What's Categorical Data?

• Categorical data consists of variables with a finite number of values (really, a small number of discrete values).
• It can be:
  • nominal
  • ordinal
  • interval
  • or ratio data,
• But it can't be continuous.

• Categorical data arise in a number of ways
  • Simple counts
  • Binary variables - yes/no, pass/fail, live/die
  • Unordered multinomial: christian, jew, muslim, atheist
  • Ordinal:
    • Pure: degree, complete second level, incomplete second level, primary only, no education.
    • Imperfect scale measurement: e.g., Likert's 5-point scale
    • Grouped variables: e.g., income in bands

• Some of these are interval-ratio variables:
  • Count of number of children
  • Income in bands (imprecise ratio measurement)
• and we can use appropriate summaries, such as means and correlations.

• However, the really powerful technique of Ordinary Least Squares Regression won't do:
  • OLS requires a dependent variable that is conditionally normally distributed
  • OLS may well predict impossible values - negative counts, probabilities outside the 0-1 range
• So something else is required

• Consider first count data: non-negative integers
• Where the mean of a count variable is sufficiently large, its distribution approximates the normal and OLS will do okay
• Counts are distributed as Poisson:
  
  $$P(Y=y) = \frac{e^{-\mu}\mu^y}{y!}$$
  
  

• This is a discrete function
• Asymmetric but approximated by the normal distribution for large values of $\mu$
• Standard deviation is $\sqrt \mu$