Regression Model¶

Linear Models (linear in the parameters)¶

Simple linear relationship: Model the conditional mean response of a continuous variable using a linear relationship to a single continuous variable assuming normal errors

Given X, Y has a normal distribution with a mean(center) of 𝛽_0+𝛽_1 𝑋 and a variance of 𝜎^2 . Also, it can be written as $$ Y|X \sim N(\beta_{0} +\beta_{1}X,\sigma^2) $$

Quadratic relationship:¶

Model the conditional mean response of a continuous variable as a quadratic relationship to a single continuous variable (this is still a linear model as it’s linear in the parameters)

Multiple linear relationships¶

Model the conditional mean response of a continuous variable as a linear relationship with each of two continuous variables (no interaction)

Non-Linear Models¶

A specific relationship: $$ Y=\beta_{0} +\beta_{1}X^{\beta_3} + \beta_2X_2^{\beta_4} \epsilon \quad \epsilon\sim N(n,\sigma^2) $$

Nonparametric regression¶

The Nonparametric regression estimate the function $$ Y_i=f(X_i)+\epsilon_i $$

LOWESS (locally weighted scatterplot smoother)

The predicted 𝑌_𝑖 for a given 𝑋_𝑖 is determined by considering only ‘local’ points in ‘window’ around 𝑋_𝑖. - Often a simple linear regression is fit to the local points, and the prediction falls in this line - Researcher chooses width of window