R语言中的回归分析

来源：互联网发布：金融资产总量数据编辑：程序博客网时间：2024/05/02 12:13

1、简单线性回归分析

满足线性回归的几个要求：

用lm()拟合回归模型

lm(formula,data)

如何对拟合的结果进行评估？

（1）简单的线性回归的例子

根据身高预测体重

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 value Pr(>|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 standard error: 1.525 on 13 degrees of freedom
Multiple R-squared: 0.991, Adjusted R-squared: 0.9903
F-statistic: 1433 on 1 and 13 DF, p-value: 1.091e-14
women$weight

115 117 120 123 126 129 132 135 139 142 146 150 154 159 164
fitted(fit)

112.5833 116.0333 119.4833 122.9333 126.3833 129.8333 133.2833 136.7333
9 10 11 12 13 14 15
140.1833 143.6333 147.0833 150.5333 153.9833 157.4333 160.8833
residuals(fit)

1 2 3 4 5 6
2.41666667 0.96666667 0.51666667 0.06666667 -0.38333333 -0.83333333
7 8 9 10 11 12
-1.28333333 -1.73333333 -1.18333333 -1.63333333 -1.08333333 -0.53333333
13 14 15
0.01666667 1.56666667 3.11666667

plot(women$height, women$weight, main = "Women Age 30-39",
xlab = "Height (in inches)", ylab = "Weight (in pounds)")
abline(fit)

（2）多项式回归

fit2 <- lm(weight ~ height + I(height^2), data = women)
summary(fit2)

Call:
lm(formula = weight ~ height + I(height^2), data = women)

Residuals:
Min 1Q Median 3Q Max
-0.50941 -0.29611 -0.00941 0.28615 0.59706

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 261.87818 25.19677 10.393 2.36e-07 ***
height -7.34832 0.77769 -9.449 6.58e-07 ***
I(height^2) 0.08306 0.00598 13.891 9.32e-09 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.3841 on 12 degrees of freedom
Multiple R-squared: 0.9995, Adjusted R-squared: 0.9994
F-statistic: 1.139e+04 on 2 and 12 DF, p-value: < 2.2e-16

plot(women$height, women$weight, main = "Women Age 30-39",
xlab = "Height (in inches)", ylab = "Weight (in lbs)")
lines(women$height, fitted(fit2))

library(car)
scatterplot(weight ~ height, data = women, spread = FALSE,
lty.smooth = 2, pch = 19, main = "Women Age 30-39", xlab = "Height (inches)",
ylab = "Weight (lbs.)")

（3）多元线性回归

当预测变量不止一个时，简单线性回归就变成了多元线性回归

以基础包中的state.x77数据集为例，我们想探究一个州的犯罪率和其他因素的关系，包括人口、文盲率、平均收入和结霜天数（温度在冰点以下的平均天数）。

states <- as.data.frame(state.x77[, c("Murder", "Population",
"Illiteracy", "Income", "Frost")])

cor()函数提供了二变量之间的
相关系数
cor(states)

library(car)

scatterplotMatrix()函数默认在非对角线区域绘制变量间的散点图，并添加平滑（loess）
和线性拟合曲线。对角线区域绘制每个变量的密度图和轴须图。
scatterplotMatrix(states, spread = FALSE, lty.smooth = 2,
main = "Scatterplot Matrix")

从上图可以看出：谋杀率是双峰的曲线，每个预测变量都一定程度上出现了偏斜。谋杀率随着人口和文盲率的增加而增加，随着收入水平和结霜天数增加而下降。同时，越冷的州府文盲率越低，收入水平越高。

fit <- lm(Murder ~ Population + Illiteracy + Income +
Frost, data = states)

summary(fit)

(4)有交互项的多元线性回归

使用mtcars数据集考虑每公里跑的英里数与马力和车中以及二个变量之间的关系

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 value Pr(>|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 standard error: 2.153 on 28 degrees of freedom
Multiple R-squared: 0.8848, Adjusted R-squared: 0.8724
F-statistic: 71.66 on 3 and 28 DF, p-value: 2.981e-13

预测mpg的模型为mpg = 49.81 -0.12×hp - 8.22×wt + 0.03×hp×wt

library(effects)
plot(effect("hp:wt", fit, list(wt = c(2.2, 3.2, 4.2))),
multiline = TRUE)

0 0