重学Statistics， Cha14 Simple Linear Regression

14.1 Simple Linear Regression Model

Simple Linear Regression Model: y = β0 + β1 x + ε

• β0 β1 are referred to as parameters of the model
• ε is a random variable referred to as the error term, which is the variability i y that cannot be explained by the linear relationship between x and y.

Simple Linear Regression Equation : E(y) = β0 + β1 x
Estimated Simple Linear Regression Equation: yˆ = b0 + b1x

14.2 Least Square Method

It is a procedure for using sample data to find the estimated regression equation.

14.3 Coefficient of Determination

怎么证明，刚刚的模型能fit the data?

SSE: SUM of squares due to error

SST = SSR + SSE

Coefficient of determination

Correlation Coefficient

Correlation Coefficient 仅能用在 a linear relationship between two variables
Coefficient of determination 可以用在 nonlinear relationship and for relationships that have two or more independent variables

14.4 Model Assumption

An important step in determining whether the assumed model is appropriate involves testing for the significance of the relationship.

14.5 Test For Significance

Estimate of σ2

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t-Test

H0: β1 = 0
H1: β1 != 0

Confidence Interval for β1

b1 +- t * sb1 = 5 +- 1.95
因为 interval 都比0大，所以可以 reject H0

F Test

F test has the same result as t test if there is only one independent variable, with more than one independent variable, only F test can be used to test for an overall significant relationship.

问题：为什么说当 β1 =0时，MSR/MSE接近于1？而且满足 F distribution？

Cautions:

Reject H0 does not enable us to conclude that the relationship between x and y is linear. We can state only that x and y are related and that a linear relationship explains a significant portion of the variability in y over the range of values for x observed in the sample.

14.6 Using estimated regression equation for estimation and prediction

Point estimation: 直接代入公式算出 y

Confidence Interval :

Prediction Interval :

14.8 Residual Analysis: Validating Model Assumptions

4个假设：
1. E(ε) = 0.
2. The variance of ε, denoted by σ2,is the same for all values of x.
3. The values of ε are independent.
4. The error term ε has a normal distribution.

Residual Plot Against x

Standardized Residuals

正好从下图中，可以看出 approximately 95% of the standardized residuals between -2 and +2

Normal Probability Plot

用 normal scores 和 standardized residuals plot一个图，看出是过原点的45度的直线。

14.9 Residual Analysis: Outliers and Influential Observations

Detecting Outliers

Identify any observation with a standardized residual of less than -2 or greater than +2 as an unusual observation.

Detecting Influential Observations