Effect of priors in logistic regression. See Chapter 13 in Regression and Other Stories.

Load packages

library("arm")
library("rstanarm")

Define a function running glm and stan_glm with simulated data

Arguments are the number of simulated observations, and prior parameters a and

bayes_sim <- function(n, a=-2, b=0.8){
  x <- runif(n, -1, 1)
  z <- rlogis(n, a + b*x, 1)
  y <- ifelse(z>0, 1, 0)
  fake <- data.frame(x, y)
  glm_fit <- glm(y ~ x, family = binomial(link = "logit"), data = fake)
  stan_fit <- stan_glm(y ~ x, family = binomial(link = "logit"),
     prior=normal(0.5, 0.5, autoscale=FALSE), data = fake)
  display(glm_fit, digits=1)
  print(stan_fit, digits=1)
}

Fit models to an increasing number of observations

set.seed(363852)
bayes_sim(10)

SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 1).
Chain 1: 
Chain 1: Gradient evaluation took 2.1e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.21 seconds.
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SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 2).
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SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 3).
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SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 4).
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glm(formula = y ~ x, family = binomial(link = "logit"), data = fake)
            coef.est coef.se
(Intercept) -1.9      1.2   
x            1.5      1.7   
---
  n = 10, k = 2
  residual deviance = 8.9, null deviance = 10.0 (difference = 1.1)
stan_glm
 family:       binomial [logit]
 formula:      y ~ x
 observations: 10
 predictors:   2
------
            Median MAD_SD
(Intercept) -1.5    0.8  
x            0.6    0.5  

------
* For help interpreting the printed output see ?print.stanreg
* For info on the priors used see ?prior_summary.stanreg
bayes_sim(100)

SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 1).
Chain 1: 
Chain 1: Gradient evaluation took 1.4e-05 seconds
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SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 2).
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glm(formula = y ~ x, family = binomial(link = "logit"), data = fake)
            coef.est coef.se
(Intercept) -1.9      0.3   
x            1.3      0.6   
---
  n = 100, k = 2
  residual deviance = 75.8, null deviance = 81.0 (difference = 5.2)
stan_glm
 family:       binomial [logit]
 formula:      y ~ x
 observations: 100
 predictors:   2
------
            Median MAD_SD
(Intercept) -1.8    0.3  
x            0.8    0.4  

------
* For help interpreting the printed output see ?print.stanreg
* For info on the priors used see ?prior_summary.stanreg
bayes_sim(1000)

SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 1).
Chain 1: 
Chain 1: Gradient evaluation took 7.1e-05 seconds
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glm(formula = y ~ x, family = binomial(link = "logit"), data = fake)
            coef.est coef.se
(Intercept) -2.0      0.1   
x            1.0      0.2   
---
  n = 1000, k = 2
  residual deviance = 735.5, null deviance = 769.0 (difference = 33.5)
stan_glm
 family:       binomial [logit]
 formula:      y ~ x
 observations: 1000
 predictors:   2
------
            Median MAD_SD
(Intercept) -2.0    0.1  
x            0.9    0.2  

------
* For help interpreting the printed output see ?print.stanreg
* For info on the priors used see ?prior_summary.stanreg
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