8.1回归的多面性
8.2 OLS回归
OLS回归拟合模型形式:
为了能够恰当地解释oLs模型的系数，数据必须满足以下统计假设。
口正态性对于固定的自变量值，因变量值成正态分布。
口独立性Yi值之间相互独立。
口线性因变量与自变量之间为线性相关。
口同方差性因变量的方差不随自变量的水平不同而变化。也可称作不变方差，但是说同方差性感觉上更犀利。
8.2.1用lm()拟合回归模型
myfit<-lm(formula,data)
formula指要拟合的模型形式，data是一个数据框，包含了用于拟合模型的数据。
表达式(formula):Y~X1 X2 … Xk
8.2.2简单线性回归
> fit<-lm(weight~height,data=women)
> summary(fit)
Call:
lm(formula = weight ~height, data = women)
Residuals:
Min 1Q Median 3Q Max
-1.7333 -1.1333-0.3833 0.7417 3.1167
Coefficients:
Estimate Std. Error t valuePr(>|t|)
(Intercept)-87.51667 5.93694 -14.74 1.71e-09 ***
height 3.45000 0.09114 37.85 1.09e-14 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’1
Residual standarderror: 1.525 on 13 degrees of freedom
MultipleR-squared: 0.991, Adjusted R-squared: 0.9903
F-statistic: 1433 on 1 and 13 DF, p-value: 1.091e-14
> plot(women$height,women$weight,xlab="h",ylab="w")
> abline(fit)
8.2.3多项式回归
> plot(women$height,women$weight,xlab="h",ylab="w")
> abline(fit)
> fit2<-lm(weight~height I(height^2),data=women)
> plot(women$height,women$weight,xlab="height(ininches)",ylab="weight (in lbs)")
> lines(women$height,fitted(fit2))
8.2.4多元线性回归
> library(car)
> states<-as.data.frame(state.x77[,c("Murder","Population","Illiteracy","Income","Frost")])
> cor(states)
Murder PopulationIlliteracy Income
Murder 1.0000000 0.3436428 0.7029752 -0.2300776
Population 0.3436428 1.0000000 0.1076224 0.2082276
Illiteracy 0.7029752 0.1076224 1.0000000 -0.4370752
Income -0.2300776 0.2082276 -0.4370752 1.0000000
Frost -0.5388834 -0.3321525 -0.6719470 0.2262822
Frost
Murder -0.5388834
Population -0.3321525
Illiteracy -0.6719470
Income 0.2262822
Frost 1.0000000
> scatterplotMatrix(states,spread=FALSE,lty.smooth=2,main="spm")
8.2.5有交互项的多元线性回归
> fit<-lm(mpg~hp wt hp:wt,data=mtcars)
> summary(fit)
Call:
lm(formula = mpg ~ hp wt hp:wt, data = mtcars)
Residuals:
Min 1Q Median 3Q Max
-3.0632 -1.6491-0.7362 1.4211 4.5513
Coefficients:
Estimate Std. Error t valuePr(>|t|)
(Intercept)49.80842 3.60516 13.816 5.01e-14 ***
hp -0.12010 0.02470 -4.863 4.04e-05 ***
wt -8.21662 1.26971 -6.471 5.20e-07 ***
hp:wt 0.02785 0.00742 3.753 0.000811 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’1
Residual standarderror: 2.153 on 28 degrees of freedom
MultipleR-squared: 0.8848, Adjusted R-squared: 0.8724
F-statistic: 71.66 on 3and 28 DF, p-value: 2.981e-13
Effects包中的effect（）函数，可以用图形展示交互项的结果
Plot(effect(term,mod,xlevels),multiline=TRUE)
term即模型要画的项，mod为通过lm ( )拟合的模型，xlevels是一个列表，指定变量要设定的常量值，multiline=TRUE选项表示添加相应直线。
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