Python for R Users

5 Statistical Modeling

5.1 Concepts in Regression

What is statistical modeling?

• It is a formalization of relationships between variables in the form of mathematical equations.
• It describes how one or more random variables are related to one or more other variables.
• The variables are not deterministically but stochastically related.

Reading Statistical Modeling: The Two Cultures http://projecteuclid.org/download/pdf_1/euclid.ss/1009213726

Example

• Height and age are probabilistically distributed among humans.
• They are stochastically related; when you know that a person is of age 30 years, this influences the chance of this person of being 4‐feet tall. When you know that a person is of age 13 years, this influences the chance of this person of being 6 feet tall.
• Model 1
  • height_i = b₀ + b₁age_i + ε_i, where b₀ is the intercept, b₁ is a parameter that age is multiplied by to get a prediction of height, ε is the error term, and i is the subject.
• Model 2
  • height_i = b₀ + b₁age_i + b₂sex_i + ε_i, where the variable sex is dichotomous.

Regression models involve the following variables:

• The unknown parameters
• The independent variables, X
• The dependent variable, Y

• Y = a + BX is the simplest form of regression
• Linear regression Y = a + Bx + (E)
• Multivariate regression Y = a + bx + cy + (E)
• Logistic regression ln(p/1 − p) = a + bX

Example

Okun’s LawThe relationship between an economy’s unemployment rate and its gross national product (GNP). Economist Arthur Okun developed this idea, ...