Problem Set 7

Problem 1

  1. Define statistical power in your own words.
  2. Explain the relationship between Type I error (\(\alpha\)), Type II error (\(\beta\)), and power.

Problem 2

2a.

Simulate power for different scenarios:

# Complete this code to simulate power for different sample sizes

simulate_power <- function(true_effect, n, sigma = 1, alpha = 0.05, n_sim = 1000) {
  # Your code here:
  # 1. Simulate n_sim datasets with given parameters
  # 2. For each dataset, run linear regression
  # 3. Calculate proportion of simulations where p < alpha
  # 4. Return power estimate
}

# Test for different sample sizes
sample_sizes <- c(50, 100, 200, 400, 800)
true_effect <- 0.2

# Create a data frame with power estimates for each sample size
power_results <- data.frame()
for (n in sample_sizes) {
  # Your code here
}

# Create visualization
library(ggplot2)
# Plot power vs sample size

Questions: 1. What sample size is needed to achieve 80% power for detecting an effect of 0.2? 2. How does changing the true effect size to 0.4 affect the required sample size? 3. What happens to power if you double the variance (sigma)?

2b.

Questions: 1. What is the “winner’s curse” and why does it occur? 2. How does sample size affect the magnitude of the winner’s curse?

Problem 3

3a.

  1. Define and distinguish between:
    • Moderator variable
    • Mediator variable
  2. Draw path diagrams for both (like in the slides)
  3. Provide a political science example of each

3b.

Using the QOG data from the slides:

library(rqog)
library(dplyr)
library(ggplot2)

# Load and prepare data
qog_data <- read_qog(which_data = "standard", data_type = "time-series")

# Create analysis dataset
analysis_data <- qog_data %>%
  select(
    country = cname,
    year = year,
    democracy = vdem_libdem,
    gdp_pc = gle_cgdpc,
    colonial = ht_colonial
  ) %>%
  filter(!is.na(democracy), !is.na(gdp_pc), !is.na(colonial)) %>%
  group_by(country) %>%
  filter(year == max(year)) %>%
  ungroup() %>%
  mutate(
    log_gdp = log(gdp_pc),
    colonized = ifelse(colonial > 0, 1, 0)
  )

# Your tasks:
# 1. Run three models:
#    a. Main effects only: democracy ~ log_gdp + colonized
#    b. With interaction: democracy ~ log_gdp * colonized
#    c. Alternative parameterization (if needed)

# 2. Calculate and interpret:
#    a. The marginal effect of log_gdp when colonized = 0
#    b. The marginal effect of log_gdp when colonized = 1
#    c. Test whether these effects are statistically different

# 3. Create visualization:
#    a. Plot with two regression lines (one for each colonized status)
#    b. Include confidence bands
#    c. Add appropriate labels and title

Questions: 1. How does the relationship between GDP and democracy differ between former colonies and never-colonized countries? 2. Is the interaction statistically significant? What does this mean substantively? 3. What are the intercepts for each group, and how do you interpret them?