We use the following packages:
library(plyr)
library(dplyr)
library(magrittr)
library(ggplot2)
Create a plot that demonstrates the following
“A replication of the procedure that generates a 95% confidence interval that is centered around the sample mean would cover the population value at least 95 out of 100 times” (Neyman, 1934)
Present a table containing all simulated samples for which the resulting confidence interval does not contain the population value.
NB: this is just one of many possible solutions!
We start by fixing the random seed. Otherwise your results will likely vary
set.seed(1234)
Then, we draw 100 samples from a standard normal distribution, i.e. a distribution with \(\mu=0\) and \(\sigma^2=1\), such that for any drawn sample \(X\) we could write \(X \sim \mathcal{N}(0, 1)\). No specification about the size of the samples is explicitly requested, so for computational reasons a mere 5000 cases would suffice to obtain a detailed approximation.
samples <- lapply(1:100, function(x) rnorm(5000, mean = 0, sd = 1))
We use the lapply()
function to draw the 100 samples and return the resulting output as a list. You could also use a for-loop. Further, we extract the following sources of information for each sample:
info <- function(x){
M <- mean(x)
DF <- length(x) - 1
SE <- 1 / sqrt(length(x))
INT <- qt(.975, DF) * SE
return(c("Mean" = M,
"Bias" = M - 0,
"Std.Err" = SE,
"Lower" = M - INT,
"Upper" = M + INT))
}
Then we can compute the info on each of the samples
results <- samples %>% sapply(info) %>% t()
Because object samples
is a list, we can execute funtion sapply()
to obtain a numerical object with the results of function info()
. Function \(t()\) is here used to obtain the transpose of sapply()
’s return - the resulting object has all information in the columns.
To create an indicator for the inclusion of the population value \(\mu=0\) in the confidence interval, we can add the following coverage column cov
to the data:
results <- results %>%
as.data.frame() %>%
mutate(Covered = Lower < 0 & 0 < Upper)
Converting the numerical object to an object of class data.frame
allows for a more convenient calling of elements. Now we can simply take the column means over dataframe results
to obtain the average of the estimates returned by info()
.
colMeans(results)
## Mean Bias Std.Err Lower Upper
## 0.001341244 0.001341244 0.014142136 -0.026383545 0.029066033
## Covered
## 0.950000000
We can see that 95 out of the 100 samples cover the population value.
To identify the samples for which the population value is not covered, we can use column cov
as it is already a logical evaluation.
noncovered <- results[!results$Covered, ]
To create a graph that would serve the purpose of the exercise, one could think about the following graph:
library(ggplot2)
ggplot(results, aes(y = Mean, x = 1:100, ymax = Upper, ymin = Lower,
colour = Covered)) +
geom_hline(yintercept = 0, color = "dark grey", size = 2) +
geom_pointrange() +
xlab("Simulations 1-100") +
ylab("Means and 95% Confidence Intervals") +
theme_minimal()
To just plot the 5 cases that do not overlap with the population parameter:
ggplot(noncovered, aes(y = Mean, x = 1:5, ymax = Upper, ymin = Lower)) +
geom_hline(aes(yintercept = 0), color = "dark grey", size = 2) +
geom_pointrange(col = "red") +
xlab("Simulations 1-100") +
ylab("Means and 95% Confidence Intervals") +
theme_minimal()
End of Practical