第九章方差分析
9.2 ANOVA 模型拟合
9.2.1 aov()函数
aov(formula, data = NULL, projections =FALSE, qr = TRUE,
contrasts = NULL, ...)
9.2.2 表达式中各项的顺序
y ~ A + B + A:B
有三种类型的方法可以分解等式右边各效应对y所解释的方差。R默认类型I
类型I(序贯型)
效应根据表达式中先出现的效应做调整。A不做调整,B根据A调整,A:B交互项根据A和
B调整。
类型II(分层型)
效应根据同水平或低水平的效应做调整。A根据B调整,B依据A调整,A:B交互项同时根
据A和B调整。
类型III(边界型)
每个效应根据模型其他各效应做相应调整。A根据B和A:B做调整,A:B交互项根据A和B
调整。
9.3 单因素方差分析
> library(multcomp)
> attach(cholesterol)
> table(trt) #各组样本大小
trt
1time 2times4times drugD drugE
10 10 10 10 10
> aggregate(response,by=list(trt),FUN=mean)#各组均值
Group.1 x
1 1time 5.78197
2 2times 9.22497
3 4times 12.37478
4 drugD 15.36117
5 drugE 20.94752
> aggregate(response,by=list(trt),FUN=sd) #各组标准差
Group.1 x
1 1time 2.878113
2 2times 3.483054
3 4times 2.923119
4 drugD 3.454636
5 drugE 3.345003
> fit<-aov(response~trt)
> summary(fit) #检验组间差异(ANOVA)
Df SumSq Mean Sq F value Pr(>F)
trt 41351.4 337.8 32.43 9.82e-13 ***
Residuals 45 468.8 10.4
---
Signif. codes:
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> library(gplots)
> plotmeans(response~trt,xlab="treatment",ylab="response",main="meanplot\nwith 95% CI")
#绘制各组均值及其置信区间的图形
> detach(cholesterol)
9.3.1 多重比较
TukeyHSD()函数提供了对各组均值差异的成对检验。但要注意TukeyHSD()函数与HH包存在兼容性问题:若载入HH包,TukeyHSD()函数将会失效。使用detach("package::HH")将它从搜寻路径中删除,然后再调用TukeyHSD()
> detach("package:HH", unload=TRUE)
> TukeyHSD(fit)
Tukey multiplecomparisons of means
95% family-wise confidence level
Fit: aov(formula = response ~ trt)
$trt
diff lwr upr p adj
2times-1time 3.44300 -0.6582817 7.5442820.1380949
4times-1time 6.59281 2.4915283 10.6940920.0003542
drugD-1time 9.57920 5.4779183 13.680482 0.0000003
drugE-1time 15.16555 11.0642683 19.266832 0.0000000
4times-2times 3.14981 -0.9514717 7.2510920.2050382
drugD-2times 6.13620 2.0349183 10.2374820.0009611
drugE-2times 11.72255 7.6212683 15.8238320.0000000
drugD-4times 2.98639 -1.1148917 7.0876720.2512446
drugE-4times 8.57274 4.4714583 12.6740220.0000037
drugE-drugD 5.58635 1.4850683 9.687632 0.0030633
> par(las=2)
> par(mar=c(5,8,4,2))
> plot(TukeyHSD(fit))
multcomp包中的glht()函数提供了多重均值比较更为全面的方法,既适用于线性模型,也适用于广义线性模型.
> library(multcomp)
> par(mar=c(5,4,6,2))
> tuk<-glht(fit,linfct=mcp(trt="Tukey"))
> plot(cld(tuk,level=.5),col="lightgrey")
9.3.2 评估检验的假设条件
> library(car)
> qqPlot(lm(response~trt,data=cholesterol),simulate=TRUE,main="Q-Qplot",labels=FALSE)
Bartlett检验:
> bartlett.test(response~trt,data=cholesterol)
Bartlett test ofhomogeneity of variances
data: response by trt
Bartlett's K-squared = 0.5797, df = 4,
p-value = 0.9653
方差齐性分析对离群点非常敏感。可利用car包中的outlierTest()函数来检测离群点:
> library(car)
> outlierTest(fit)
No Studentized residuals withBonferonni p < 0.05
Largest |rstudent|:
rstudent unadjusted p-value Bonferonni p
19 2.251149 0.029422 NA
9.4 单因素协方差分析
单因素协方差分析(ANCOVA)扩展了单因素方差分析(ANOVA),包含一个或多个定量的
协变量
> data(litter,package="multcomp")
> attach(litter)
> table(dose)
dose
0 5 50 500
20 19 18 17
> aggregate(weight,by=list(dose),FUN=mean)
Group.1 x
1 0 32.30850
2 5 29.30842
3 50 29.86611
4 500 29.64647
> fit<-aov(weight~gesttime+dose)
> summary(fit)
Df Sum Sq Mean Sq Fvalue Pr(>F)
gesttime 1 134.3 134.30 8.049 0.00597 **
dose 3 137.1 45.71 2.739 0.04988 *
Residuals 69 1151.3 16.69
---
Signif. codes:
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’0.1 ‘ ’ 1
由于使用了协变量,要获取调整的组均值——即去除协变量效应后的组均值。可使
用effects包中的effects()函数来计算调整的均值:
> library(effects)
> effect("dose",fit)
dose effect
dose
0 5 50 500
32.35367 28.87672 30.56614 29.33460
对用户定义的对照的多重比较
> library(multcomp)
> contrast<-rbind("no drugvs. drug"=c(3,-1,-1,-1))
> summary(glht(fit,linfct=mcp(dose=contrast)))
Simultaneous Tests for General LinearHypotheses
Multiple Comparisons of Means: User-definedContrasts
Fit: aov(formula = weight ~ gesttime +dose)
Linear Hypotheses:
Estimate Std. Error tvalue
no drug vs. drug == 0 8.284 3.209 2.581
Pr(>|t|)
no drug vs. drug == 0 0.012 *
---
Signif. codes:
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’0.1 ‘ ’ 1
(Adjusted p values reported --single-step method)
9.4.1 评估检验的假设条件
检验回归斜率的同质性
> library(multcomp)
> fit2<-aov(weight~gesttime*dose,data=litter)
> summary(fit2)
Df Sum Sq Mean Sq F value Pr(>F)
gesttime 1 134.3 134.30 8.289 0.00537 **
dose 3 137.1 45.71 2.821 0.04556 *
gesttime:dose 3 81.9 27.29 1.684 0.17889
Residuals 66 1069.4 16.20
---
Signif. codes:
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’0.1 ‘ ’ 1
9.4.2 结果可视化
HH包中的ancova()函数可以绘制因变量、协变量和因子之间的关系图。
> library(HH)
> ancova(weight~gesttime+dose,data=litter)
Analysis of Variance Table
Response: weight
Df Sum Sq Mean Sq F value Pr(>F)
gesttime 1 134.30 134.304 8.0493 0.005971 **
dose 3 137.12 45.708 2.7394 0.049883 *
Residuals 69 1151.27 16.685
---
Signif. codes:
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’0.1 ‘ ’ 1
9.5 双因素方差分析
双因素ANOVA
> attach(ToothGrowth)
> table(supp,dose)
dose
supp 0.5 1 2
OJ 10 10 10
VC 10 10 10
> aggregate(len,by=list(supp,dose),FUN=mean)
Group.1 Group.2 x
1 OJ 0.5 13.23
2 VC 0.5 7.98
3 OJ 1.0 22.70
4 VC 1.0 16.77
5 OJ 2.0 26.06
6 VC 2.0 26.14
> aggregate(len,by=list(supp,dose),FUN=sd)
Group.1 Group.2 x
1 OJ 0.5 4.459709
2 VC 0.5 2.746634
3 OJ 1.0 3.910953
4 VC 1.0 2.515309
5 OJ 2.0 2.655058
6 VC 2.0 4.797731
> fit<-aov(len~supp*dose)
> summary(fit)
Df Sum Sq Mean Sq F value Pr(>F)
supp 1 205.3 205.3 12.317 0.000894 ***
dose 1 2224.3 2224.3 133.415 < 2e-16 ***
supp:dose 1 88.9 88.9 5.333 0.024631 *
Residuals 56 933.6 16.7
---
Signif. codes:
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’0.1 ‘ ’ 1
table语句的预处理表明该设计是均衡设计(各设计单元中样本大小都相同),aggregate
语句处理可获得各单元的均值和标准差。用summary()函数得到方差分析表,可以看到主效应
(supp和dose)和交互效应都非常显著。有多种方式对结果进行可视化处理。interaction.plot()函数来展示双因素方差分析的交互效应。
> interaction.plot(dose,supp,len,type="b",col=c("red","blue"),pch=c(16,18),main="interactionbetween dose and supplement type")
还可以用gplots包中的plotmeans()函数来展示交互效应。
> library(gplots)
> plotmeans(len~interaction(supp,dose,sep=""),connect=list(c(1,3,5),c(2,4,6)),col=c("red","darkgreen"),main="interactionplot with 95%CIs",xlab="treatment and dose combination")
用HH包中的interaction2wt()函数来可视化结果,图形对任意顺序的因子
设计的主效应和交互效应都会进行展示
> library(HH)
> interaction2wt(len~supp*dose)
9.6 重复测量方差分析
含一个组间因子和一个组内因子的重复测量方差分析
w1b1<-subset(CO2,Treatment=="chilled")
w1b1
fit<-aov(uptake~conc*Type+Error(Plant/(conc)),w1b1)
summary(fit)
par(las=2)
par(mar=c(10,4,4,2))
with(w1b1,interaction.plot(conc,Type,uptake,type="b",col=c("red","blue"),pch=c(16,18),main="InteractionPlot for Plant Type and Concentration"))
boxplot(uptake~Type*conc,data=w1b1,col=c("gold","green"),
main="Chilled Quebec andMississippi Plants",
ylab="Carbon dioxide uptake rateumol/m^2 sec")
9.7 多元方差分析
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)
head(UScereal)
attach(UScereal)
y<-cbind(calories,fat,sugars)
aggregate(y,by=list(shelf),FUN=mean)
cov(y)
fit<-manova(y~shelf)
summary(fit)
summary.aov(fit)
9.7.1 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。
理论补充
若有一个p*1的多元正态随机向量x,均值为μ,协方差矩阵为Σ,那么x与μ的马氏距离
的平方服从*度为p的卡方分布。Q-Q图展示卡方分布的分位数,横纵坐标分别是样本量与
马氏距离平方值。如果点全部落在斜率为1、截距项为0的直线上,则表明数据服从多元正态
分布。
检验多元正态性
> 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="q-qplot assessing multivariate normality",ylab="mahalanobis d2")
> identify(coord$x,coord$y,labels=row.names(UScereal))
9.7.2 稳健多元方差分析
如果多元正态性或者方差—协方差均值假设都不满足,又或者你担心多元离群点,那么可以
考虑用稳健或非参数版本的MANOVA检验。稳健单因素MANOVA可通过rrcov包中的
Wilks.test()函数实现。vegan包中的adonis()函数则提供了非参数MANOVA的等同形式。
> library(rrcov)
> Wilks.test(y,shelf,method="mcd")
9.8 用回归来做ANOVA
> library(multcomp)
> levels(cholesterol$trt)
[1] "1time" "2times" "4times""drugD"
[5] "drugE"
> fit.aov<-aov(response~trt,data=cholesterol)
> summary(fit.aov)
Df Sum Sq Mean Sq F value Pr(>F)
trt 4 1351.4 337.8 32.43 9.82e-13 ***
Residuals 45 468.8 10.4
---
Signif. codes:
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’0.1 ‘ ’ 1
用lm()函数拟合同样的模型
> fit.lm<-lm(response~trt,data=cholesterol)
> summary(fit.lm)
Call:
lm(formula = response ~ trt, data =cholesterol)
Residuals:
Min 1Q Median 3Q Max
-6.5418 -1.9672 -0.0016 1.8901 6.6008
Coefficients:
Estimate Std. Error t valuePr(>|t|)
(Intercept) 5.782 1.021 5.665 9.78e-07 ***
trt2times 3.443 1.443 2.385 0.0213 *
trt4times 6.593 1.443 4.568 3.82e-05 ***
trtdrugD 9.579 1.443 6.637 3.53e-08 ***
trtdrugE 15.166 1.443 10.507 1.08e-13 ***
---
Signif. codes:
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’0.1 ‘ ’ 1
Residual standard error: 3.227 on 45degrees of freedom
Multiple R-squared: 0.7425, AdjustedR-squared: 0.7196
F-statistic: 32.43 on 4 and 45DF, p-value: 9.819e-13