5 Day 5 (June 9)
5.2 Introduction to linear models
What is a model?
What is a linear model?
Most widely used model in science, engineering, and statistics
Vector form: \(\mathbf{y}=\beta_{0}+\beta_{1}\mathbf{x}_{1}+\beta_{2}\mathbf{x}_{2}+\ldots+\beta_{p}\mathbf{x}_{p}+\boldsymbol{\varepsilon}\)
Matrix form: \(\mathbf{y}=\mathbf{X}\boldsymbol{\beta}+\boldsymbol{\varepsilon}\)
Which part of the model is the mathematical model
Which part of the model makes the linear model a “statistical” model
Visual
Which of the four below are a linear model \[\mathbf{y}=\beta_{0}+\beta_{1}\mathbf{x}_{1}+\beta_{2}\mathbf{x}^{2}_{1}+\boldsymbol{\varepsilon}\] \[\mathbf{y}=\beta_{0}+\beta_{1}\mathbf{x}_{1}+\beta_{2}\text{log(}\mathbf{x}_{1}\text{)}+\boldsymbol{\varepsilon}\] \[\mathbf{y}=\beta_{0}+\beta_{1}e^{\beta_{2}\mathbf{x}_{1}}+\boldsymbol{\varepsilon}\] \[\mathbf{y}=\beta_{0}+\beta_{1}\mathbf{x}_{1}+\text{log(}\beta_{2}\text{)}\mathbf{x}_{1}+\boldsymbol{\varepsilon}\]
Why study the linear model?
- Building block for more complex models (e.g., GLMs, mixed models, machine learning, etc)
- We know the most about it