#Lab 10

#274-Wilcox (Fall 2019)

#Name:

#Student ID:

rm(list=ls())

source(‘Rallfun-v33.txt’)

#1) Import the dataset lab10hw1.txt in table form:

#2) For this dataset, what is our dependent variable?

#3) How many independent variables do we have?

#4) How many levels does each independent variable have (use the function unique(x) to check)?

#5) Make a boxplot for this set of data (submit the image). What problem do you see?

#6) What is our null hypothesis?

#7) Now use the classic method to analyze this dataset using the format aov(x~factor(g)).

# Save this as an object called hw1.anova.

#NOTE: MAKE SURE TO USE factor() AROUND YOUR GROUPING VARIABLE SO IT IS TREATED AS A FACTOR, NOT AS A NUMERIC VARIABLE.

# Then summarize these results using summary(hw1.anova).

#8) Do we reject or do we fail to reject the null hypothesis?

#9) Now let’s use the t1way() function, which is based on trimmed means and can deal with heteroscedasticity.

#Hint 1: First, reorganize your data using fac2list(x, g). Save your new list as hw1.list.

#Hint 2: You will need to have loaded in the source code to use the t1way function.

#10) Do we reject or do we fail to reject the null hypothesis from 1.9?

———————————————————————————————————————————————————-

Lab 10 lecture notes:

#Lab 10

#Lab 10-Contents

#1. One-Way Independent Groups ANOVA (Equal Variance)

#2. One-Way Independent Groups ANOVA (Unequal Variance-Welch’s Test)

#———————————————————————————

# 1. One-Way Independent Groups ANOVA (Equal Variance)

#———————————————————————————

#Scenario for first exercise:

# A professor is interested in the effect of visualization strategies

#on test performance. In order to study this, he tells students in

#his statistics class that they will have a 15 question exam in

#two weeks. Then, he randomly assigns students to three groups.

#

# The first group is told to spend 15 min each day vizualizing

#the outcome of getting an A on the test to vividly imagine

#the exam with an “A” written on it and how great it will feel.

#

# The second group is a control group that does no visualization.

#

# The third group is told to spend 15 min each day visualizing

#the process of studying for the exam: imagine the hours of studying,

#reviewing their chapters, working through chapter problems,

# quizzing themeselves, etc.

# Two weeks later, the students take the exam and the professor

# records how many questions the students answer correctly out of 15.

#So, the groups are:

#Group 1: Visualize Outcome (Grade)

#Group 2: No visualization (Control)

#Group 3: Visiualize Process (Studying)

######################################################

#Question: Are the groups here Independent?

######################################################

#We’ll instroduce a few new terms:

#Factor: A variable that consists of categories.

#Levels: The categories of the Factor variable.

#In our example above, the variable that contains

#the groups is called “Group”.

#So, our factor is the variable “Group”

#How many levels are there for the Group Factor?

#Let’s read in LAB10A.txt

lab10a=read.table(‘LAB10A.txt’, header=T)

#While we can easily see the levels for the Group

#factor we could also use a new command to figure out

#the number of unique levels.

#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^#

# Number of Unique Levels: unique(data$variable)

#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^#

unique(lab10a$Group)

#As we can see, there are 3 levels. 1, 2, and 3

#Look at boxplot of each group using

#boxplot(y~group, data=data)

par(mfrow=c(1,1))

boxplot(Score~Group, data=lab10a)

#Do you think the means will be different (statistically)

#between the groups?

#Before we begin to test for differences between

#the means, let’s wrtie out our NUll

#and Alternative Hyhpotheses

#H0: The means are equal (mu1=mu2=mu3)

#HA: At least one mean is different.

#(eg. mu1 != mu2 OR mu1 != mu3 OR mu2 != mu3 )

#To test the Hypothesis we can use the ANOVA function aov():

#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^#

# One-Way ANOVA: aov(y~factor(g), data)

#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^#

#The aov() function assumes that the

#variance is the same within each of the groups.

mod1=aov(Score ~ factor(Group), data=lab10a)

summary(mod1)

#A) If pval < alpha, then Reject the Null Hypothesis

#B) If pval > alpha, then Fail to Reject the Null Hypothesis

#Do we Reject or Fail to Reject the Null?

#Reject 0.00129 < .05 then Reject H0

#What does this tell us? That the groups are different?

#If so, how do we know which groups?

#P-value we just got is called the Omnibus P-value,

#which tells us that there are differences somewhere

#With this P-value we often use the term

#”Main Effect” to say that there is an effect of the

#factor on the outcome.

#In this instance we’d say that there is a Main Effect

#of Group on the Score.

#To Answer which groups are different, we need to first

#conver the data into List Mode (a different way

#of storing the data). We can convert the factor Group

#to a list using the function fac2list(y, g)

#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^#

# Convert Factors to List Data: fac2list(data$y, data$g)

#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^#

listA=fac2list(lab10a$Score, lab10a$Group)

listA

#Once the data is in List Mode we have to use the

#lincon() command from Dr. Wilcox’s source code.

#The lincon() package is used to compare the groups while

#controlling for the experimentwise Type 1 error rate.

#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^#

# Compare Groups: lincon(list_name, tr=0.2)

#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^#

#By default lincon() compares groups using 20% trimming.

#We will set this to 0 for now:

lincon(listA, tr=0)

#result:

# H0_1: mu1=mu2 — p=0.32 —Fail to reject

# H0_2: mu1=mu3 — p=0.0009 —Reject

# H0_3: mu2=mu3 — p=0.008 —Reject

#———————————————————————————

# 2. One-Way Independent Groups ANOVA (Unequal Variance-Welch’s Test)

#———————————————————————————

# We just learned how to conduct a One-Way ANOVA

# when the variances are equal within each group.

# Now, we will learn how to conduct a One-Way ANOVA

#for then the variance is not equal.

# Let’s start by reading in the LAB10B.txt datafile.

lab10b=read.table(‘LAB10B.txt’, header=T)

# Then examine a boxplot of all of it.

boxplot(Score~factor(Group), data=lab10b)

# What do we notice about this boxplot?

#—–

# Let’s start by running the equal variance ANOVA

#on the data (which of course is WRONG!)

mod2=aov(Score ~ factor(Group), data=lab10b) #—DON’T

summary(mod2)

#A) If pval < alpha, then Reject the Null Hypothesis

#B) If pval > alpha, then Fail to Reject the Null Hypothesis

# Do we Reject or Fail to Reject the Null?

#Fail to reject: p-value=0.0895 > .05 !!!INCORRECT—-

#—-

# Now let’s try to run the correct test that assumes

#unequal variance.

#We call this the Welch’s test (just like in the t-test)

#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^#

# Welch’s One-Way ANOVA: t1way(list_name, tr=0.20)

#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^#

#In order to use this t1way function,

#we will first need to convert the data to

#List Mode using fac2list()

listB=fac2list(lab10b$Score, lab10b$Group)

t1way(listB, tr=0.2)

# Do we Reject or Fail to Reject the Null?

#Reject: p-value:0.04966583 <.05

#Again, we can use the lincon() command to

#find out Where the group differences are.

#This time we will use the 20% trimming.

lincon(listB, tr=0.2)

# G1 and G2: p-value=0.92210409 > .05 Fail to reject

# G1 and G3: p-value=0.19451518 > .05 Fail to reject

#G2 and G3: p-value=0.03227316 < .05 Reject

#