Linear Regression
M. Drew LaMar
February 4, 2019
Class announcements
Errors in Hypothesis Testing
Definition:
Type I error is rejecting a true null hypothesis. The probability of a Type I error is given by \[ \mathrm{Pr[Reject} \ H_{0} \ | \ H_{0} \ \mathrm{is \ true}] = \alpha \]
Definition:
Type II error is failing to reject a false null hypothesis. The probability of a Type II error is given by \[ \mathrm{Pr[Do \ not \ reject} \ H_{0} \ | \ H_{0} \ \mathrm{is \ false}] = \beta \]
Errors in Hypothesis Testing - Power
Definition: The
power of a statistical test (denoted \( 1-\beta \)) is given by \[ \begin{align*} \mathrm{Pr[Reject} \ H_{0} \ | \ H_{0} \ \mathrm{is \ false}] & = 1-\beta \\ & = 1 - \mathrm{Pr[Type \ II \ error]} \end{align*} \]
Probability of errors in hypothesis testing
- \( \alpha \) is the significance level
- \( 1-\beta \) is the power
Statistical power example
https://qubeshub.org/tools/statpowerviz/
Power analysis
Regression
Linear regression
Definition:
Linear regression draws a straight line through the data to predict the response variable from the explanatory variable.
Formula for the line
Graph of the line
- \( a \): intercept
- \( b \): slope
Assumptions of linear regression
Note: At each value of \( X \), there is a population of \( Y \)-values whose mean lies on the true regression line (this is the linear assumption).
Assumptions of linear regression
- Linearity
- Residuals are normally distributed
- Constant variance of residuals
- Independent observations
