# Chapter 2 - Stats331 Simple Linear Regression Chapter 2...

• Notes
• 101
• 100% (6) 6 out of 6 people found this document helpful

This preview shows page 1 - 11 out of 101 pages.

Stats331 Simple Linear Regression Chapter 2
Chapter objectives The model and important assumptions. Estimation of Parameters. Inferences about the Regression Parameters. Prediction. Analysis of Variance Approach to Regression. Coefficient of Determination. Related Models. 25/05/2015 Chapter 2 -Simple Linear Regression 2 2
The Simple Linear Regression Model Frequently also called straight line regression. For n pairs of observation (x i , y i ), i =1, 2,…, n. The noise or error is random y i = β 0 + β 1 x i + ε i Linear relation is deterministic(non-random) 25/05/2015 Chapter 2 -Simple Linear Regression 3 μ
Noise account for the variability of the observations about the straight line. No noise relation is deterministic. Increased noise increased variability. 25/05/2015 Chapter 2 -Simple Linear Regression 4
It is assumed that β 0 , β 1 and σ 2 are unknown parameters. The x’s may be random or deterministic but it is assumed they are obtained without significant measurement error. 25/05/2015 Chapter 2 -Simple Linear Regression 5
Important Assumptions about x i and ε i x i , i=1,2,.., n are not a random variables(can be taken as a constant), they are under the experimenter’s control. E( ε i )=0, for all i =1, 2, . . . , n μ i = E( y i ) = β 0 + β 1 x i V( ε i )= σ 2 is constant for all i =1, 2, . . . , n V( y i ) = σ 2 is constant for all i =1, 2, . . . , n All observations have the same precision. ε i and ε j are independent random variables for i ≠ j y i , y j are independent Cov(ε i , ε j ) =0 for i ≠ j 25/05/2015 Chapter 2 -Simple Linear Regression 6
1. The relationship between the independent and dependent variable is linear. It is also important to check for outliers since linear regression is sensitive to outlier effects. 25/05/2015 Chapter 2 -Simple Linear Regression 7 x x y y
2. y is distributed Normally at each value of x. 3. The variance of y at every value of x is the same. (homogeneity of variances) 4. The observations are independent. 25/05/2015 Chapter 2 -Simple Linear Regression 8
# Experiment with this simulation program simple.sim <- function(intercept=0,slope=1, x=seq(1,10), sigma= sigma){ noise <-rnorm(length(x),sd=sigma) y <- intercept+ slope*x + noise title <- paste("sigma=",sigma) plot(x, y, pch=16, main=title) abline(intercept,slope,col=4, lwd=2) } par(mfrow=c(2,2)) simple.sim(sigma=0.00) simple.sim(sigma=0.5) simple.sim(sigma=2) simple.sim(sigma=10) 25/05/2015 Chapter 2 -Simple Linear Regression 9
Simulation Example 25/05/2015 Chapter 2 -Simple Linear Regression 10 2 4 6 8 10 2 4 6 8 10 sigma= 0 x y 2 4 6 8 10 2 4 6 8 sigma= 0.5 x y 2 4 6 8 10 2 4 6 8 10 sigma= 2 x y 2 4 6 8 10 -10 0 5 15 sigma= 10 x y