AIC/AICc/BIC Formula in R for GLM

4
votes

I'm trying to check that I understand how R calculates the statistic AIC, AICc (corrected AIC) and BIC for a glm() model object (so that I can perform the same calculations on revoScaleR::rxGlm() objects - particularly the AICc, which isn't available by default)

I had understood that these were defined as follows:

let p = number of model parameters

let n = number of data points

AIC  = deviance + 2p
AICc = AIC + (2p^2 + 2p)/(n-p-1)
BIC  = deviance + 2p.log(n)

So I tried to replicate these numbers and compare them to the corresponding R function calls. It didn't work:

library(AICcmodavg) # for the AICc() function

data(mtcars)

glm_a1 <- glm(mpg ~ cyl + disp + hp + drat + wt + qsec + vs + am + gear + carb,
              data = mtcars,
              family = gaussian(link = "identity"),
              trace = TRUE)

summary(glm_a1)

n   <- nrow(glm_a1$data) # 32
p   <- glm_a1$rank # 11
dev <- glm_a1$deviance# 147.49

my_AIC  <- dev + 2 * p
my_AICc <- my_AIC + (2 * p^2 + 2 * p)/(n - p - 1)
my_BIC  <- dev + 2 * p * log(n)

AIC(glm_a1) # 163.71
my_AIC # 169.49

AICc(glm_a1) # 180.13 (from AICcmodavg package)
my_AICc # 182.69

BIC(glm_a1) # 181.30
my_BIC # 223.74

By using debug(AIC) I can see that the calculation is different. It's based on 12 parameters (one extra for the estimated dispersion/scale parameter?). Also the log likelihood is obtained using logLik() which brings back a number -69.85, which suggests to me that the model deviance would be -2*-69.85 = 139.71 (which it isn't).

Does anyone know what I've done wrong please? Thank you.

2
rglm

2 Answers

5
votes

in the extractAIC manual page

Where :

  • L is the likelihood and edf the equivalent degrees of freedom (i.e., the number of parameters for usual parametric models) of fit.
  • For generalized linear models (i.e., for lm, aov, and glm), -2log L is the deviance, as computed by deviance(fit).
  • k = 2 corresponds to the traditional AIC, using k = log(n) provides the BIC (Bayes IC) instead.

Thus

Edits following discussion in the comments and input of @user20650

  • glm_a1$ranks returns the number of fitted parameter without accounting for the fitted variance used in gaussian families.

  • ?glm states

    deviance: up to a constant, minus twice the maximized log-likelihood. Where sensible, the constant is chosen so that a saturated model has deviance zero.

    that's why -2*logLik(glm_a1) - deviance(glm_a1) = 7.78 > 0

  • summary(glm_a1) returns the following line Dispersion parameter for gaussian family taken to be 7.023544 approximately the difference between -2 log likelihood and the deviance.


library(AICcmodavg)
#> Warning: package 'AICcmodavg' was built under R version 3.6.2
#> Warning: no function found corresponding to methods exports from 'raster' for:
#> 'wkt'

data(mtcars)

glm_a1 <- glm(mpg ~ cyl + disp + hp + drat + wt + qsec + vs + am + gear + carb,
              data = mtcars,
              family = gaussian(link = "identity"),
              trace = TRUE)
#> Deviance = 147.4944 Iterations - 1
#> Deviance = 147.4944 Iterations - 2

(loglik <- logLik(glm_a1))
#> 'log Lik.' -69.85491 (df=12)

# thus the degrees of freedom r uses are 12 instead of 11

n   <- attributes(loglik)$nobs # following user20650 recommendation 
p   <- attributes(loglik)$df # following user20650 recommendation
dev <- -2*as.numeric(loglik)
my_AIC  <- dev + 2 * p
my_AICc <- my_AIC + (2 * p^2 + 2 * p)/(n - p - 1)
my_BIC  <- dev +  p * log(n)

BIC(glm_a1) 
#> [1] 181.2986
my_BIC 
#> [1] 181.2986
AIC(glm_a1)
#> [1] 163.7098
my_AIC 
#> [1] 163.7098
AICc(glm_a1)
#> [1] 180.1309
my_AICc
#> [1] 180.1309
0
votes

Function to calculate these quantities for an rxGlm() object consistent with treatment of glm() (adjusting for the "up to a constant" difference in deviance):

wrc_information_criteria <- function(rx_glm) # an object created by rxGlm()
{

  # add 1 to parameter count for cases where the GLM scale parameter needs to be estimated (notably Gamma/gaussian)
  
  extra_parameter_flag <- case_when(
    rx_glm$family$family == "gaussian" ~ 1,
    rx_glm$family$family == "Gamma" ~ 1,
    rx_glm$family$family == "poisson" ~ 0,
    rx_glm$family$family == "binomial" ~ 0,
    TRUE ~ 999999999
  )
  
  n   <- rx_glm$nValidObs
  p   <- rx_glm$rank + extra_parameter_flag 
  dev <- rx_glm$deviance
  
  cat("\n")
  cat("n       :", n, "\n")
  cat("p       :", p, "\n")
  cat("deviance:", dev, "\n")
  
  AIC  <- dev + 2 * p
  AICc <- AIC + (2 * p^2 + 2 * p)/(n - p - 1)
  BIC  <- dev + p * log(n)
  
  # make a constant adjustment to AIC/AICc/BIC to give consistency with R's built in AIC/BIC functions applied to glm objects
  # can do this because rxGlm() supplies AIC already (consistent with R/glm()) - as long as computeAIC = TRUE in the function call
  
  deviance_constant_adjustment <- rx_glm$aic[1] - AIC
  
  AIC  <- AIC  + deviance_constant_adjustment
  AICc <- AICc + deviance_constant_adjustment
  BIC  <- BIC  + deviance_constant_adjustment
  
  cat("\n")  
  cat("AIC: ", AIC , "\n")
  cat("AICc:", AICc, "\n")
  cat("BIC: ", BIC , "\n")
  
}

Let's test it...

data(mtcars)

glm_a1 <- glm(mpg ~ cyl + disp + hp + drat + wt + qsec + vs + am + gear + carb,
              data = mtcars,
              family = gaussian(link = "identity"),
              trace = TRUE)

glm_b1 <- rxGlm(mpg ~ cyl + disp + hp + drat + wt + qsec + vs + am + gear + carb,
                data = mtcars,
                family = gaussian(link = "identity"),
                verbose = 1,
                computeAIC = TRUE)

AIC(glm_a1)
AICc(glm_a1)
BIC(glm_a1)

wrc_information_criteria(glm_b1) # gives same results for glm_b1 as I got for glm_a1