#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 lets 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?
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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-Welchs 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?
######################################################
#Well 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?
#Lets 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, lets 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 wed 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. Wilcoxs 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-Welchs 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. # Lets 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? # # Lets start by running the equal variance ANOVA #on the data (which of course is WRONG!) mod2=aov(Score ~ factor(Group), data=lab10b) #DONT 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 lets try to run the correct test that assumes
#unequal variance.
#We call this the Welchs test (just like in the t-test)
#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^#
# Welchs 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 #
