**List of figures**

**List of tables**

**List of listings**

**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 R

^{2} 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 R

^{2} interpretations

4.6.3.1 Efron’s pseudo-R^{2}

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 R

^{2} measures

4.6.4.1 The count R^{2}

4.6.4.2 The adjusted count R^{2}

4.6.4.3 Veall and Zimmermann R^{2}

4.6.4.4 Cameron–Windmeijer R^{2}

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 R^{2}_{Efron}
in Poisson regression

**A Tables**

**References**