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1. Wallis Universal Manual Pdf
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The Kruskal–Wallis test is a rank-based test that is similarto the Mann–Whitney U test, but can be applied to one-way data with more thantwo groups.

Without further assumptions about the distribution of thedata, the Kruskal–Wallis test does not address hypotheses about the medians ofthe groups. Instead, the test addresses if it is likely that an observation inone group is greater than an observation in the other. This is sometimesstated as testing if one sample has stochastic dominance compared with theother.

The test assumes that the observations are independent. That is, it is not appropriate for paired observations or repeated measuresdata.

It is performed with the kruskal.test function.

Appropriate effect size statistics include maximum Varghaand Delaney’s A, maximum Cliff’s delta, Freeman’s theta,and epsilon-squared.

Post-hoc tests

The outcome of the Kruskal–Wallis test tells you if thereare differences among the groups, but doesn’t tell you which groups aredifferent from other groups. In order to determine which groups are differentfrom others, post-hoc testing can be conducted. Probably the most commonpost-hoc test for the Kruskal–Wallis test is the Dunn test, here conducted withthe dunnTest function in the FSA package.

Appropriate data

• One-way data

• Dependent variable is ordinal, interval, or ratio

• Independent variable is a factor with two or more levels. Thatis, two or more groups

• Observations between groups are independent. That is, notpaired or repeated measures data

• In order to be a test of medians, the distributions ofvalues for each group need to be of similar shape and spread. Otherwise thetest is typically a test of stochastic equality.

Hypotheses

• Null hypothesis: The groups are sampled from populationswith identical distributions. Typically, that the sampled populations exhibitstochastic equality.

• Alternative hypothesis (two-sided): The groups are sampledfrom populations with different distributions. Typically, that one sampledpopulation exhibits stochastic dominance.

Interpretation

Significant results can be reported as “There was a significantdifference in values among groups.”

Post-hoc analysis allows you to say “There was a significantdifference in values between groups A and B.”, and so on.

Other notes and alternative tests

Mood’s median test compares the medians of groups.

Packages used in this chapter

The packages used in this chapter include:

• psych

• FSA

• lattice

• multcompView

• rcompanion

The following commands will install these packages if theyare not already installed:

if(!require(psych)){install.packages('psych')}
if(!require(FSA)){install.packages('FSA')}
if(!require(lattice)){install.packages('lattice')}
if(!require(multcompView)){install.packages('multcompView')}
if(!require(rcompanion)){install.packages('rcompanion')}

Kruskal–Wallis test example

This example re-visits the Pooh, Piglet, and Tigger datafrom the Descriptive Statistics with the likert Package chapter.

It answers the question, “Are the scores significantlydifferent among the three speakers?”

The Kruskal–Wallis test is conducted with the kruskal.testfunction, which produces a p-value for the hypothesis. First the dataare summarized and examined using bar plots for each group.

Input =('
Speaker Likert
Pooh 3
Pooh 5
Pooh 4
Pooh 4
Pooh 4
Pooh 4
Pooh 4
Pooh 4
Pooh 5
Pooh 5
Piglet 2
Piglet 4
Piglet 2
Piglet 2
Piglet 1
Piglet 2
Piglet 3
Piglet 2
Piglet 2
Piglet 3
Tigger 4
Tigger 4
Tigger 4
Tigger 4
Tigger 5
Tigger 3
Tigger 5
Tigger 4
Tigger 4
Tigger 3
')
Data = read.table(textConnection(Input),header=TRUE)

### Order levels of the factor; otherwise R will alphabetize them
Data$Speaker = factor(Data$Speaker,
levels=unique(Data$Speaker))

### Create a new variable which is the likert scores as an ordered factor
Data$Likert.f = factor(Data$Likert,
ordered = TRUE)
### Check the data frame
library(psych)
headTail(Data)
str(Data)

summary(Data)
### Remove unnecessary objects
rm(Input)

Summarize data treating Likert scores as factors

xtabs( ~ Speaker + Likert.f,
data = Data)

Likert.f
Speaker 1 2 3 4 5
Pooh 0 0 1 6 3
Piglet 1 6 2 1 0
Tigger 0 0 2 6 2

XT = xtabs( ~ Speaker + Likert.f,
data = Data)

prop.table(XT,
margin = 1)

Likert.f
Speaker 1 2 3 4 5
Pooh 0.0 0.0 0.1 0.6 0.3
Piglet 0.1 0.6 0.2 0.1 0.0
Tigger 0.0 0.0 0.2 0.6 0.2

Bar plots of data by group

library(lattice)
histogram(~ Likert.f Speaker,
data=Data,
layout=c(1,3) # columns and rows ofindividual plots
)

Summarize data treating Likert scores as numeric

library(FSA)
Summarize(Likert ~ Speaker,
data=Data,
digits=3)

Speaker n mean sd min Q1 median Q3 max percZero
1 Pooh 10 4.2 0.632 3 4 4 4.75 5 0
2 Piglet 10 2.3 0.823 1 2 2 2.75 4 0
3 Tigger 10 4.0 0.667 3 4 4 4.00 5 0

Kruskal–Wallis test example

This example uses the formula notation indicating that Likertis the dependent variable and Speaker is the independent variable. The data=option indicates the data frame that contains the variables. For the meaningof other options, see ?kruskal.test.

kruskal.test(Likert ~ Speaker,
data = Data)

Kruskal-Wallis rank sum test
Kruskal-Wallis chi-squared = 16.842, df = 2, p-value = 0.0002202

Effect size

Statistics of effect size for the Kruskal–Wallis testprovide the degree to which one group has data with higher ranks than anothergroup. They are related to the probability that a value from one group will begreater than a value from another group. Unlike p-values, they are notaffected by sample size.

Appropriate effect size statistics for the Kruskal–Wallistest include Freeman’s theta and epsilon-squared. epsilon-squaredis probably the most common. For Freeman’s theta, an effect size of 1indicates that the measurements for each group are entirely greater or entirelyless than some other group, and an effect size of 0 indicates that there is noeffect; that is, that the groups are absolutely stochastically equal.

Another option is to use the maximum Cliff’s delta orVargha and Delaney’s A (VDA) from pairwise comparisons of all groups. VDA is the probability that an observation from one group is greater than anobservation from the other group. Because of this interpretation, VDA is an effectsize statistic that is relatively easy to understand.

Interpretation of effect sizes necessarily varies bydiscipline and the expectations of the experiment. The following guidelinesare based on my personal intuition or published values. They should not beconsidered universal.

Technical note: The values for the interpretations forFreeman’s theta to epsilon-squared below were derived by keepingthe interpretation for epsilon-squared constant and equal to that forthe Mann–Whitney test. Interpretation values for Freeman’s theta weredetermined through comparing Freeman’s theta to epsilon-squaredfor simulated data (5-point Likert items, n per group between 4 and 25).

Interpretations for Vargha and Delaney’s A andCliff’s delta come from Vargha and Delaney (2000).

epsilon-squared

library(rcompanion)
epsilonSquared(x = Data$Likert,
g = Data$Speaker)

epsilon.squared
0.581

Freeman’s theta

library(rcompanion)
freemanTheta(x = Data$Likert,
g = Data$Speaker)

Freeman.theta
0.64

Maximum Vargha and Delaney’s A or Cliff’s delta

Here, the multiVDA function is used to calculateVargha and Delaney’s A (VDA), Cliff’s delta (CD), and rbetween all pairs of groups. The function identifies the comparison with themost extreme VDA statistic (0.95 for Pooh – Piglet). That is, itidentifies the most disparate groups.

source('http://rcompanion.org/r_script/multiVDA.r')
library(rcompanion)
library(coin)
multiVDA(x = Data$Likert,
g = Data$Speaker)

$pairs
Comparison VDA CD r VDA.m CD.m r.m
1 Pooh - Piglet = 0 0.95 0.90 0.791 0.95 0.90 0.791
2 Pooh - Tigger = 0 0.58 0.16 0.154 0.58 0.16 0.154
3 Piglet - Tigger = 0 0.07 -0.86 -0.756 0.93 0.86 0.756
$comparison
Comparison
'Pooh - Piglet = 0'
$statistic
VDA
0.95
$statistic.m
VDA.m
0.95

Post-hoc test: Dunn test for multiple comparisons ofgroups

If the Kruskal–Wallis test is significant, a post-hocanalysis can be performed to determine which groups differ from each othergroup.

Probably the most popular post-hoc test for theKruskal–Wallis test is the Dunn test. The Dunn test can be conducted with the dunnTestfunction in the FSA package.

Because the post-hoc test will produce multiple p-values,adjustments to the p-values can be made to avoid inflating thepossibility of making a type-I error. There are a variety of methods for controllingthe familywise error rate or for controlling the false discovery rate. See ?p.adjustfor details on these methods.

When there are many p-values to evaluate, it isuseful to condense a table of p-values to a compact letter displayformat. In the output, groups are separated by letters. Groups sharing thesame letter are not significantly different. Compact letter displays are aclear and succinct way to present results of multiple comparisons.

### Order groups by median

Data$Speaker = factor(Data$Speaker,
levels=c('Pooh', 'Tigger','Piglet'))
levels(Data$Speaker)
### Dunn test
library(FSA)
DT = dunnTest(Likert ~ Speaker,
data=Data,
method='bh') # Adjustsp-values for multiple comparisons;
# See ?dunnTestfor options
DT

Dunn (1964) Kruskal-Wallis multiple comparison
p-values adjusted with the Benjamini-Hochberg method.
Comparison Z P.unadj P.adj
1 Pooh - Tigger 0.4813074 0.6302980448 0.6302980448
2 Pooh - Piglet 3.7702412 0.0001630898 0.0004892695
3 Tigger - Piglet 3.2889338 0.0010056766 0.0015085149

### Compact letter display

PT = DT$res
PT
library(rcompanion)
cldList(P.adj ~ Comparison,
data = PT,
threshold = 0.05)

Group Letter MonoLetter
1 Pooh a a
2 Tigger a a
3 Piglet b b
Groups sharing a letter not signficantly different(alpha = 0.05).

Post-hoc test: pairwise Mann–Whitney U-tests formultiple comparisons

I don’t recommend using pairwise Mann–Whitney U-tests for post-hoctesting for the Kruskal–Wallis test, but the following example shows how thiscan be done.
The pairwise.wilcox.test function produces a table of p-values comparingeach pair of groups.
To prevent the inflation of type I error rates, adjustments to the p-valuescan be made using the p.adjust.method option. Here the fdrmethod is used. See ?p.adjust for details on available p-valueadjustment methods.
When there are many p-values to evaluate, it is useful to condense atable of p-values to a compact letter display format. This can beaccomplished with a combination of the fullPTable function in the rcompanionpackage and the multcompLetters function in the multcompViewpackage.
In a compact letter display, groups sharing the same letter are notsignificantly different.

Here the fdrp-value adjustment method is used. See ?p.adjustfor details on available methods.

The code creates a matrix of p-values called PT, then convertsthis to a fuller matrix called PT1. PT1 is then passed to the multcompLettersfunction to be converted to a compact letter display.

Note that the p-value results of the pairwiseMann–Whitney U-tests differ somewhat from those of the Dunn test.

### Order groups by median

Data$Speaker = factor(Data$Speaker,
levels=c('Pooh', 'Tigger', 'Piglet'))
Data
### Pairwise Mann–Whitney
PT = pairwise.wilcox.test(Data$Likert,
Data$Speaker,
p.adjust.method='fdr')
# Adjusts p-values formultiple comparisons;
# See ?p.adjust foroptions

PT

Pairwise comparisons using Wilcoxon rank sum test
Pooh Tigger
Tigger 0.5174 -
Piglet 0.0012 0.0012
P value adjustment method: fdr
### Note that the values in the table are p-valuescomparing each
### pair of groups.

### Convert PT to a full table and callit PT1
PT = PT$p.value ### Extract p-value table

library(rcompanion)
PT1 = fullPTable(PT)
PT1

Pooh Tigger Piglet
Pooh 1.000000000 0.517377650 0.001241095
Tigger 0.517377650 1.000000000 0.001241095
Piglet 0.001241095 0.001241095 1.000000000

### Produce compact letter display
library(multcompView)
multcompLetters(PT1,
compare='<',
threshold=0.05, # p-value to useas significance threshold
Letters=letters,
reversed = FALSE)

Pooh Tigger Piglet
'a' 'a' 'b'
### Values sharing a letter are not significantlydifferent

Plot of medians and confidence intervals

The following code uses the groupwiseMedian functionto produce a data frame of medians for each speaker along with the 95%confidence intervals for each median with the percentile method. These mediansare then plotted, with their confidence intervals shown as error bars. Thegrouping letters from the multiple comparisons (Dunn test or pairwiseMann–Whitney U-tests) are added.

Wallis Universal Manual Pdf

Note that bootstrapped confidence intervals may not bereliable for discreet data, such as the ordinal Likert data used in theseexamples, especially for small samples.

library(rcompanion)
Sum = groupwiseMedian(Likert ~ Speaker,
data = Data,
conf = 0.95,
R = 5000,
percentile = TRUE,
bca = FALSE,
digits = 3)
Sum

Speaker n Median Conf.level Percentile.lower Percentile.upper
1 Pooh 10 4 0.95 4.0 5.0
2 Piglet 10 2 0.95 2.0 3.0
3 Tigger 10 4 0.95 3.5 4.5

X = 1:3
Y = Sum$Percentile.upper + 0.2
Label = c('a', 'b', 'a')
library(ggplot2)
ggplot(Sum, ### The data frame touse.
aes(x = Speaker,
y = Median)) +
geom_errorbar(aes(ymin = Percentile.lower,
ymax = Percentile.upper),
width = 0.05,
size = 0.5) +
geom_point(shape = 15,
size = 4) +
theme_bw() +
theme(axis.title = element_text(face = 'bold')) +
ylab('Median Likert score') +
annotate('text',
x = X,
y = Y,
label = Label)

Wallis Universal Cm101 Manual

Plot of median Likert score versus Speaker. Error bars indicate the 95% confidenceintervals for the median with the percentile method.

References

Cohen, J. 1988. Statistical Power Analysis for the BehavioralSciences, 2nd Edition. Routledge.

Vargha, A. and H.D. Delaney. A Critique and Improvement of theCL Common Language Effect Size Statistics of McGraw and Wong. 2000. Journal ofEducational and Behavioral Statistics 25(2):101–132.

Exercises L

1. Considering Pooh, Piglet, and Tigger’s data,

a. What was the median score for each instructor?

Wallis Universal Cm 103 Manual

b. According to the Kruskal–Wallis test, is there a statisticaldifference in scores among the instructors?

c. What is the value of maximum Vargha and Delaney’s A forthese data?

d. How do you interpret this value? (What does it mean? And isthe standard interpretation in terms of “small”, “medium”, or “large”?)

e. Looking at the post-hoc analysis, which speakers’ scoresare statistically different from which others? Who had the statisticallyhighest scores?

f. How would you summarize the results of the descriptivestatistics and tests? Include practical considerations of any differences.

2. Brian, Stewie, and Meg want to assess the education level of students intheir courses on creative writing for adults. They want to know the medianeducation level for each class, and if the education level of the classes weredifferent among instructors.

They used the following table to code his data.

Code Abbreviation Level
1 < HS Less than high school
2 HS High school
3 BA Bachelor’s
4 MA Master’s
5 PhD Doctorate

The following are the course data.

Instructor Student Education
'Brian Griffin' a 3
'Brian Griffin' b 2
'Brian Griffin' c 3
'Brian Griffin' d 3
'Brian Griffin' e 3
'Brian Griffin' f 3
'Brian Griffin' g 4
'Brian Griffin' h 5
'Brian Griffin' i 3
'Brian Griffin' j 4
'Brian Griffin' k 3
'Brian Griffin' l 2
'Stewie Griffin' m 4
'Stewie Griffin' n 5
'Stewie Griffin' o 4
'Stewie Griffin' p 4
'Stewie Griffin' q 4
'Stewie Griffin' r 4
'Stewie Griffin' s 3
'Stewie Griffin' t 5
'Stewie Griffin' u 4
'Stewie Griffin' v 4
'Stewie Griffin' w 3
'Stewie Griffin' x 2
'Meg Griffin' y 3
'Meg Griffin' z 4
'Meg Griffin' aa 3
'Meg Griffin' ab 3
'Meg Griffin' ac 3
'Meg Griffin' ad 2
'Meg Griffin' ae 3
'Meg Griffin' af 4
'Meg Griffin' ag 2
'Meg Griffin' ah 3
'Meg Griffin' ai 2
'Meg Griffin' aj 1

Wallis Universal Cm 101 Manual Pdf

For each of the following, answer the question, and showthe output from the analyses you used to answer the question.

a. What was the median education level for each instructor’sclass? (Be sure to report the education level, not just the numeric code!)

b. According to the Kruskal–Wallis test, is there a differencein the education level of students among the instructors?

c. What is the value of maximum Vargha and Delaney’s A forthese data?

d. How do you interpret this value? (What does it mean? And isthe standard interpretation in terms of “small”, “medium”, or “large”?)

Wallis Universal Cm 101 Manuale Pdf

e. Looking at the post-hoc analysis, which classes educationlevels are statistically different from which others? Who had the statisticallyhighest education level?

f. Plot Brian, Stewie, and Meg’s data in a way that helps youvisualize the data. Do the results reflect what you would expect from lookingat the plot?

g. How would you summarize the results of the descriptive statisticsand tests? What do you conclude practically?