Week 8 Worksheet Exercises

Study design: Simulation-based power analysis for study design

Aims

By the end of the session you will learn how to perform:

  • Simulation of random variables
  • Power analysis via simulation

Setup

Create a worksheet document

In Week 1 you set up R and RStudio, and an RProject folder (we called it “HSS8005_labs”) with an .R script and a .qmd or .Rmd document in it (we called these “Lab_1”). Ideally, you saved this on a cloud drive so you can access it from any computer (e.g. OneDrive). You will be working in this folder. If it’s missing, complete Exercise 3 from the Week 1 Worksheet.

Create a new Quarto markdown file (.qmd) for this session (e.g. “Lab_7.qmd”) and work in it to complete the exercises and report on your final analysis.

R packages

Code
# Just in case we get errors asking that the package repositories be explicitly set when installing new packages:

options(repos = list(CRAN = "http://cran.rstudio.com/"))

# Install and load required packages

# We can use the {pacman} package to easily install and load several packages:
# ({pacman} itself may need installing if not yet installed)

pacman::p_load(
  tidyverse, sjlabelled, easystats, 
  ggformula, ggeffects, marginaleffects, 
  modelsummary, gtsummary,
  MASS                           # for the function `mvrnorm` to generate variables with pre-specified covariance structure
  )

Exercise 1: basic data simulation

Code
rand_norms_10000 <- rnorm(n = 10000, mean = 0, sd = 1)

hist(rand_norms_10000, xlab = "Random value (X)", col = "grey",
     main = "", cex.lab = 1.5, cex.axis = 1.5)

  • Compare the above distribution with a normal distribution that has a standard deviation of 2 instead of 1.

  • Sample 10,000 new values in rnorm with sd = 2 instead of sd = 1 and create a new histogram with hist.

  • To see what the distribution of sampled data might look like given a low sample size (e.g., 10), repeat the process of sampling from rnorm(n = 10, mean = 0, sd = 1) multiple times and look at the shape of the resulting histogram.

Exercise 2: Simple randomised experiment

Work through Example 1 in the Notes, making the following adjustments:

  • change the seed number for the simulations to 8005
  • draw 300 samples instead of 100
  • once you have done the exercise with a normally distributed grade outcome, try another simulation for a Fail/Pass binary outcome and adjust the model functions accordingly

Exercise 3: (advanced) Multivariate distributions

Work through Example 2 in the Notes, and as an exercise, try to expand the work in Exercise 2 above by simulating an additional covariate and modelling joint effects on that data.