Frequency of Error Type 1 in R: A Detailed Explanation

In this article, we will explore the concept of type I error and how to calculate its frequency in R using a statistical model.

What is a Type I Error?

A type I error occurs when a true null hypothesis is incorrectly rejected. In other words, it happens when we conclude that there is an effect or difference when, in fact, there is none. The probability of committing a type I error is denoted by α (alpha) and is typically set to 0.05.

Background

The problem presented in the Stack Overflow question involves estimating the frequency of type I errors in a statistical model using Monte Carlo simulations. The model estimates the relationship between two variables, X and Y, using Ordinary Least Squares (OLS). We will delve into the details of this model and explain how to calculate the frequency of type I errors.

Statistical Model

The given code implements an OLS regression model with two variables: vetor_x and vetor_y. The goal is to estimate the coefficients (Beta1 and Beta2) that represent the relationships between these variables. However, we are not interested in estimating the coefficients themselves but rather in calculating the frequency of type I errors.

To do this, we need to simulate a large number of datasets with different values for vetor_x and vetor_y. For each simulation, we will estimate the coefficients using OLS and then calculate the p-value associated with each coefficient. If the p-value is less than the significance level (0.05), we reject the null hypothesis that the corresponding coefficient is zero.

Monte Carlo Simulations

The code uses Monte Carlo simulations to generate multiple datasets for vetor_x and vetor_y. For each simulation, it estimates the coefficients using OLS and calculates the p-value associated with each coefficient. The frequency of type I errors is then calculated as the proportion of simulations where the p-value is less than 0.05.

Calculating Type I Error Frequency

To calculate the frequency of type I errors, we need to analyze the output of the Monte Carlo simulations. In this case, the code provides a table that shows the mean values and p-values for each simulation.

To extract the frequency of type I errors, we can look at the proportion of simulations where the p-value is less than 0.05. This can be done by summing up the number of simulations where the p-value is less than 0.05 and dividing it by the total number of simulations.

Code Explanation

The code uses several functions to perform the Monte Carlo simulations:

Estima_Beta: estimates the coefficients using OLS
Estima_T: calculates the p-values associated with each coefficient
t.test: performs a two-sided t-test to compare the observed value to the null hypothesis

The code also uses the sapply function to apply these functions to multiple simulations.

Example Code

Here is an example of how to calculate the frequency of type I errors using the provided code:

set.seed(1237)
mu <- 1
sd <- 2
trial <- 10
sim <- sapply(1:trial, function(x) {
  rnd <- rnorm(100, mu, sd)
  p <- t.test(rnd, mu = mu,
              alternative = "two.sided", conf.level = 0.95)$p.value
  c(mean = mean(rnd), pvalue=p)
})
sum(sim[,2] < 0.05) / ncol(sim)

This code generates multiple simulations and calculates the frequency of type I errors as the proportion of simulations where the p-value is less than 0.05.

Conclusion

In this article, we explored the concept of type I error and how to calculate its frequency in R using a statistical model. We discussed the details of the OLS regression model and explained how to perform Monte Carlo simulations to estimate the frequency of type I errors. The example code demonstrates how to calculate the frequency of type I errors using the provided code.

Last modified on 2023-10-15