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Generalized Linear Models and Extensions

James Hardin, Joseph Hilbe

Table of contents

List of Tables
List of Figures
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

Part I - Foundations of Generalized Linear Models

2 Generalized Linear Models

  • 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 (OPG)
    • 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 Checks for systematic departure from the model
  • 4.5 Residual analysis
    • 4.5.1 Response residuals
    • 4.5.2 Working residuals
    • 4.5.3 Pearson residuals
    • 4.5.4 Partial residuals
    • 4.5.5 Anscombe residuals
    • 4.5.6 Deviance residuals
    • 4.5.7 Adjusted deviance residuals
    • 4.5.8 Likelihood residuals
    • 4.5.9 Score residuals
  • 4.6 Model statistics
    • 4.6.1 Criterion measures
      • 4.6.1.1 Akaike information criterion (AIC)
      • 4.6.1.2 Bayesian information criterion (BIC)
    • 4.6.2 The interpretation of R2 in linear regression
      • 4.6.2.1 Percent 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 Additional 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

Part 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 (non-binomial)
  • 5.4 Maximum likelihood estimation
  • 5.5 GLM log-normal 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 Maximum likelihood 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 The inverse Gaussian algorithm
  • 7.3 Maximum likelihood algorithm
  • 7.4 Example: The canonical inverse Gaussian
  • 7.5 Non-canonical links

8 The power family and link

  • 8.1 Power links
  • 8.2 Example: Power link
  • 8.3 The power family

Part 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 Goodness-of-fit statistics (GOF)
  • 9.6 Interpretation of parameter estimates

10 The general binomial family

  • 10.1 Non-canonical binomial models
  • 10.2 Non-canonical binomial links (binary form)
  • 10.3 The probit model
  • 10.4 The complementary log-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 Summary
  • 10.7 Generalized binomial regression

11 The problem of overdispersion

  • 11.1 Overdispersion
  • 11.2 Scaling of standard errors
  • 11.3 Williams' procedure
  • 11.4 Robust standard errors

Part 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
  • 13.7 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 hurdle models
  • 14.5 Heterogeneous negative binomial models
  • 14.6 Generalized Poisson regression models
  • 14.7 Censored count response models

Part V - Multinomial Response Models

15 The ordered-response family

  • 15.1 Ordered outcomes for general link
  • 15.2 Ordered outcomes for specific links
    • 15.2.1 Ordered logit
    • 15.2.2 Ordered probit
    • 15.2.3 Ordered clog-log
    • 15.2.4 Ordered log-log
    • 15.2.5 Ordered cauchit
  • 15.3 Generalized ordered outcome models
  • 15.4 Example: Synthetic data
  • 15.5 Example: Automobile data
  • 15.6 Partial proportional-odds models
  • 15.7 Continuation ratio models

16 Unordered-response family

  • 16.1 The multinomial logit model
    • 16.1.1 Example: Relation to logistic regression
    • 16.1.2 Example: Relation to conditional logistic regression
    • 16.1.3 Example: Extensions with conditional logistic regression
    • 16.1.4 The independence of irrelevant alternatives
    • 16.1.5 Example: Assessing the IIA
    • 16.1.6 Interpretation of coefficients
    • 16.1.7 Example : Medical admissions - introduction
    • 16.1.8 Example : Medical admissions - summary
  • 16.2 The multinomial probit models
    • 16.2.1 Example : A comparison of the models
    • 16.2.2 Example : comparing probit and multinomial probit
    • 16.2.3 Example : Concluding remarks

Part VI - Extensions to the GLM

17 Extending the likelihood

  • 17.1 The quasi-likelihood
  • 17.2 Example: Wedderburn's leaf blotch data
  • 17.3 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
    • 17.3.2 Conditional fixed effects estimators
  • 18.4 Random effects
    • 18.4.1 Maximum likelihood estimation
    • 17.4.2 Gibbs sampling
  • 18.5 GEEs
  • 18.6 Other models

Part VII - Stata Software

19 Programs for Stata

  • 19.1 The glm command
    • 19.1.1 Syntax
    • 19.1.2 description
    • 19.1.3 Options
  • 19.2 the predict command after glm
    • 19.2.1 Syntax
    • 19.2.2 Options
  • 19.3 User-written programs
    • 19.3.1 Global macros available for user-written programs
    • 19.3.2 User-written variance functions
    • 19.3.3 User-written programs for link functions
    • 19.3.4 User-written programs for Newey–West weights
  • 19.4 Remarks
    • 19.4.1 Equivalent comments
    • 19.4.2 Special comments on family (Gaussian) models
    • 19.4.3 Special comments on family (binomial) models
    • 19.4.4 Special comments on family (nbinomial) models
    • 19.4.5 Special comment on family (gamma) link (log) models

A Tables
References
Author index
Subject index