Confidence intervals for the mean for non-normal data. See https://tonyladson.wordpress.com/2016/02/05/confidence-interval-for-the-mean-of-non-normal-data/

CI_NonNormal.R

# Confidence intervals for the mean for non-normal data
# See https://tonyladson.wordpress.com/2016/01/11/plotting-water-quality-samples-on-a-hydrograph/


# Modified Cox function for calculating confidence intervals for the mean
# Based on Olsson, 2005
# http://www.amstat.org/publications/jse/v13n1/olsson.html

# x is a set of log-normal data

ModifiedCox <- function(x){
  
  n <- length(x)
  y <- log(x)
  y.m <- mean(y)
  y.var <- var(y)
  
  my.t <- qt(0.975, df = n-1)
  
  my.mean <- y.m + y.var/2
  upper <- y.m + y.var/2 + my.t*sqrt(y.var/n + y.var^2/(2*(n - 1)))
  lower <- y.m + y.var/2 - my.t*sqrt(y.var/n + y.var^2/(2*(n - 1)))
  
  return(list(upper = exp(upper), mean = exp(my.mean), lower = exp(lower)))
  
}


Singleton <- c(76.26, 171.87, 218.21, 668.79, 1374.42, 124.12,
               276.30, 895.50, 1374.42, 280.18, 202.62, 4052.42,
               2323.77, 2536.31, 3315.62, 1232.73,1391.43,
               12525.66, 1099.54, 447.75, 478.92, 180.52, 164.36,
               229.54, 2125.4, 966.35, 2751.68, 49.03, 79.51, 912.5, 926.67)


ModifiedCox(Singleton)

$upper
[1] 2973.871

$mean
[1] 1401.69

$lower
[1] 763.9697

# Bootstrapping
# See http://stats.stackexchange.com/questions/112829/how-do-i-calculate-confidence-intervals-for-a-non-normal-distribution/193940#193940

# function to obtain the mean
Bmean <- function(data, i) {
  d <- data[i] # allows boot to select sample 
  return(mean(d))
} 

# bootstrapping with 1000 replications 
results <- boot(data=Singleton, statistic=Bmean, R=1000)

# view results
results 
plot(results)

# get 95% confidence interval 
boot.ci(results, type=c("norm", "basic", "perc", "bca"))