Generalized Linear Models

Andy Grogan-Kaylor

25 Oct 2020 08:12:02

Introduction

This discussion closely follows the Stata help for the generalized linear model, see help glm.

Briefly, per Stata documentation, in the generalized linear model framework, we consider models of the form:

\[ g(E(y)) = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... \]

where y is distributed as \(F\) i.e. \(y \sim F\).

\(g\) is called the link function; \(F\) is called the distribution. Hence¹:

Link	Distribution	Standard Command	glm
identity	normal	`regress y x`	`glm y x, link(identity) family(gaussian)`
logit	bernoulli	`logit y x`	`glm y x, link(logit) family(binomial)`
probit	bernoulli	`probit y x`	`glm y x, link(probit) family(binomial)`

Though not the subject of this discussion, it is worth noting here, that count regression models follow a similar logic.

Link	Distribution	Standard Command	glm
log	poisson	`poisson y x`	`glm y x, link(log) family(poisson)`
log	nbinomial	`nbinomial y x`	`glm y x, link(log) family(nbinomial)`

Palmer Penguins

Thes examples use the Palmer Penguins data set: https://github.com/allisonhorst/palmerpenguins.

. clear all

. use penguins.dta, clear

Palmer Penguins Illustration from @allison_horst

Models

I use the Stata prefix quietly to run the models without output. I then store the results using estimates store. Finally, I present all the results together in compact form using estimates table.

What Predicts Culmen Depth?

. quietly: regress culmen_depth_mm body_mass_g flipper_length_mm

. est store usual_OLS // store estimates usual OLS

. quietly: glm culmen_depth_mm body_mass_g flipper_length_mm, link(identity) family(gaussian)

. est store glm_OLS // store estimates glm OLS

What Predicts That A Penguin Lives on Dream Island?

. tabulate island

     island │      Freq.     Percent        Cum.
────────────┼───────────────────────────────────
     Biscoe │        168       48.84       48.84
      Dream │        124       36.05       84.88
  Torgersen │         52       15.12      100.00
────────────┼───────────────────────────────────
      Total │        344      100.00

. generate dream = island == 2

. label variable dream "Penguin Lives on Dream Island"

. quietly: logit dream flipper_length_mm body_mass_g

. est store usual_logit // store estimates usual logit

. quietly: glm dream flipper_length_mm body_mass_g, link(logit) family(binomial)

. est store glm_logit // store estimates glm logit

. quietly: probit dream flipper_length_mm body_mass_g

. est store usual_probit // store estimates usual probit

. quietly: glm dream flipper_length_mm body_mass_g, link(probit) family(binomial)

. est store glm_probit // store estimates glm probit

Results

. est table usual_OLS glm_OLS usual_logit glm_logit usual_probit glm_probit, star

─────────────┬────────────────────────────────────────────────────────────────────────────────
    Variable │   usual_OLS        glm_OLS       usual_logit      glm_logit     usual_probit   
─────────────┼────────────────────────────────────────────────────────────────────────────────
_            │
 body_mass_g │  .00037535                                                                     
flipper_le~m │  -.1006443***                                                                  
       _cons │  35.794997***                                                                  
─────────────┼────────────────────────────────────────────────────────────────────────────────
culmen_dep~m │
 body_mass_g │                  .00037535                                                     
flipper_le~m │                  -.1006443***                                                  
       _cons │                  35.794997***                                                  
─────────────┼────────────────────────────────────────────────────────────────────────────────
dream        │
flipper_le~m │                                  -.0160116       -.0160116      -.01114532     
 body_mass_g │                                  -.0013785***    -.0013785***   -.00082575***  
       _cons │                                   8.193819**      8.193819**     5.2018764**   
─────────────┴────────────────────────────────────────────────────────────────────────────────
                                                      legend: * p<0.05; ** p<0.01; *** p<0.001


─────────────┬────────────────
    Variable │  glm_probit    
─────────────┼────────────────
_            │
 body_mass_g │                
flipper_le~m │                
       _cons │                
─────────────┼────────────────
culmen_dep~m │
 body_mass_g │                
flipper_le~m │                
       _cons │                
─────────────┼────────────────
dream        │
flipper_le~m │ -.01114532     
 body_mass_g │ -.00082575***  
       _cons │  5.2018764**   
─────────────┴────────────────
legend: * p<0.05; ** p<0.01; *** p<0.001

This table and the table below draw heavily on the Stata documentation.↩︎