Adding uncorrelated variables to glm increases AIC

Question

I am trying to understand how AIC works. I am using data from this tutorial: https://www.jaredknowles.com/journal/2013/11/25/getting-started-with-mixed-effect-models-in-r

library(lme4) # load library
library(arm) # convenience functions for regression in R
lmm.data <- read.table("http://bayes.acs.unt.edu:8083/BayesContent/class/Jon/R_SC/Module9/lmm.data.txt",
                       header=TRUE, sep=",", na.strings="NA", dec=".", strip.white=TRUE)

When fitting the first model we get this AIC:

MLexamp <- glm(extro ~ open + agree + social, data=lmm.data)
AIC(MLexamp)
[1] 8774.291

From my understanding, if I add a variable which is totally uncorrelated to the model, the AIC should compensate the overfitting to stay the same in mean, but it appears to correct more than that:

res.rand <- replicate(1000, {
  lmm.data$rand.cont <- rnorm(nrow(lmm.data))
  list(aic = AIC(glm(extro ~ open + agree + social + rand.cont, data=lmm.data )),
       adj.r2 = summary(lm(extro ~ open + agree + social + rand.cont, 
                           data=lmm.data))$adj.r.squared)
}, simplify=F)

mean(sapply(res.rand, "[[", "aic"))
[1] 8775.331

sd(sapply(res.rand, "[[", "aic"))
[1] 1.267697

The AIC is in mean 1 point higher than in the first model.

If I estimate adjusted R squared from lm, adding the uncorrelated variable has approximately no effect in mean:

OLSexamp <- lm(extro ~ open + agree + social, data = lmm.data)
summary(OLSexamp)$adj.r.squared
[1] -0.001984873

mean(sapply(res.rand, "[[", "adj.r2"))
[1] -0.00202206

sd(sapply(res.rand, "[[", "adj.r2"))
[1] 0.001057333

Can you figure out what I missed?

Answer 1

"From my understanding, if I add a variable which is totally uncorrelated to the model, the AIC should compensate the overfitting to stay the same in mean, but it appears to correct more than that:"

This is incorrect in multiple ways. AIC does not 'compensate' anything.

As you add more parameters to a model, its $R^2$ will increase. At some point, though, you will start to overfit your data, meaning the continued increase in $R^2$ is misleading (predictive performance on a holdout dataset gets worse beyond this point, even though the $R^2$ continues to increase). AIC is a tool that can be used to diagnose this misleading increase in model complexity. It is a number that quantifies relative model fit, and it works in part by penalising models for being more complex i.e. having more parameters. As long as an additional parameter adds some degree of explanatory/predictive power to the model, the AIC will improve (decrease) in spite of the parameter penalty.

And if you add an uncorrelated variable as a predictor, you are making the model more complex while adding essentially zero explanatory/predictive power to it. So the AIC gets worse (increases). Which is exactly what it is designed to do.