial, hurdle, and finite mixture models.

目录

    List of figures
    List of tables
    List of listings
    Preface
    1 Introduction
    1.1 Origins and motivation
    1.2 Notational conventions
    1.3 Applied or theoretical?
    1.4 Road map
    1.5 Installing the support materials
    I Foundations of Generalized Linear Models
    2 GLMs
    2.1 Components
    2.2 Assumptions
    2.3 Exponential family
    2.4 Example: Using an offset in a GLM
    2.5 Summary
    3 GLM estimation algorithms
    3.1 Newton–Raphson (using the observed Hessian)
    3.2 Starting values for Newton–Raphson
    3.3 IRLS (using the expected Hessian)
    3.4 Starting values for IRLS
    3.5 Goodness of fit
    3.6 Estimated variance matrices
    3.6.1 Hessian
    3.6.2 Outer product of the gradient
    3.6.3 Sandwich
    3.6.4 Modified sandwich
    3.6.5 Unbiased sandwich
    3.6.6 Modified unbiased sandwich
    3.6.7 Weighted sandwich: Newey–West
    3.6.8 Jackknife
    3.6.8.1 Usual jackknife
    3.6.8.2 One-step jackknife
    3.6.8.3 Weighted jackknife
    3.6.8.4 Variable jackknife
    3.6.9 Bootstrap
    3.6.9.1 Usual bootstrap
    3.6.9.2 Grouped bootstrap
    3.7 Estimation algorithms
    3.8 Summary
    4 Analysis of fit
    4.1 Deviance
    4.2 Diagnostics
    4.2.1 Cook’s distance
    4.2.2 Overdispersion
    4.3 Assessing the link function
    4.4 Residual analysis
    4.4.1 Response residuals
    4.4.2 Working residuals
    4.4.3 Pearson residuals
    4.4.4 Partial residuals
    4.4.5 Anscombe residuals
    4.4.6 Deviance residuals
    4.4.7 Adjusted deviance residuals
    4.4.8 Likelihood residuals
    4.4.9 Score residuals
    4.5 Checks for systematic departure from the model
    4.6 Model statistics
    4.6.1 Criterion measures
    4.6.1.1 AIC
    4.6.1.2 BIC
    4.6.2 The interpretation of R2 in linear regression
    4.6.2.1 Percentage variance explained
    4.6.2.2 The ratio of variances
    4.6.2.3 A transformation of the likelihood ratio
    4.6.2.4 A transformation of the F test
    4.6.2.5 Squared correlation
    4.6.3 Generalizations of linear regression R2 interpretations
    4.6.3.1 Efron’s pseudo-R2
    4.6.3.2 McFadden’s likelihood-ratio index
    4.6.3.3 Ben-Akiva and Lerman adjusted likelihood-ratio index
    4.6.3.4 McKelvey and Zavoina ratio of variances
    4.6.3.5 Transformation of likelihood ratio
    4.6.3.6 Cragg and Uhler normed measure
    4.6.4 More R2 measures
    4.6.4.1 The count R2
    4.6.4.2 The adjusted count R2
    4.6.4.3 Veall and Zimmermann R2
    4.6.4.4 Cameron–Windmeijer R2
    4.7 Marginal effects
    4.7.1 Marginal effects for GLMs
    4.7.2 Discrete change for GLMs
    II Continuous Response Models
    5 The Gaussian family
    5.1 Derivation of the GLM Gaussian family
    5.2 Derivation in terms of the mean
    5.3 IRLS GLM algorithm (nonbinomial)
    5.4 ML estimation
    5.5 GLM log-Gaussian models
    5.6 Expected versus observed information matrix
    5.7 Other Gaussian links
    5.8 Example: Relation to OLS
    5.9 Example: Beta-carotene
    6 The gamma family
    6.1 Derivation of the gamma model
    6.2 Example: Reciprocal link
    6.3 ML estimation
    6.4 Log-gamma models
    6.5 Identity-gamma models
    6.6 Using the gamma model for survival analysis
    7 The inverse Gaussian family
    7.1 Derivation of the inverse Gaussian model
    7.2 Shape of the distribution
    7.3 The inverse Gaussian algorithm
    7.4 Maximum likelihood algorithm
    7.5 Example: The canonical inverse Gaussian
    7.6 Noncanonical links
    8 The power family and link
    8.1 Power links
    8.2 Example: Power link
    8.3 The power family
    III Binomial Response Models
    9 The binomial–logit family
    9.1 Derivation of the binomial model
    9.2 Derivation of the Bernoulli model
    9.3 The binomial regression algorithm
    9.4 Example: Logistic regression
    9.4.1 Model producing logistic coefficients: The heart data
    9.4.2 Model producing logistic odds ratios
    9.5 GOF statistics
    9.6 Grouped data
    9.7 Interpretation of parameter estimates
    10 The general binomial family
    10.1 Noncanonical binomial models
    10.2 Noncanonical binomial links (binary form)
    10.3 The probit model
    10.4 The clog-log and log-log models
    10.5 Other links
    10.6 Interpretation of coefficients
    10.6.1 Identity link
    10.6.2 Logit link
    10.6.3 Log link
    10.6.4 Log complement link
    10.6.5 Log-log link
    10.6.6 Complementary log-log link
    10.6.7 Summary
    10.7 Generalized binomial regression
    10.8 Beta binomial regression
    10.9 Zero-inflated models
    11 The problem of overdispersion
    11.1 Overdispersion
    11.2 Scaling of standard errors
    11.3 Williams’ procedure
    11.4 Robust standard errors
    IV Count Response Models
    12 The Poisson family
    12.1 Count response regression models
    12.2 Derivation of the Poisson algorithm
    12.3 Poisson regression: Examples
    12.4 Example: Testing overdispersion in the Poisson model
    12.5 Using the Poisson model for survival analysis
    12.6 Using offsets to compare models
    12.7 Interpretation of coefficients
    13 The negative binomial family
    13.1 Constant overdispersion
    13.2 Variable overdispersion
    13.2.1 Derivation in terms of a Poisson–gamma mixture
    13.2.2 Derivation in terms of the negative binomial probability function
    13.2.3 The canonical link negative binomial parameterization
    13.3 The log-negative binomial parameterization
    13.4 Negative binomial examples
    13.5 The geometric family
    13.6 Interpretation of coefficients
    14 Other count-data models
    14.1 Count response regression models
    14.2 Zero-truncated models
    14.3 Zero-inflated models
    14.4 General truncated models
    14.5 Hurdle models
    14.6 Negative binomial(P) models
    14.7 Negative binomial(Famoye)
    14.8 Negative binomial(Waring)
    14.9 Heterogeneous negative binomial models
    14.10 Generalized Poisson regression models
    14.11 Poisson inverse Gaussian models
    14.12 Censored count response models
    14.13 Finite mixture models
    14.14 Quantile regression for count outcomes
    14.15 Heaped data models
    V Multinomial Response Models
    15 Unordered-response family
    15.1 The multinomial logit model
    15.1.1 Interpretation of coefficients: Single binary predictor
    15.1.2 Example: Relation to logistic regression
    15.1.3 Example: Relation to conditional logistic regression
    15.1.4 Example: Extensions with conditional logistic regression
    15.1.5 The independence of irrelevant alternatives
    15.1.6 Example: Assessing the IIA
    15.1.7 Interpreting coefficients
    15.1.8 Example: Medical admissions—introduction
    15.1.9 Example: Medical admissions—summary
    15.2 The multinomial probit model
    15.2.1 Example: A comparison of the models
    15.2.2 Example: Comparing probit and multinomial probit
    15.2.3 Example: Concluding remarks
    16 The ordered-response family
    16.1 Interpretation of coefficients: Single binary predictor
    16.2 Ordered outcomes for general link
    16.3 Ordered outcomes for specific links
    16.3.1 Ordered logit
    16.3.2 Ordered probit
    16.3.3 Ordered clog-log
    16.3.4 Ordered log-log
    16.3.5 Ordered cauchit
    16.4 Generalized ordered outcome models
    16.5 Example: Synthetic data
    16.6 Example: Automobile data
    16.7 Partial proportional-odds models
    16.8 Continuation-ratio models
    16.9 Adjacent category model
    VI Extensions to the GLM
    17 Extending the likelihood
    17.1 The quasilikelihood
    17.2 Example: Wedderburn’s leaf blotch data
    17.3 Example: Tweedie family variance
    17.4 Generalized additive models
    18 Clustered data
    18.1 Generalization from individual to clustered data
    18.2 Pooled estimators
    18.3 Fixed effects
    18.3.1 Unconditional fixed-effects estimators
    18.3.2 Conditional fixed-effects estimators
    18.4 Random effects
    18.4.1 Maximum likelihood estimation
    18.4.2 Gibbs sampling
    18.5 Mixed-effect models
    18.6 GEEs
    18.7 Other models
    19 Bivariate and multivariate models
    19.1 Bivariate and multivariate models for binary outcomes
    19.2 Copula functions
    19.3 Using copula functions to calculate bivariate probabilities
    19.4 Synthetic datasets
    19.5 Examples of bivariate count models using copula functions
    19.6 The Famoye bivariate Poisson regression model
    19.7 The Marshall–Olkin bivariate negative binomial regression model
    19.8 The Famoye bivariate negative binomial regression model
    20 Bayesian GLMs
    20.1 Brief overview of Bayesian methodology
    20.1.1 Specification and estimation
    20.1.2 Bayesian analysis in Stata
    20.2 Bayesian logistic regression
    20.2.1 Bayesian logistic regression—noninformative priors
    20.2.2 Diagnostic plots
    20.2.3 Bayesian logistic regression—informative priors
    20.3 Bayesian probit regression
    20.4 Bayesian complementary log-log regression
    20.5 Bayesian binomial logistic regression
    20.6 Bayesian Poisson regression
    20.6.1 Bayesian Poisson regression with noninformative priors
    20.6.2 Bayesian Poisson with informative priors
    20.7 Bayesian negative binomial likelihood
    20.7.1 Zero-inflated negative binomial logit
    20.8 Bayesian normal regression
    20.9 Writing a custom likelihood
    20.9.1 Using the llf() option
    20.9.1.1 Bayesian logistic regression using llf()
    20.9.1.2 Bayesian zero-inflated negative binomial logit regression using llf()
    20.9.2 Using the llevaluator() option
    20.9.2.1 Logistic regression model using llevaluator()
    20.9.2.2 Bayesian clog-log regression with llevaluator()
    20.9.2.3 Bayesian Poisson regression with llevaluator()
    20.9.2.4 Bayesian negative binomial regression using llevaluator()
    20.9.2.5 Zero-inflated negative binomial logit using llevaluator()
    20.9.2.6 Bayesian gamma regression using llevaluator()
    20.9.2.7 Bayesian inverse Gaussian regression using llevaluator()
    20.9.2.8 Bayesian zero-truncated Poisson using llevaluator()
    20.9.2.9 Bayesian bivariate Poisson using llevaluator()
    VII Stata Software
    21 Programs for Stata
    21.1 The glm command
    21.1.1 Syntax
    21.1.2 Description
    21.1.3 Options
    21.2 The predict command after glm
    21.2.1 Syntax
    21.2.2 Options
    21.3 User-written programs
    21.3.1 Global macros available for user-written programs
    21.3.2 User-written variance functions
    21.3.3 User-written programs for link functions
    21.3.4 User-written programs for Newey–West weights
    21.4 Remarks
    21.4.1 Equivalent commands
    21.4.2 Special comments on family(Gaussian) models
    21.4.3 Special comments on family(binomial) models
    21.4.4 Special comments on family(nbinomial) models
    21.4.5 Special comment on family(gamma) link(log) models
    22 Data synthesis
    22.1 Generating correlated data
    22.2 Generating data from a specified population
    22.2.1 Generating data for linear regression
    22.2.2 Generating data for logistic regression
    22.2.3 Generating data for probit regression
    22.2.4 Generating data for complimentary log-log regression
    22.2.5 Generating data for Gaussian variance and log link
    22.2.6 Generating underdispersed count data
    22.3 Simulation
    22.3.1 Heteroskedasticity in linear regression
    22.3.2 Power analysis
    22.3.3 Comparing fit of Poisson and negative binomial
    22.3.4 Effect of missing covariate on R2Efron in Poisson regression
    A Tables
    References
    Author index
    Subject index