Background:

• Simple Linear Regression assume errors (and response) are normal, but when normality isn't the case, predictions may fall outside of meaningful range (maybe not a big deal) and inference is not reliable (probably a bigger deal)
• Generalized Linear Models (GLM) assume some other distribution than normality, for example for:

  • Count data (e.g., number of defects on a product)

  • Skewed data (e.g., salaries)

  • Proportions

  • Labels (e.g., good/neutral/bad or yellow/blue/green)

• Generalized Linear Models (GLM) have three ingredients
  • A distribution for the response given the predictors (the random piece)
  • A linear predictor (the systematic piece)
  • A link function (the piece that random and systematic pieces)

• Use the R-square to compare models within a distribution. 

• Use AICc and BIC (information criteria.) to compare between distributions

  • AICc and BIC estimate the Kullback-Leibler divergence, which is the distance from the fitted model to the truth

  • Use them to compare models within the same distribution and across different distributions

  • Rule of thumb: AIC tends to overfit and BIC tends to underfit

• General guidelines for choosing Distributions for Continuous Response

  • Do we have negative values? Use normal

  • Is it bound to (0,1)?  Use beta

  • Does variance increase with the mean?  Use gamma, Weibull, lognormal

  • Is it time-to-event/censored? Probably use Weibull or lognormal

  • A pretty good catch-all? Use normal

  • Do we suspect that we have outliers? Use Cauchy or t(5)

• Choosing Distribution when response isn’t numeric
  • Is it two-level? Use the binomia; (e.g., Yes/No or A/B)
  • Is it 3+ levels and order matters? Use Ordinal logistic. (eg., Low/Medium/High or Small/Medium/Large
  • Is it 3+ levels and order doesn’t matter?  Use the Multinomial (e.g.,Pizza/Hamburger/Burrito or Red/Blue/Green/Orange)

See how to choose, specify and build, compare, and evaluate models using Generalized Regression in JMP Pro. Q&A and is included throughout the presentation.

Resources:

• Developer Tutorial: Using JMP Pro Generalized Regression to Better Understand Observational Data
• Developer Tutorial: Using JMP Pro Generalized Regression to Analyze Designed Experiments
• Generalized Regression documentation