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# Remove most objects from the working environment
rm(list = ls())
options(stringsAsFactors = F)
# 9.3. One-way ANOVA
# code listing 9.1
library(multcomp)
attach(cholesterol)
colnames(cholesterol)
table(trt) # group sample sizes
# trt
# 1time 2times 4times drugD drugE
# 10 10 10 10 10
aggregate(response, by=list(trt), mean) # group means
# 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), sd) # group standard deviations
# 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) # Test for group differences(ANOVA)
# 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
library(gplots)
plotmeans(response ~ trt,
xlab = "Treatment",
ylab = "Response",
main = "Mean Plot \nwith 95% CI") # figure 9.1
detach(cholesterol)
# 9.3.1. Multiple comparisons
# code listing 9.2. Tukey HSD pairwise group comparisons
TukeyHSD(fit)
# Tukey multiple comparisons of means
# 95% family-wise confidence level
#
# Fit: aov(formula = response ~ trt)
#
# $trt
# diff lwr upr p adj
# 2times-1time 3.44300 -0.6582817 7.544282 0.1380949
# 4times-1time 6.59281 2.4915283 10.694092 0.0003542
# 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 7.251092 0.2050382
# drugD-2times 6.13620 2.0349183 10.237482 0.0009611
# drugE-2times 11.72255 7.6212683 15.823832 0.0000000
# drugD-4times 2.98639 -1.1148917 7.087672 0.2512446
# drugE-4times 8.57274 4.4714583 12.674022 0.0000037
# drugE-drugD 5.58635 1.4850683 9.687632 0.0030633
par(las=2)
par(mar=c(5,8,4,2))
plot(TukeyHSD(fit)) # figure 9.2
library(multcomp)
par(mar=c(5,4,6,2))
tuk <- glht(fit, linfct=mcp(trt="Tukey"))
plot(cld(tuk, level=.05),col="lightgrey") # figure 9.3
# 9.3.2. Assessing test assumptions
# figure 9.4
library(car)
qqPlot(lm(response ~ trt, data=cholesterol),
simulate=TRUE, main="Q-Q Plot", labels=FALSE)
# code
bartlett.test(response ~ trt, data=cholesterol)
# Bartlett test of homogeneity of variances
#
# data: response by trt
# Bartlett's K-squared = 0.57975, df = 4, p-value = 0.9653
outlierTest(fit)
# No Studentized residuals with Bonferroni p < 0.05
# Largest |rstudent|:
# rstudent unadjusted p-value Bonferroni p
# 19 2.251149 0.029422 NA
#=============================================================
# 9.4. One-way ANCOVA
# code listing 9.3. One-way ANCOVA
data("litter", package = "multcomp")
attach(litter)
table(dose)
# dose
# 0 5 50 500
# 20 19 18 17
aggregate(weight, by=list(dose), 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 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
library(effects)
effect("dose", fit)
# dose effect
# dose
# 0 5 50 500
# 32.35367 28.87672 30.56614 29.33460
# code listing 9.4 Multiple comparisons employing user-supplied contrasts
library(multcomp)
contrast <- rbind("no drug vs. drug" = c(3, -1, -1, -1))
summary(glht(fit, linfct=mcp(dose=contrast)))
# Simultaneous Tests for General Linear Hypotheses
#
# Multiple Comparisons of Means: User-defined Contrasts
#
#
# Fit: aov(formula = weight ~ gesttime + dose)
#
# Linear Hypotheses:
# Estimate Std. Error t value Pr(>|t|)
# no drug vs. drug == 0 8.284 3.209 2.581 0.012 *
# ---
# Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
# (Adjusted p values reported -- single-step method)
help("glht")
# 9.4.1. Assessing test assumptions
# code listing 9.5. Testing for homogeneity of regression slopes
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. Visualizing the results
# install.packages("HH")
library(HH)
ancova(weight ~ gesttime + dose, data=litter) # figure 9.5
# 9.5. Two-way factorial ANOVA
# code listing 9.6 Two-way ANOVA
attach(ToothGrowth)
head(ToothGrowth)
# len supp dose
# 1 4.2 VC 0.5
# 2 11.5 VC 0.5
# 3 7.3 VC 0.5
# 4 5.8 VC 0.5
# 5 6.4 VC 0.5
# 6 10.0 VC 0.5
class(dose)
colnames(ToothGrowth)
# [1] "len" "supp" "dose"
table(supp,dose)
# dose
# supp 0.5 1 2
# OJ 10 10 10
# VC 10 10 10
aggregate(len, by=list(supp, dose), 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), 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 1 205.4 205.4 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
dose <- factor(dose)
fit2 <- aov(len ~ supp*dose)
summary(fit2)
# Df Sum Sq Mean Sq F value Pr(>F)
# supp 1 205.4 205.4 15.572 0.000231 ***
# dose 2 2426.4 1213.2 92.000 < 2e-16 ***
# supp:dose 2 108.3 54.2 4.107 0.021860 *
# Residuals 54 712.1 13.2
# ---
# Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
detach(ToothGrowth)
# figure 9.6
interaction.plot(dose, supp, len, type="b",
col=c("red","blue"), pch=c(16, 18),
main = "Interaction between Dose and Supplement Type")
# figure 9.7
library(gplots)
plotmeans(len ~ interaction(supp, dose, sep=" "),
connect=list(c(1,3,5),c(2,4,6)),
col=c("red", "darkgreen"),
main = "Interaction Plot with 95% CIs",
xlab="Treatment and Dose Combination")
# figure 9.8
library(HH)
interaction2wt(len~supp*dose)
# 9.6. Repeated measures ANOVA
# code listing 9.7. Repeated measures ANOVA with one between- and within-groups factor
CO2$conc <- factor(CO2$conc)
w1b1 <- subset(CO2, Treatment=='chilled')
fit <- aov(uptake ~ conc*Type + Error(Plant/(conc)), w1b1)
summary(fit)
# Error: Plant
# Df Sum Sq Mean Sq F value Pr(>F)
# Type 1 2667.2 2667.2 60.41 0.00148 **
# Residuals 4 176.6 44.1
# ---
# Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#
# Error: Plant:conc
# Df Sum Sq Mean Sq F value Pr(>F)
# conc 6 1472.4 245.40 52.52 1.26e-12 ***
# conc:Type 6 428.8 71.47 15.30 3.75e-07 ***
# Residuals 24 112.1 4.67
# ---
# Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
par(las=2)
par(mar=c(10, 4, 4, 2))
# figure 9.9
with(w1b1, interaction.plot(conc,Type,uptake,
type="b", col=c("red","blue"), pch=c(16,18),
main="Interaction Plot for Plant Type and Concentration"))
# figure 9.10
boxplot(uptake ~ Type*conc, data=w1b1, col=(c("gold", "green")),
main="Chilled Quebec and Mississippi Plants",
ylab="Carbon dioxide uptake rate (umol/m^2 sec)")
# 9.7. Multivariate analysis of variance (MANOVA)
# code listing 9.8. One-way MANOVA
library(MASS)
attach(UScereal)
shelf <- factor(shelf)
y <- cbind(calories, fat, sugars)
aggregate(y, by=list(shelf), 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 2 0.4021 5.1167 6 122 0.0001015 ***
# Residuals 62
# ---
# 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 2 50435 25217.6 7.8623 0.0009054 ***
# Residuals 62 198860 3207.4
# ---
# Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#
# Response fat :
# Df Sum Sq Mean Sq F value Pr(>F)
# shelf 2 18.44 9.2199 3.6828 0.03081 *
# Residuals 62 155.22 2.5035
# ---
# Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#
# Response sugars :
# Df Sum Sq Mean Sq F value Pr(>F)
# shelf 2 381.33 190.667 6.5752 0.002572 **
# Residuals 62 1797.87 28.998
# ---
# Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
# 9.7.1. Assessing test assumptions
# code listing 9.9. Assessing multivariate normality
center <- colMeans(y)
n <- nrow(y)
p <- ncol(y)
cov <- cov(y)
d <- mahalanobis(y, center, cov)
# figure 9.11
coord <- qqplot(qchisq(ppoints(n), df=p),
d, main="Q-Q Plot Assessing Multivariate Normality",
ylab="Mahalanobis D2")
abline(a=0,b=1)
identify(coord$x, coord$y, labels = row.names(UScereal))
#================================================================
# install.packages("mvoutlier")
library(mvoutlier)
outliers <- aq.plot(y)
outliers
# 9.7.2. Robust MANOVA
# code listing 9.10. Robust one-way MANOVA
library(rrcov)
Wilks.test(y, shelf, method = "mcd")
# Robust One-way MANOVA (Bartlett Chi2)
#
# data: x
# Wilks' Lambda = 0.51073, Chi2-Value = 25.3745, DF = 5.1632, p-value = 0.0001377
# sample estimates:
# calories fat sugars
# 1 119.8210 0.7010828 5.663143
# 2 128.0407 1.1849576 12.537533
# 3 160.8604 1.6524559 10.352646
# 9.8. ANOVA as regression
library(multcomp)
levels(cholesterol$trt)
# [1] "1time" "2times" "4times" "drugD" "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
# code listing 9.11. A regression approach to the ANOVA problem in section 9.3
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 value Pr(>|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 45 degrees of freedom
# Multiple R-squared: 0.7425, Adjusted R-squared: 0.7196
# F-statistic: 32.43 on 4 and 45 DF, p-value: 9.819e-13
contrasts(cholesterol$trt)
# 2times 4times drugD drugE
# 1time 0 0 0 0
# 2times 1 0 0 0
# 4times 0 1 0 0
# drugD 0 0 1 0
# drugE 0 0 0 1
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