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# Remove most objects from the working environment
rm(list = ls())
options(stringsAsFactors = F)
# 12.2.1. Independent two-sample and k-sample tests
# code listing 12.1. t-test versus one-way permutation test for the hypothetical data
library(coin)
score <- c(40, 57, 45, 55, 58, 57, 64, 55, 62, 65)
treatment <- factor(c(rep("A", 5), rep("B", 5)))
mydata <- data.frame(treatment, score)
t.test(score~treatment, data = mydata, var.equal=T)
# Two Sample t-test
#
# data: score by treatment
# t = -2, df = 8, p-value = 0.05
# alternative hypothesis: true difference in means is not equal to 0
# 95 percent confidence interval:
# -19.04 -0.16
# sample estimates:
# mean in group A mean in group B
# 51 61
oneway_test(score~treatment, data = mydata, distribution="exact")
# Exact Two-Sample Fisher-Pitman Permutation Test
#
# data: score by treatment (A, B)
# Z = -2, p-value = 0.07
# alternative hypothesis: true mu is not equal to 0
library(MASS)
UScrime <- transform(UScrime, So = factor(So))
wilcox_test(Prob ~ So, data = UScrime, distribution="exact")
# Exact Wilcoxon-Mann-Whitney Test
#
# data: Prob by So (0, 1)
# Z = -4, p-value = 8e-05
# alternative hypothesis: true mu is not equal to 0
library(multcomp)
set.seed(1234)
oneway_test(response~trt, data = cholesterol,
distribution=approximate(nresample=9999))
# Approximative K-Sample Fisher-Pitman Permutation Test
#
# data: response by trt (1time, 2times, 4times, drugD, drugE)
# chi-squared = 36, p-value <1e-04
#===============================================================
# 12.2.2. Independence in contingency tables
library(coin)
library(vcd)
Arthritis <- transform(Arthritis,
Improved=as.factor(as.numeric(Improved)))
set.seed(1234)
chisq_test(Treatment~Improved, data = Arthritis,
distribution=approximate(nresample=9999))
# Approximative Pearson Chi-Squared Test
#
# data: Treatment by Improved (1, 2, 3)
# chi-squared = 13, p-value = 0.001
Arthritis <- transform(Arthritis,
Improved=as.factor(Improved))
set.seed(1234)
chisq_test(Treatment~Improved, data = Arthritis,
distribution=approximate(nresample=9999))
# Approximative Pearson Chi-Squared Test
#
# data: Treatment by Improved (1, 2, 3)
# chi-squared = 13, p-value = 0.002
#===============================================
# 12.2.3. Independence between numeric variables
states <- as.data.frame(state.x77)
set.seed(1234)
spearman_test(Illiteracy~Murder, data=states,
distribution=approximate(nresample=9999))
# Approximative Spearman Correlation Test
#
# data: Illiteracy by Murder
# Z = 5, p-value <1e-04
# alternative hypothesis: true rho is not equal to 0
#====================================================
# 12.2.4. Dependent two-sample and k-sample tests
library(coin)
library(MASS)
wilcoxsign_test(U1~U2, data = UScrime, distribution="exact")
# Exact Wilcoxon-Pratt Signed-Rank Test
#
# data: y by x (pos, neg)
# stratified by block
# Z = 6, p-value = 1e-14
# alternative hypothesis: true mu is not equal to 0
#====================================================
# 12.3.1. Simple and polynomial regression
# code listing 12.2. Permutation tests for simple linear regression
# install.packages("lmPerm")
library(lmPerm)
set.seed(1234)
fit <- lmp(weight~height, data=women, perm="Prob")
# [1] "Settings: unique SS : numeric variables centered"
summary(fit)
# Call:
# lmp(formula = weight ~ height, data = women, perm = "Prob")
#
# Residuals:
# Min 1Q Median 3Q Max
# -1.733 -1.133 -0.383 0.742 3.117
#
# Coefficients:
# Estimate Iter Pr(Prob)
# height 3.45 5000 <2e-16 ***
# ---
# Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#
# Residual standard error: 1.5 on 13 degrees of freedom
# Multiple R-Squared: 0.991, Adjusted R-squared: 0.99
# F-statistic: 1.43e+03 on 1 and 13 DF, p-value: 1.09e-14
#===============================================================
# code listing 12.3. Permutation tests for polynomial regression
library(lmPerm)
set.seed(1234)
fit <- lmp(weight~height + I(height^2), data=women, perm="Prob")
# [1] "Settings: unique SS : numeric variables centered"
summary(fit)
# Call:
# lmp(formula = weight ~ height + I(height^2), data = women, perm = "Prob")
#
# Residuals:
# Min 1Q Median 3Q Max
# -0.50941 -0.29611 -0.00941 0.28615 0.59706
#
# Coefficients:
# Estimate Iter Pr(Prob)
# height -7.3483 5000 <2e-16 ***
# I(height^2) 0.0831 5000 <2e-16 ***
# ---
# Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#
# Residual standard error: 0.38 on 12 degrees of freedom
# Multiple R-Squared: 0.999, Adjusted R-squared: 0.999
# F-statistic: 1.14e+04 on 2 and 12 DF, p-value: <2e-16
#=======================================================
# 12.3.2. Multiple regression
# code listing 12.4. Permutation tests for multiple regression
library(lmPerm)
set.seed(1234)
states <- as.data.frame(state.x77)
fit <- lmp(Murder~Population + Illiteracy+Income+Frost,
data=states, perm="Prob")
# [1] "Settings: unique SS : numeric variables centered"
summary(fit)
# Call:
# lmp(formula = Murder ~ Population + Illiteracy + Income + Frost,
# data = states, perm = "Prob")
#
# Residuals:
# Min 1Q Median 3Q Max
# -4.7960 -1.6495 -0.0811 1.4815 7.6210
#
# Coefficients:
# Estimate Iter Pr(Prob)
# Population 2.24e-04 51 1.0000
# Illiteracy 4.14e+00 5000 0.0004 ***
# Income 6.44e-05 51 1.0000
# Frost 5.81e-04 51 0.8627
# ---
# Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#
# Residual standard error: 2.5 on 45 degrees of freedom
# Multiple R-Squared: 0.567, Adjusted R-squared: 0.528
# F-statistic: 14.7 on 4 and 45 DF, p-value: 9.13e-08
#==============================================================
# 12.3.3. One-way ANOVA and ANCOVA
# code listing 12.5. Permutation test for One-Way ANOVA
library(lmPerm)
library(multcomp)
set.seed(1234)
fit <- aovp(response~trt, data=cholesterol, perm="Prob")
# [1] "Settings: unique SS "
summary(fit)
# Component 1 :
# Df R Sum Sq R Mean Sq Iter Pr(Prob)
# trt 4 1351.37 337.84 5000 < 2.2e-16 ***
# Residuals 45 468.75 10.42
# ---
# Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#==============================================================
# code listing 12.6. Permutation test for one-way ANCOVA
library(lmPerm)
set.seed(1234)
fit <- aovp(weight ~ gesttime + dose, data=litter, perm="Prob")
# [1] "Settings: unique SS : numeric variables centered"
summary(fit)
# Component 1 :
# Df R Sum Sq R Mean Sq Iter Pr(Prob)
# gesttime 1 161.49 161.493 5000 0.0006 ***
# dose 3 137.12 45.708 5000 0.0392 *
# Residuals 69 1151.27 16.685
# ---
# Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#==============================================================
# 12.3.4. Two-way ANOVA
# code listing 12.7. Permutation test for two-way ANOVA
library(lmPerm)
set.seed(1234)
fit <- aovp(len~supp*dose, data=ToothGrowth, perm="Prob")
# [1] "Settings: unique SS : numeric variables centered"
summary(fit)
# Component 1 :
# Df R Sum Sq R Mean Sq Iter Pr(Prob)
# supp 1 205.35 205.35 5000 < 2e-16 ***
# dose 1 2224.30 2224.30 5000 < 2e-16 ***
# supp:dose 1 88.92 88.92 2032 0.04724 *
# Residuals 56 933.63 16.67
# ---
# Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#==============================================================
# 12.6.1. Bootstrapping a single statistic
rsq <- function(formula, data, indices) {
d <- data[indices,]
fit <- lm(formula, data=d)
return(summary(fit)$r.square)
}
library(boot)
set.seed(1234)
results <- boot(data=mtcars, statistic=rsq,
R=1000, formula=mpg~wt+disp)
print(results)
# ORDINARY NONPARAMETRIC BOOTSTRAP
#
#
# Call:
# boot(data = mtcars, statistic = rsq, R = 1000, formula = mpg ~
# wt + disp)
#
#
# Bootstrap Statistics :
# original bias std. error
# t1* 0.7809306 0.01379126 0.05113904
plot(results) # figure 12.1
boot.ci(results, type = c("perc", "bca"))
# BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
# Based on 1000 bootstrap replicates
#
# CALL :
# boot.ci(boot.out = results, type = c("perc", "bca"))
#
# Intervals :
# Level Percentile BCa
# 95% ( 0.6753, 0.8835 ) ( 0.6344, 0.8561 )
# Calculations and Intervals on Original Scale
# Some BCa intervals may be unstable
#=============================================
# 12.6.2. Bootstrapping several statistics
bs <- function(formula, data, indices) {
d <- data[indices,]
fit <- lm(formula, data=d)
return(coef(fit))
}
library(boot)
set.seed(1234)
results <- boot(data=mtcars, statistic=bs,
R=1000, formula=mpg~wt+disp)
print(results)
# ORDINARY NONPARAMETRIC BOOTSTRAP
#
#
# Call:
# boot(data = mtcars, statistic = bs, R = 1000, formula = mpg ~
# wt + disp)
#
#
# Bootstrap Statistics :
# original bias std. error
# t1* 34.96055404 4.715497e-02 2.546106756
# t2* -3.35082533 -4.908125e-02 1.154800744
# t3* -0.01772474 6.230927e-05 0.008518022
plot(results, index = 2) # figure 12.2
boot.ci(results, type = "bca", index = 2) # t2: car weight
# BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
# Based on 1000 bootstrap replicates
#
# CALL :
# boot.ci(boot.out = results, type = "bca", index = 2)
#
# Intervals :
# Level BCa
# 95% (-5.477, -0.937 )
# Calculations and Intervals on Original Scale
boot.ci(results, type = "bca", index = 3) # t3: engine displacement
# BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
# Based on 1000 bootstrap replicates
#
# CALL :
# boot.ci(boot.out = results, type = "bca", index = 3)
#
# Intervals :
# Level BCa
# 95% (-0.0334, -0.0011 )
# Calculations and Intervals on Original Scale
plot(results, index = 3) # figure 12.3
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