1. Simple Linear Regression
Chapter 11
Simple Linear Regression
Slides for Optional Sections
No optional sections

2. Probabilistic Models General form of Probabilistic Models
Y = Deterministic Component + Random Error
where
E(y) = Deterministic Component

3. Probabilistic Models
First Order (Straight-Line) Probabilistic Model

4. Probabilistic Models 5 steps of Simple Linear Regression
Hypothesize the deterministic component
Use sample data to estimate unknown model parameters
Specify probability distribution of , estimate standard deviation of the distribution
Statistically evaluate model usefulness
Use for prediction, estimatation, once model is useful

5. Fitting the Model: The Least Squares Approach
Reaction Time versus Drug Percentage
Subject
Amount of Drug x (%)
Reaction Time y (seconds) 1 2 3 4 5

6. Fitting the Model: The Least Squares Approach
Least Squares Line has:
Sum of errors (SE) = 0
Sum of Squared errors (SSE) is smallest of all straight line models
Formulas:
Slope: y-intercept

7. Fitting the Model: The Least Squares Approach

8. Model Assumptions Mean of the probability distribution of ε is 0
Variance of the probability distribution of ε is constant for all values of x
Probability distribution of ε is normal
Values of ε are independent of each other

9. An Estimator of 2
Estimator of 2 for a straight-line model

10. Assessing the Utility of the Model: Making Inferences about the Slope 1
Sampling Distribution of

11. Assessing the Utility of the Model: Making Inferences about the Slope 1
A Test of Model Utility: Simple Linear Regression
One-Tailed Test
Two-Tailed Test
H0: β1=0
Ha: β1<0 (or Ha: β1>0)
Ha: β1≠0
Rejection region: t< -tα
(or t< -tα when Ha: β1>0)
Rejection region: |t|> tα/2
Where tα and tα/2 are based on (n-2) degrees of freedom

12. Assessing the Utility of the Model: Making Inferences about the Slope 1
A 100(1-α)% Confidence Interval for 1
where

13. The Coefficient of Correlation
A measure of the strength of the linear relationship between two variables x and y

14. The Coefficient of Determination

15. Using the Model for Estimation and Prediction
Sampling errors and confidence intervals will be larger for Predictions than for Estimates
Standard error of
Standard error of the prediction

16. Using the Model for Estimation and Prediction
100(1-α)% Confidence interval for Mean Value of y at x=xp
100(1-α)% Confidence interval for an Individual New Value of y at x=xp
where tα/2 is based on (n-2) degrees of freedom