G892229 Multiple Regression Week 6(Wednesday)

Regression Analysis
Chapter 11
Regression and Correlation
Techniques that are used to establish whether there is a mathematical relationship between two or more variables, so that the behavior of one variable can be used to predict the behavior of others. Applicable to “Variables” data only.
run
axis.
b
0
X
A simple linear relationship can be described mathematically by
Y = mX + b
Simple Linear Regression
slope =
rise run
=
(6 - 3)
1
=
(10 - 4)
2
Y
rise
5
run intercept = 1
Rent
Step 1: Scatter plot
2500 2300 2100 1900 1700 1500 1300 1100 900 700 500
500 700 900 1100 1300 1500 1700 1900 2100
Size
Scatter plot suggests that there is a ‘linear’ relationship between Rent and Size
High

0.000 0.000 ** ** R-Sq==100.0% 100.0% R-Sq(adj) R-Sq(adj)==100.0% 100.0% R-Sq SS SS 330.00 330.00 0.00 0.00 330.00 330.00 MS MS FF 330.00 ** 330.00 0.00 0.00 PP **
290
280 0 1 2 3 4 5 6 7 8 9 10
price
SS==00 Analysisof ofVariance Variance Analysis Source DF Source DF Regression 11 Regression Error Error 88 Total Total 99
X 12 L X 1 p X 22 L X 2 p X 32 L X 3 p M L M X n 2 L X np
n
矩阵方程的解 Q =| ε |2 = (Y − XB)`(Y − XB) 广义逆矩阵
∂Q = −2 X `(Y − XB) = 0 ∂B X `Y = ( X ` X )B
多元回归的矩阵形式
n
矩阵表示
1 X11 B0 Y1 1 X 21 B1 Y2 Y = B = B2 X = 1 X 31 M M M M Y n 1 X B p n1 Y = XB + U
实例10.1 Tab 10-1，p200
n
收益对价格的回归
Regression Analysis The regression equation is
returns = 300 - 2.01 price
Predictor Constant price S = 0.9878 Coef 299.920 -2.0055 StDev 0.675 0.1088 T P

1 X
x12
x22
xk21
x12
x122
x1k2
1 x1n x2n xkn 1 x1n x12n x1kn
和
y 1
Y
y2
y n
求得 XX、XY和( XX)-1，并由
b=( XX)-1( XY)获得相应的多项式回归统计数。
(四) 多项式回归方程的估计标准误
y 的总平方和 SSy 可分解为回归和离回归两部分：
曲线回归分析方法的主要内容有：
① 确定两个变数间数量变化的某种特定的规则或规 律；
② 估计表示该种曲线关系特点的一些重要参数，如 回归参数、极大值、极小值和渐近值等；
③ 为生产预测或试验控制进行内插，或在论据充足 时作出理论上的外推。
第一节 曲线的类型与特点
一、指数函数曲线 二、对数函数曲线 三、幂函数曲线 四、双曲函数曲线 五、S型曲线
F
Qk
Uk /k /[n(k 1
)]
(11·24)
可测验多项式回归关系的真实性。
相关指数：Ry·x，x2， ，，kxk次多项式的回归平方
和占Y总平方和的比率的平方根值，可用来表示Y与X
的多项式的相关密切程度。
Ry· x，x2， ，xk Uk /SSy
(11·25)
决定系数：在Y 的总变异中，可由X 的k 次多项式
3 162.5 204.4 238.9 275.1 237.9 204.5 192.5
6 216.4 276.7 295.9 325.3 320.5 286.9 219.9
氮肥 9
274.7 342.8 363.3 336.3 353.7 322.5 278.0
12 274.3 343.4 361.7 381.0 369.5 345.9 319.1

Estimated (or predicted) value of Y Estimated intercept Estimated slope coefficients
ˆ b b X b X b X Y i 0 1 1i 2 2i k ki
In this chapter we will use Excel to obtain the regression slope coefficients and other regression summary measures.
14
15
450
300
5.00
7.00
3.5
2.7
Copyright © 2015, 2012, 2009 Pearson Education, Inc.
Chapter 14, Slide 7
Excel Multiple Regression Output
DCOVA
Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.72213 0.52148 0.44172 47.46341 15
Predictor Coef SE Coef T P Constant 306.50 114.30 2.68 0.020 Price -24.98 10.83 -2.31 0.040 Advertising 74.13 25.97 2.85 0.014 S = 47.4634 R-Sq = 52.1% R-Sq(adj) = 44.2% Analysis of Variance
Source DF SS MS F P Regression 2 29460 14730 6.54 0.012 Residual Error 12 27033 2253 Total 14 56493