Regression model A prediction approach

Prediction Independent variable (input/operating variable) Dependent variable (response variable) Prediction error The prediction model: the linear regression line, the linear regression equation

Simple regression model The regression line- Expected response Error of simple regression model Actual response ∵

Graphic explanation

LSE estimators By definition

Assumptions of regression model The distribution of residual: e Normality (~N(0, σ2)) Equality of variance (the same variance σ2 for every σi2 ) Independence from X i

Distribution of the β estimators

S xx

Distribution of the β estimators

Residual analysis Sum of Square of Residuals

Inference of β ∵ ∴ Therefore,, and

Confidence interval for β

Decomposition of Syy, Syy=SSr+SSm X Y Y- Y^ Yi (Yi-Y-) (Yi-Y^) (Y^-Y-) 如果離差越大, 表示 Y^ 不太可能是水平線, 因 為若是水平線, 則差的 平方和將會很小 如果離差越小, 表示 迴歸線越接近真實 值, 預測得越準確 ! Y 的離差, 因給 定 sample 之後 固定不變

Inference of the mean response α+β x 0

Confidence interval for the mean response α+β x 0 分母代入 X 2 /(n-2) 之後除去σ2

Inference of the response at the input level x 0

Confidence interval for the response at the input level x 0 Normality & equality testing Plotting the random data and regression line Plotting the residuals along the predictive variable

Determination and sample correlation coefficient By definition,

More regression models Multiple regression More than one predictive variables Transforming to linearity Log, square Logistic regression For binary response data

Homework #2 Problem 5,12,26,39,42, optional 48