Type I Error in R

A Type I error, also known as a false positive or alpha error, occurs when a null hypothesis is rejected when it is actually true. In statistical hypothesis testing, the null hypothesis (H0) often represents a baseline assumption, such as no effect or no difference between groups. A Type I error is the probability of incorrectly rejecting H0 when it is true, and this probability is represented by the significance level (alpha).

In R programming, you can calculate the Type I error rate (also known as the alpha level or false positive rate) by simulating data and comparing the proportion of false positives to the total number of tests conducted.

Here’s an example of how you can calculate the Type I error rate for a t-test using R:

# Set the parameters

alpha <- 0.05

sample_size <- 30

num_simulations <- 10000

# Set the seed for reproducibility

set.seed(123)

# Initialize the counter for false positives

false_positives <- 0

# Perform the simulations

for (i in 1:num_simulations) {

# Generate two samples from the same normal 

# distribution (null hypothesis is true)

sample1 <- rnorm(sample_size, mean = 0, sd = 1)

sample2 <- rnorm(sample_size, mean = 0, sd = 1)

# Conduct a t-test

test_result <- t.test(sample1, sample2)

# Check if the p-value is less than the alpha level

if (test_result$p.value < alpha) {

false_positives <- false_positives + 1

}

}

# Calculate the Type I error rate

type1_error_rate <- false_positives / num_simulations

# Print the Type I error rate

cat("Type I Error Rate:", type1_error_rate)

Output

> # Print the Type I error rate

> cat("Type I Error Rate:", type1_error_rate)

Type I Error Rate: 0.0481

In this example, we run 10,000 simulations where we draw two samples from the same normal distribution, and conduct a t-test for each pair of samples. We count the number of times we reject the null hypothesis when it is true (false positives) and divide it by the total number of simulations to estimate the Type I error rate.

Keep in mind that this approach can be adapted for other statistical tests and scenarios as needed.

Example – 2

Here’s another example, where we calculate the Type I error rate for a chi-squared test using R:

# Set the parameters

alpha <- 0.05

num_simulations <- 10000

# Set the seed for reproducibility

set.seed(123)

# Initialize the counter for false positives

false_positives <- 0

# Define the true proportions for the null hypothesis

true_proportions <- c(0.25, 0.25, 0.25, 0.25)

# Perform the simulations

for (i in 1:num_simulations) {

# Generate a sample from a multinomial distribution with 

# the same proportions (null hypothesis is true)

sample <- rmultinom(1, size = 100, prob = true_proportions)

# Conduct a chi-squared test

test_result <- chisq.test(sample)

# Check if the p-value is less than the alpha level

if (test_result$p.value < alpha) {

false_positives <- false_positives + 1

}

}

# Calculate the Type I error rate

type1_error_rate <- false_positives / num_simulations

# Print the Type I error rate

cat("Type I Error Rate:", type1_error_rate)

Output

> # Print the Type I error rate

> cat("Type I Error Rate:", type1_error_rate)

Type I Error Rate: 0.0481

In this example, we run 10,000 simulations where we draw a sample from a multinomial distribution with the same true proportions specified in true_proportions. We conduct a chi-squared test for each sample to compare the observed frequencies to the expected frequencies under the null hypothesis. We count the number of times we reject the null hypothesis when it is true (false positives) and divide it by the total number of simulations to estimate the Type I error rate.

Example -3

Here’s another example of how to calculate the Type I error in R using a one-sample t-test:

1. Generate some sample data:

set.seed(123)

data <- rnorm(n = 100, mean = 0, sd = 1)

2. Perform the one-sample t-test and obtain the p-value:

t.test(data, mu = 0)

The output should look something like this:

The p-value is 0.5017.

3. Determine the significance level (alpha) of the test. Let’s say you choose a significance level of 0.05.

4. Compare the p-value to the significance level. If the p-value is less than or equal to the significance level, reject the null hypothesis. If the p-value is greater than the significance level, do not reject the null hypothesis.

In this case, the p-value (0.5017) is greater than the significance level (0.05), so you do not reject the null hypothesis.

Assuming the null hypothesis is true, you have made the correct decision. There is no Type I error in this case. However, if the p-value had been less than or equal to the significance level, you would have rejected the null hypothesis when it was actually true, resulting in a Type I error.

Type II Error in R

Type II error, also known as a false negative, occurs when you fail to reject the null hypothesis when it’s actually false. In hypothesis testing, this error is denoted as β (beta). To calculate Type II error in R, you need to know the effect size (difference between the null and alternative hypotheses), sample size, standard deviation, and the desired significance level (alpha).

Here’s an example code to calculate Type II error in R:

# Install and load required packages

if (!require(pwr))

install.packages("pwr")

library(pwr)

# Parameters

effect_size <-

0.5 # The difference between null and alternative hypotheses

sample_size <- 100 # The number of observations in each group

sd <- 15 # The standard deviation

alpha <- 0.05 # The significance level

# Calculate Type II Error

pwr_result <-

pwr.t.test(

n = sample_size,

d = effect_size / sd,

sig.level = alpha,

type = "two.sample",

alternative = "two.sided"

)

type_II_error <- 1 - pwr_result$power

# Print Type II Error

print(type_II_error)

In this example, we are using the pwr package to calculate the power of the test, and then subtracting it from 1 to obtain the Type II error (β). Remember to adapt the parameters according to your specific problem.

Output

> # Print Type II Error

> print(type_II_error)

[1] 0.9436737

Another example code to calculate Type II error in R

To calculate the Type II error in R, you need to perform a power analysis, which requires several inputs such as sample size, effect size, significance level, and power. Here is an example of how to do it:

# define the sample size

n <- 50

# define the effect size

d <- 0.5

# define the significance level

alpha <- 0.05

# define the power

power <- 0.8

# calculate the critical t-value for the given 

# significance level and degrees of freedom

df <- n - 1

t_crit <- qt(1 - alpha/2, df)

# calculate the non-centrality parameter

ncp <- d * sqrt(n) / sqrt(1 + d^2 / (2*(n-1)))

# calculate the Type II error rate

1 - pt(t_crit, df, ncp, lower.tail=FALSE)

In this example, we first defined the sample size, effect size, significance level, and power. We then calculated the critical t-value using the qt function, which returns the t-value corresponding to the given significance level and degrees of freedom. We then calculated the non-centrality parameter using the formula ncp = d * sqrt(n) / sqrt(1 + d^2 / (2*(n-1))), which represents the distance between the null hypothesis and the alternative hypothesis in terms of standard errors. Finally, we used the pt function to calculate the probability of observing a t-value greater than the critical t-value under the alternative hypothesis, which represents the Type II error rate.