R语言中的多元方差分析

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 fat sugars

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

fat 60.67438 2.713399 3.995474

sugars 180.38032 3.995474 34.050018

fit <- manova(y ~ shelf)

summary(fit)

Df Pillai approx F num Df den Df Pr(>F)

shelf 1 0.19594 4.955 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 value Pr(>F)

shelf 1 45313 45313 13.995 0.0003983 ***

Residuals 63 203982 3238

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Response fat :

Df Sum Sq Mean Sq F value Pr(>F)

shelf 1 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)

shelf 1 183.34 183.34 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