如何针对重复测量设计计算方差分析：R中的aov（）vs lm（）

标题说明了一切，我很困惑。下面的代码在R中运行重复的aov（），并运行我认为是等效的lm（）的调用，但是它们返回不同的误差残差（尽管平方和相同）。

显然，来自aov（）的残差和拟合值是模型中使用的残差和拟合值，因为它们的平方和加到summary（my.aov）中报告的每个模型/残差平方和。那么，应用于重复测量设计的实际线性模型是什么？

set.seed(1)
# make data frame,
# 5 participants, with 2 experimental factors, each with 2 levels
# factor1 is A, B
# factor2 is 1, 2
DF <- data.frame(participant=factor(1:5), A.1=rnorm(5, 50, 20), A.2=rnorm(5, 100, 20), B.1=rnorm(5, 20, 20), B.2=rnorm(5, 50, 20))

# get our experimental conditions
conditions <- names(DF)[ names(DF) != "participant" ]

# reshape it for aov
DFlong <- reshape(DF, direction="long", varying=conditions, v.names="value", idvar="participant", times=conditions, timevar="group")

# make the conditions separate variables called factor1 and factor2
DFlong$factor1 <- factor( rep(c("A", "B"), each=10) )
DFlong$factor2 <- factor( rep(c(1, 2), each=5) )

# call aov
my.aov <- aov(value ~ factor1*factor2 + Error(participant / (factor1*factor2)), DFlong)

# similar for an lm() call
fit <- lm(value ~ factor1*factor2 + participant, DFlong)

# what's aov telling us?
summary(my.aov)

# check SS residuals
sum(residuals(fit)^2)       # == 5945.668

# check they add up to the residuals from summary(my.aov)
2406.1 + 1744.1 + 1795.46   # == 5945.66

# all good so far, but how are the residuals in the aov calculated?
my.aov$"participant:factor1"$residuals

#clearly these are the ones used in the ANOVA:
sum(my.aov$"participant:factor1"$residuals ^ 2)

# this corresponds to the factor1 residuals here:
summary(my.aov)


# but they are different to the residuals reported from lm()
residuals(fit)
my.aov$"participant"$residuals
my.aov$"participant:factor1"$residuals
my.aov$"participant:factor1:factor2"$residuals

想一想一种方法是用工具变量来对待的情况作为3因子方差分析学科之间participant，factor1，factor2，和1单元大小anova(lm(value ~ factor1*factor2*participant, DFlong))计算所有的SS在这3单因素方差分析（3个主效应，3所有的影响一阶互动，1个二阶互动）。由于每个单元中只有1个人，因此完整模型没有错误，并且上述调用anova()无法计算F检验。但是SS与设计中的2因数相同。

anova()实际上如何计算效果的SS？通过顺序模型比较（类型I）：它适合没有问题的受限模型，也包含包含该效应的非受限模型。与此效应相关的SS是两个模型之间的误差SS之差。

# get all SS from the 3-way between subjects ANOVA
anova(lm(value ~ factor1*factor2*participant, DFlong))

dfL <- DFlong   # just a shorter name for your data frame
names(dfL) <- c("id", "group", "DV", "IV1", "IV2")   # shorter variable names

# sequential model comparisons (type I SS), restricted model is first, then unrestricted
# main effects first
anova(lm(DV ~ 1,      dfL), lm(DV ~ id,         dfL))  # SS for factor id
anova(lm(DV ~ id,     dfL), lm(DV ~ id+IV1,     dfL))  # SS for factor IV1
anova(lm(DV ~ id+IV1, dfL), lm(DV ~ id+IV1+IV2, dfL))  # SS for factor IV2

# now first order interactions
anova(lm(DV ~ id+IV1+IV2, dfL), lm(DV ~ id+IV1+IV2+id:IV1,  dfL))  # SS for id:IV1
anova(lm(DV ~ id+IV1+IV2, dfL), lm(DV ~ id+IV1+IV2+id:IV2,  dfL))  # SS for id:IV2
anova(lm(DV ~ id+IV1+IV2, dfL), lm(DV ~ id+IV1+IV2+IV1:IV2, dfL))  # SS for IV1:IV2

# finally the second-order interaction id:IV1:IV2
anova(lm(DV ~ id+IV1+IV2+id:IV1+id:IV2+IV1:IV2,            dfL),
      lm(DV ~ id+IV1+IV2+id:IV1+id:IV2+IV1:IV2+id:IV1:IV2, dfL))

现在，id:IV1通过从受限模型的误差SS中减去非受限模型的误差SS来检查与交互相关的效果SS 。

sum(residuals(lm(DV ~ id+IV1+IV2,        dfL))^2) -
sum(residuals(lm(DV ~ id+IV1+IV2+id:IV1, dfL))^2)

现在，您已经拥有所有“原始”效果SS，您只需选择正确的误差项来测试效果SS，就可以构建对象内部测试。例如，factor1针对的交互作用SS 测试效果SS participant:factor1。

为了对模型比较方法进行出色的介绍，我建议使用Maxwell＆Delaney（2004）。设计实验和分析数据。