Goal: After completing this lab, you should be able to…

• Use simulation to verify significance levels.
• Use simulation to estimate power.

In this lab we will use, but not focus on…

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# Exercise 0 - Verifying $$\alpha$$, Power

The following function will be used throughout this lab:

sim_p_value = function(mu = 0, sigma = 1, n = 25, mu_0 = 0) {

# sample from population
sample_data = rnorm(n = n, mean = mu, sd = sigma)

# calculate statistics
x_bar = mean(sample_data)
s = sd(sample_data)
t = (x_bar - mu_0) / (s / sqrt(n))

# calculate p-value
p_value = 2 * pt(-abs(t), df = n - 1)

# return p-value
p_value
}

This function simulates p-values for a two-sided, one sample $$t$$-test. That is, we assume that the data is sampled from a normal distribution, $$N(\mu, \sigma^2)$$, and we are performing the test:

$H_0: \mu = \mu_0 \text{ vs } H_1: \mu \neq \mu_0$

The function takes as input four arguments:

• mu: The true value of $$\mu$$ in the population
• sigma: The true value of $$\sigma$$ in the population
• n: The sample size
• mu_0: The hypothesized value of $$\mu$$ which we call $$\mu_0$$

Let’s run the function and discuss what is happening.

set.seed(1)
sim_p_value(mu = 10, sigma = 3, n = 25, mu_0 = 13)
##  0.0002035751

First, internally the function samples 25 observations from a normal distribution with a mean of 10 and a standard deviation of 3. Then, the function calculates and returns the p-value for testing

$H_0: \mu = 13 \text{ vs } H_1: \mu \neq 13$

Note that this is a “small” p-value, so with a significance level of say, $$\alpha = 0.05$$ we would reject $$H_0$$. (Which makes sense, the true value of $$\mu$$ is not 13!)

Recall that $$\alpha$$, the significance level is the probably of rejecting the null hypothesis when it is true.

$\alpha = P(\text{Reject } H_0 \mid H_0 \text{ True})$

Lets verify the significance level of the test

$H_0: \mu = 0 \text{ vs } H_1: \mu \neq 0$

The true distribution will be normal with a mean of 0 and a standard deviation of 1. We will use a sample size of 25. We will simulate this test 25000 times and use the simulated p-values to estimate the true $$\alpha$$ of the test.

set.seed(42)
p_values = replicate(25000, sim_p_value(mu = 0, sigma = 1, n = 25, mu_0 = 0))
head(p_values)
##  0.4798228 0.1843076 0.4695465 0.7457961 0.4990247 0.3441031

Here we use the replicate() function to repeatedly run the sim_p_value() function. This avoids needing to write a for loop.

Notice that these p-values appear roughly uniform.

hist(p_values, probability = TRUE, col = "darkgrey",
main = "Histogram of P-Values, Null True", xlab = "p-values")
box()
grid()