1、当因变量（结果变量）不止一个时，可用多元方差分析（MANOVA）对它们同时进行分析。
library(MASS)
attach(UScereal)
y <- cbind(calories, fat, sugars)
aggregate(y, by = list(shelf), FUN = mean)
Group.1 calories fatsugars
1 1 119.4774 0.6621338 6.295493
2 2 129.8162 1.3413488 12.507670
3 3 180.1466 1.9449071 10.856821
cov(y)
calories fat sugars
calories 3895.24210 60.674383 180.380317
fat60.67438 2.713399 3.995474
sugars180.38032 3.995474 34.050018
fit <- manova(y ~ shelf)
summary(fit)
Df Pillai approx F num Df den Df Pr(>F)
shelf 1 0.195944.955 3 61 0.00383 **
Residuals 63
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
summary.aov(fit)
Response calories :
Df Sum Sq Mean Sq F valuePr(>F)
shelf1 45313 45313 13.995 0.0003983 ***
Residuals 63 2039823238
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Response fat :
Df Sum Sq Mean Sq F value Pr(>F)
shelf1 18.421 18.4214 7.476 0.008108 **
Residuals 63 155.236 2.4641
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Response sugars :
Df Sum Sq Mean Sq F value Pr(>F)
shelf1 183.34 183.34 5.787 0.01909 *
Residuals 63 1995.87 31.68
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
2、评估假设检验
单因素多元方差分析有两个前提假设，一个是多元正态性，一个是方差—协方差矩阵同质性。
（1）多元正态性
第一个假设即指因变量组合成的向量服从一个多元正态分布。可以用Q-Q图来检验该假设条件。
center <- colMeans(y)
n <- nrow(y)
p <- ncol(y)
cov <- cov(y)
d <- mahalanobis(y, center, cov)
coord <- qqplot(qchisq(ppoints(n), df = p), d, main = "QQ
Plot Assessing Multivariate Normality",
ylab = "Mahalanobis D2")
abline(a = 0, b = 1)
identify(coord$x, coord$y, labels = row.names(UScereal))
如果所有的点都在直线上，则满足多元正太性。
2、方差—协方差矩阵同质性即指各组的协方差矩阵相同，通常可用Box’s M检验来评估该假设
3、检测多元离群点
library(mvoutlier)
outliers <- aq.plot(y)
outliers
1、RMSE（均方根误差）即标准误差：假如数据在A1：Z1
标准方差用函数=STDEV（A1:Z1）
方差用函数=VARA（A1:Z1）
2、MRE（平均相对误差）
Excel/函数/统计/STDEV（Sd）
计算出标准偏差Sd值，然后除以平均数再×100％就可以了。
为了找到均方根误差，我们首先需要找到残差（也称为误差，我们需要对这些值均方根），然后需要计算这些残差的均方根。因此，如果我们有一个线性回归模型对象说M，则均方根误差可以找到为sqrt（mean（M $residuals ^ 2））。
示例
x1<-rnorm(500,50,5)
y1<-rnorm(500,50,2)
M1<-lm(y1~x1)
summary(M1)
输出结果
Call:
lm(formula = y1 ~ x1)
Residuals:
Min 1QMedian3QMax
-5.6621 -1.2257 -0.0272 1.4151 6.6421
Coefficients:
EstimateStd.Errort value Pr(>|t|)
(Intercept) 50.178943 0.915473 54.812 <2e-16 ***
x1 -0.002153 0.018241 -0.118 0.906
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 1.966 on 498 degrees of freedom
Multiple R-squared: 2.798e-05, Adjusted R-squared: -0.00198
F-statistic: 0.01393 on 1 and 498 DF, p-value: 0.9061
从模型M1中找到均方根误差-
示例
sqrt(mean(M1$residuals^2))
输出结果
[1] 1.961622
示例
x2<-rnorm(5000,125,21)
y2<-rnorm(5000,137,10)
M2<-lm(y2~x2)
summary(M2)
输出结果
Call:
lm(formula = y2 ~ x2)
Residuals:
Min 1QMedian3QMax
-37.425 -7.005 -0.231 6.836 36.627
Coefficients:
Estimate Std.Error t value Pr(>|t|)
(Intercept) 138.683501 0.851247 162.918 <2e-16 ***
x2 -0.014386 0.006735 -2.136 0.0327 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 10.06 on 4998 degrees of freedom
Multiple R-squared: 0.0009121, Adjusted R-squared: 0.0007122
F-statistic: 4.563 on 1 and 4998 DF, p-value: 0.03272
从模型M2中找到均方根误差：
示例
sqrt(mean(M2$residuals^2))
输出结果
[1] 10.05584
示例
x37<-rpois(500,5)
y3<-rpois(500,10)
M3<-lm(y3~x3)
summary(M3)
输出结果
Call:
lm(formula = y3 ~ x3)
Residuals:
Min 1QMedian3QMax
-7.9004 -1.9928 -0.2155 2.1921 9.3770
Coefficients:
EstimateStd.Error t value Pr(>|t|)
(Intercept) 10.17770 0.3233031.481<2e-16 ***
x3 -0.09244 0.06145-1.5040.133
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 3.027 on 498 degrees of freedom
Multiple R-squared: 0.004524, Adjusted R-squared: 0.002525
F-statistic: 2.263 on 1 and 498 DF, p-value: 0.1331
从模型M3查找均方根误差-
示例
sqrt(mean(M3$residuals^2))
输出结果
[1] 3.020734
示例
x4<-runif(50000,5,10)
y4<-runif(50000,2,10)
M4<-lm(y4~x4)
summary(M4)
输出结果
Call:
lm(formula = y4 ~ x4)
Residuals:
Min1Q Median 3QMax
-4.0007 -1.9934 -0.0063 1.9956 3.9995
Coefficients:
EstimateStd.Error t value Pr(>|t|)
(Intercept) 5.9994268 0.0546751 109.729 <2e-16 ***
x40.0001572 0.0071579 0.0220.982
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 2.309 on 49998 degrees of freedom
Multiple R-squared: 9.646e-09, Adjusted R-squared: -1.999e-05
F-statistic: 0.0004823 on 1 and 49998 DF, p-value: 0.9825
从模型M4找到均方根误差-
示例
sqrt(mean(M4$residuals^2))
输出结果
[1] 2.308586
示例
x5<-sample(5001:9999,100000,replace=TRUE)
y5<-sample(1000:9999,100000,replace=TRUE)
M5<-lm(y5~x5)
summary(M5)
输出结果
Call:
lm(formula = y5 ~ x5)
Residuals:
Min 1QMedian 3Q Max
-4495 -2242-42230 4512
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 5.504e+03 4.342e+01 126.765 <2e-16 ***
x5-1.891e-03 5.688e-03 -0.333 0.74
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 2594 on 99998 degrees of freedom
Multiple R-squared: 1.106e-06, Adjusted R-squared: -8.895e-06
F-statistic: 0.1106 on 1 and 99998 DF, p-value: 0.7395
从模型M5中找到均方根误差<
示例
sqrt(mean(M5$residuals^2))
输出结果
[1] 2593.709
