Standard errors for lm and glm

lm_glm.R

# From https://stats.stackexchange.com/questions/224045/using-lm-for-2-sample-proportion-test

## prop.test gives pooled 2-sample proportion result
## glm w/ binomial family gives unpooled 2-sample proportion result
## lm and glm w/ gaussian family give unknown result

library(dplyr)
library(broom)
set.seed(12345)

## set up dataframe -------------------------
n_A <- 5000
n_B <- 5000

outcome <- rbinom(
  n = n_A + n_B,
  size = 1,
  prob = 0.5
)
treatment <- c(
  rep("A", n_A),
  rep("B", n_B)
)

df <- tbl_df(data.frame(outcome = outcome, treatment = treatment))


## by hand, 2-sample prop tests ---------------------------------------------
p_A <- sum(df$outcome[df$treatment == "A"])/n_A
p_B <- sum(df$outcome[df$treatment == "B"])/n_B

p_pooled <- sum(df$outcome)/(n_A + n_B)
z_pooled <- (p_B - p_A) / sqrt( p_pooled * (1 - p_pooled) * (1/n_A + 1/n_B) )
pvalue_pooled <- 2*(1-pnorm(abs(z_pooled)))

z_unpooled <- (p_B - p_A) / sqrt( (p_A * (1 - p_A))/n_A + (p_B * (1 - p_B))/n_B )
pvalue_unpooled <- 2*(1-pnorm(abs(z_unpooled)))


## using prop.test --------------------------------------
res_prop_test <- tidy(prop.test(
  x = c(sum(df$outcome[df$treatment == "A"]), 
        sum(df$outcome[df$treatment == "B"])),
  n = c(n_A, n_B),
  correct = FALSE
))
res_prop_test # same as pvalue_pooled
all.equal(res_prop_test$p.value, pvalue_pooled)
# [1] TRUE


# using glm with identity link -----------------------------------
res_glm_binomial <- df %>%
  do(tidy(glm(outcome ~ treatment, family = binomial(link = "identity")))) %>%
  filter(term == "treatmentB")
res_glm_binomial # same as p_unpooled
all.equal(res_glm_binomial$p.value, pvalue_unpooled)
# [1] TRUE


## glm and lm gaussian --------------------------------

res_glm <- df %>%
  do(tidy(glm(outcome ~ treatment))) %>%
  filter(term == "treatmentB")
res_glm 
all.equal(res_glm$p.value, pvalue_unpooled)
all.equal(res_glm$p.value, pvalue_pooled)

res_lm <- df %>%
  do(tidy(lm(outcome ~ treatment))) %>% 
  filter(term == "treatmentB")
res_lm
all.equal(res_lm$p.value, pvalue_unpooled)
all.equal(res_lm$p.value, pvalue_pooled)

all.equal(res_lm$p.value, res_glm$p.value)
# [1] TRUE

# Explore differences

## calculate lm standard error --------------------------------

res_lm_nobroom <- lm(outcome ~ treatment, data = df)

y <- res_lm_nobroom$model$outcome
y_hat_lm <- res_lm_nobroom$fitted.values
x <- ifelse(res_lm_nobroom$model$treatment == "A", 0, 1)
x_bar <- mean(x)
n <- nrow(df)

se_lm <- sqrt(sum((y - y_hat_lm)^2) / (n - 2)) / sqrt(sum((x - x_bar)^2))

## calculate glm_binomial standard error --------------------------------

res_glm_binomial_nobroom <- glm(outcome ~ treatment, data = df, family = binomial(link = "identity"))

y_hat_glm_binomial <- res_glm_binomial_nobroom$fitted.values
se_glm_binomial <- sqrt(sum((y - y_hat_glm_binomial)^2) / (n)) / sqrt(sum((x - x_bar)^2))

## calculate se_unpooled --------------------------------

se_unpooled <- sqrt( (p_A * (1 - p_A))/n_A + (p_B * (1 - p_B))/n_B )

## compare standard errors --------------------------------

all.equal(se_lm, res_lm$std.error) # TRUE
all.equal(se_lm, res_glm$std.error) # TRUE

all.equal(se_unpooled, res_glm_binomial$std.error) # TRUE
all.equal(se_glm_binomial, res_glm_binomial$std.error) # TRUE

all.equal(se_lm, se_glm_binomial) # FALSE
all.equal(res_lm$std.error, res_glm_binomial$std.error) # FALSE
all.equal(res_glm$std.error, res_glm_binomial$std.error) # FALSE