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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) - 1.96*summary(lin_mod)$coefficients[2,2] CI_upper <- coefficients(lin_mod) + 1.96*summary(lin_mod)$coefficients[2,2]
coefficients(lin_mod) 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) - 1.96*summary(model)$coefficients[2,2] # = 0.4539815 CI_upper <- coefficients(model) + 1.96*summary(model)$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
(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.