Shiny Project Presentation

To explore the interaction between type I error (\( \alpha \)) and type II error (\( \beta \)), I built a shiny app that allows the user to manipulate \( \sigma \), \( \mu_a \), n, and \( \alpha \). The app then computes the resulting beta (and power) and shades in the appropriate area under each curve to represent alpha and beta.

• Type I error is incorrectly rejecting a true null hypothesis
  • such a “false positive” is denoted by \( \alpha \)
    • detecting an effect that is not present
  • \( \alpha \) = P(type I error) = P(reject \( H_0 \)|\( H_0 \) is true)
• Type II error is failing to reject a false null hypothsis
  • such a “false negative” is denoted by \( \beta \)
    • failing to detect an effect that is present
  • \( \beta \) = P(type II error) = P(FTR \( H_0 \)|\( H_0 \) is false)
• How about confidence and power?
  • confidence = 1 - \( \alpha \) = rejecting an effect that is not present
  • power = 1 - \( \beta \) = detecting an effect that is present

• \( \alpha \) is usually controlled by the experimenter
  • represents the amount of error that is tolerable
  • affected by:
    • sample size (n)
    • spread in the data (\( \sigma \))
• \( \beta \) is much harder to control
  • affected by the true (but unkown) alt hypotheis (\( \mu_a \))
    • the greater \( \mu_a \) is from the null hypothesis (\( \mu_0 \)) the better
    • it's easier to detect a bigger effect (\( \Delta\mu \))
• \( \alpha \) and \( \beta \) work against each other
  • to increase both, increase n, \( \sigma \), and/or \( \Delta\mu \)

• Type I error (\( \alpha \)) is controlled by the experimenter
  • \( \alpha \) = area under red curve to the right of the line
• Type II error (\( \beta \)) depends on \( \sigma \), \( \mu_a \), n, and \( \alpha \)
  • \( \beta \) = area under blue curve to the left of the line
• In the app, line location is set by the \( \alpha \) slider
  • \( \beta \) = lower tail of a pnorm function call
  • for example:
    • if \( \mu_0 \) = 30, \( \sigma \) = 4, \( \mu_a \) = 32, n = 16, and \( \alpha \) = 0.05
    • then \( \beta \) = 0.361
    • and power = 63.9 %