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Summary and Analysis of Extension Program Evaluation in R

Salvatore S. Mangiafico

Two-sample Mann–Whitney U Test

When to use this test

 

The two-sample Mann–Whitney U test compares values for two groups.  A significant result suggests that the values for the two groups are different.  It is equivalent to a two-sample Wilcoxon rank-sum test.

 

In the context of this book, the test is useful to compare the scores or ratings from two speakers, two different presentations, or two groups of audiences.

 

If the shape and spread of the distributions of values of each group is similar, then the test compares the medians of the two groups.  Otherwise, the test is really testing if there is a systematic difference in the values of the two groups.

 

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

 

The test is performed with the wilcox.test function.

 

If the distributions of values of each group are similar in shape, but have outliers, then Mood’s median test is an appropriate alternative.

 

Appropriate data

•  Two-sample data.  That is, one-way data with two groups only

•  Dependent variable is ordinal, interval, or ratio

•  Independent variable is a factor with two levels.  That is, two groups

•  Observations between groups are independent.  That is, not paired or repeated measures data

•  In order to be a test of medians, the distributions of values for each group need to be of similar shape and spread; outliers affect the spread.  Otherwise the test is a test of distributions.

 

Hypotheses

If the distributions of the two groups are similar in shape and spread:

•  Null hypothesis:  The medians of values for each group are equal.

•  Alternative hypothesis (two-sided): The medians of values for each group are not equal.

 

If the distributions of the two groups are not similar in shape and spread:

•  Null hypothesis:  The distribution of values for each group are equal.

•  Alternative hypothesis (two-sided): There is systematic difference in the distribution of values for the groups.

 

Interpretation

If the distributions of the two groups are similar in shape:

Significant results can be reported as e.g. “The median value of group A was significantly different from that of group B.”

 

If the distributions of the two groups are not similar in shape:

Significant results can be reported as e.g. “Values for group A were significantly different from those for group B.” 

 

Other notes and alternative tests

The Mann–Whitney U test can be considered equivalent to the Kruskal–Wallis test with only two groups.

 

Mood’s median test compares the medians of two groups.  It is described in the next chapter.

 

Another alternative is to use cumulative link models for ordinal data, which are described later in this book.

 

Packages used in this chapter

 

The packages used in this chapter include:

•  psych

•  FSA

•  lattice

 

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


if(!require(psych)){install.packages("psych")}
if(!require(FSA)){install.packages("FSA")}
if(!require(lattice)){install.packages("lattice")}


Two-sample Mann–Whitney U test example

 

This example re-visits the Pooh and Piglet data from the Descriptive Statistics with the likert Package chapter.

 

It answers the question, “Are Pooh's scores significantly different from those of Piglet?”

 

The Mann–Whitney U test is conducted with the wilcox.test function, which produces a p-value for the hypothesis.  First the data are summarized and examined using bar plots for each group.

 

Because the bar plots show that the distributions of scores for Pooh and Piglet are relatively similar in shape, the Mann–Whitney U test can be interpreted as a test of medians.

 

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
")

Data = read.table(textConnection(Input),header=TRUE)


### 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
  Piglet 1 6 2 1 0
  Pooh   0 0 1 6 3


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


prop.table(XT,
           margin = 1)


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


Bar plots of data by group


library(lattice)

histogram(~ Likert.f | Speaker,
          data=Data,
          layout=c(1,2)      #  columns and rows of individual plots
          )


image


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  Piglet 10  2.3 0.823   1  2      2 2.75   4        0
2    Pooh 10  4.2 0.632   3  4      4 4.75   5        0


Two-sample Mann–Whitney U test example

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

 

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

 

Wilcoxon rank sum test with continuity correction
W = 5, p-value = 0.0004713
alternative hypothesis: true location shift is not equal to 0

### You may get a "cannot compute exact p-value with ties" error.
###    You can ignore this or use the exact=FALSE option.

Exercises J


1. Considering Pooh and Piglet’s data,

a.  What was the median score for each instructor?

b.  What were the first and third quartiles for each instructor’s scores?

c.  Are the data for both instructors reasonably symmetric about their medians?

d.  Based on your previous answer, what is the null hypothesis for the Mann–Whitney test?

e.  According to the Mann–Whitney test, is there a difference in scores between the instructors?

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


2. Brian and Stewie Griffin want to assess the education level of students in their courses on creative writing for adults.  They want to know the median education level for each class, and if the education level of the classes were different between 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


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

 

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

 

b.  What were the first and third quartiles for each instructor’s scores?

c.  Are the data for both instructors reasonably symmetric about their medians?

d.  Based on your previous answer, what is the null hypothesis for the Mann–Whitney test?

e.  According to the Mann–Whitney test, is there a difference in scores between the instructors?

f.  Plot Brian and Stewie’s data in a way that helps you visualize the data.  Do the results reflect what you would expect from looking at the plot? 

 

g.  How would you summarize the results of the descriptive statistics and tests?  Include your practical interpretation.