My (edited) questions are:
- Does glm() provide Std.Errors for the transformed or untransformed parameters?
- If Std.Errors are provided on the scale of the link function, how to obtain them for the parameters on the original (i.e. untransformed) scale?
As an example, given the dataset:
set.seed(100) dd <- data.frame( group = c(rep("group1", 100),rep("group2", 100)), values = c(rpois(n=100, lambda=2), rpois(n=100, lambda=7)) ) 
I computed the following summary statistics:
library(doBy) logmean <- function(x){ log(mean(x)) } ss <- summaryBy(values ~ group, data=dd, FUN=c(length, logmean, mean, var, sd) ) ss$SE.normal <- sqrt(ss$values.var/(ss$values.length-1)) ss$sd.Poiss <- sqrt(ss$values.mean) ss$se.Poiss <- sqrt(ss$values.mean/ss$values.length) ss <- cbind(ss[,"group"], round(ss[,-1],3)) names(ss) <- c("group", "N", "log.mean", "mean", "variance", "sd", "SE","sd.Poiss","se.Poiss") Object ss looks like this:
group N log.mean mean variance sd SE sd.Poiss se.Poiss 1 group1 100 0.723 2.06 1.653 1.286 0.129 1.435 0.144 2 group2 100 1.937 6.94 8.340 2.888 0.290 2.634 0.263 I then compared these results with those computed by glm():
glm1 <- glm(values ~ group, data=dd, family="poisson"(link = "log")) ss$log.mean[1] coincides with glm1$coef[1] and ss$log.mean[2] coincides with glm1$coef[1] + glm1$coef[2], but the Std.Errors provided by glm() do not have any correspondence with those I calculated in table ss:
summary(glm1)$coefficients[, 2] (Intercept) groupgroup2 0.06967330 0.07934287 [Note: while mean = exp(log(mean)), the Std.Error of the mean does not equals to exp(summary(glm1)$coefficients[,2])].
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Best Answer
This can be done using the package 'emmeans'
library(emmeans) df = emmeans (glm1, ~ group, type = "response") data.frame(df)[,3] [1] 0.1435270 0.2634388