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.