After fitting a logistic regression model in R using model <- glm(y~x,family='binomial') I can obtain the confidence intervals for the fitted coefficients using confint(model), but I want to know how to manually compute these values. In the case of a linear model lin_mod <- lm(y~x) I can just do the following to obtain a 95% confidence interval for the slope coefficient:
CI_lower <- coefficients(lin_mod)[2] - 1.96*summary(lin_mod)$coefficients[2,2]
CI_upper <- coefficients(lin_mod)[2] + 1.96*summary(lin_mod)$coefficients[2,2]
Where coefficients(lin_mod)[2] is the estimated value of the coefficient, and summary(lin_mod)$coefficients[2,2] is corresponding standard error.
However when I use this same process to compute the confidence interval of the fitted coefficients of a logistic regression, the values don't agree with the output from confint. Below is an example using some randomly generated data:
x <- rnorm(n=100, mean=5, sd=2)
y_prob <- plogis(x, location=5, scale=1)
y <- sapply(y_prob, function(p) rbinom(1, 1, p))
model <- glm(y~x, family='binomial')
summary(model)$coefficients
# Estimate Std. Error z value Pr(>|z|)
# (Intercept) -3.8998231 0.8838826 -4.412150 1.023490e-05
# x 0.7963213 0.1746632 4.559183 5.135303e-06
CI_lower <- coefficients(model)[2] - 1.96*summary(model)$coefficients[2,2] # = 0.4539815
CI_upper <- coefficients(model)[2] + 1.96*summary(model)$coefficients[2,2] # = 1.138661
confint(model)
# 2.5 % 97.5 %
# (Intercept) -5.8044657 -2.313925
# x 0.4843258 1.173998
As you can see, manually computing the 95% CI around the x-coefficient yielded (0.4539815,1.138661) whereas computing it using confint yielded (0.4843258,1.173998). So my question is, how is confint computing this confidence interval, and why does my estimate differ? From some additional tests on larger samples I can see that the two estimates converge in the large-N limit, but I'm interested in what's going on for small N, in particular why the CI produced by confint is not symmetric about the coefficient estimate.
