Lecture 8: Putting it all together

Question

Consider the simple linear regression model

\[Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i\]

Assumptions:

• \(\varepsilon_i \sim N(0, \sigma^2)\) (normality, constant variance, and zero mean)
• Shape (linearity)
• Independence
• Randomness

Question

\[Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i\]

Assumptions:

• \(\varepsilon_i \sim N(0, \sigma^2)\) (normality, constant variance, and zero mean)
• Shape (linearity)
• Independence
• Randomness

Question: How do we assess importance of the normality assumption?

Simulation study plan

• Aims:
• Data generation:
• Estimand/target:
• Methods:
• Performance measures:

Implementation

assess_coverage <- function(n, nsim, beta0, beta1, noise_dist){
  results <- rep(NA, nsim)

  for(i in 1:nsim){
    x <- runif(n, min=0, max=1)
    noise <- noise_dist(n)
    y <- beta0 + beta1*x + noise

    lm_mod <- lm(y ~ x)
    ci <- confint(lm_mod, "x", level = 0.95)
    results[i] <- ci[1] < beta1 & ci[2] > beta1
  }
  return(mean(results))
}

Implementation

Iterating over different distributions

set.seed(45)

noise_dists <- list(rnorm, rexp, 
                    function(m) {return(rchisq(m, df=1))})
ci_coverage <- rep(NA, length(noise_dists))

for(i in 1:length(noise_dists)){
  ci_coverage[i] <- assess_coverage(n = 100, nsim = 1000, 
                                    beta0 = 0.5, beta1 = 1, 
                                    noise_dist = noise_dists[[i]])
}
ci_coverage

[1] 0.949 0.960 0.946

R tools and topics (so far)

• Vectors and indexing; creating vectors
• Random sampling (from vectors and from distributions)
• Iteration (for loops and while loops)
• Functions; function defaults, anonymous functions, function scope
• Lists

What’s next

• Introduction to Python
• File management, version control, and GitHub
• Advanced data wrangling

Class activity

https://sta279-f23.github.io/class_activities/ca_lecture_8.html