|
Index to Module 4 Notes
|
In Module Notes 4.1, we
indicated that there are two assumptions when doing statistical tests involving
the comparison of two samples to make an inference about population means. The
first is that the samples are drawn from normally distributed populations (or
samples are large without extreme skew); and the second is that the samples
have equal variances. When these assumptions are met, we use the t-Test for Two
Independent Samples Assuming Equal Variances. In the situation where the
variances are not equal, but the parent populations are normally distributed
(or samples are large without extreme skew) , we use the t-Test for Two
independent Samples Assuming Unequal Variances.
What if we cannot make the assumption that the parent populations from which
the samples were drawn are normal, especially when we are working with small
samples? Then a safe alternative is to use a nonparametric test, rather
than the parametric tests we have been using. Parametric tests are those tests
such as the t-Test and the F-Test, which rely upon statistical distributions
and assumptions about the distributions for analysis about population parameters.
Nonparametric tests do not rely upon statistical distributions. Rather,
nonparametric tests for numerical data, such as the one discussed in this
module, make inferences about populations by studying sample measures of
location, such as the median. Rather than using all of the information
available in samples, these nonparametric tests typically use only the ranks
of the observations (Aczel, p. 625). As such, these type of tests work well
with ordinal as well as interval or ratio scale numerical data.
This set of Module notes introduces a nonparametric test for the study of two
independent samples called the Wilcoxon Rank Sum Test (also goes by Mann
Whitney U Test, Aczel, p. 644). This will be followed by a nonparametric test
for the study of two related (dependent) samples called the Wilcoxon
Signed-Rank Test.
Wilcoxon
Rank-Sum Test for Comparing Two Independent Samples
I will continue with
the Brand of Gas data. I am interested in comparing cars tested with Brand A
against those tested with Brand B. The only assumptions required for the
Wilcoxon Rank Sum Test are that the samples are random, and that observations
drawn for each sample are independent from each other. Therefore, this test
could not be used for the paired sample situation of Module 4.2.
The general hypotheses for the Wilcoxon Rank-Sum Test are (Aczel, p. 644):
H0:
The distributions of the two populations are identical
Ha: The distributions of the two populations are not identical
If we want to state the hypotheses in terms of population medians (Levine, 1999), then we are stating that if there is a difference in the populations, it is a difference in location (Aczel, p. 645):
H0:
MedianA = Median B
Ha: MedianA =/= MedianB
Worksheet 4.3.1 provides the
original data. Note I added two columns and ranked the data within each group.
Worksheet 4.3.1
|
Brand A |
Brand B |
Brand A |
Brand B |
|
20 |
20 |
14 |
18 |
|
20 |
20.5 |
15 |
18.5 |
|
19 |
18.5 |
16 |
19 |
|
16 |
20 |
17 |
19 |
|
15 |
19 |
19 |
20 |
|
17 |
19 |
20 |
20 |
|
14 |
18 |
20 |
20.5 |
To perform the Wilcoxon Rank-Sum Test, we replace the observations in the two
samples of size n1 and n2 with their combined ranks.
There will be n = n1 + n2 ranks. We can let Brand A be
group 1 and Brand B be group 2. So, for our example:
Eq. 4.3.1: n = n1 + n2 = nA + nB = 7 + 7 = 14 ranks
We assign the ranks so that
rank 1 is given to the smallest of the combined number, rank 2 to the
second-smallest, and so forth. If several values are tied, a common practice is
to assign each tied observation "the average of the ranks that would
otherwise have been assigned had there been no ties. "
A simple way of assigning the ranks in Excel is shown in Worksheet 4.3.2.
Worksheet 4.3.2
|
Brand A |
Brand B |
A Ranks |
B Ranks |
|
14 |
1 |
||
|
15 |
2 |
||
|
16 |
3 |
||
|
17 |
4 |
||
|
18 |
5 |
||
|
18.5 |
6 |
||
|
19 |
8 |
||
|
19 |
8 |
||
|
19 |
8 |
||
|
20 |
11.5 |
||
|
20 |
11.5 |
||
|
20 |
11.5 |
||
|
20 |
11.5 |
||
|
20.5 |
14 |
||
|
T1 = 41 |
T2 = 64 |
The smallest number is 14, so it gets rank 1, followed by 15, 16, 17, 18 and
18.5 which get ranks 2, 3, 4, 5, and 6 respectively. The next three numbers are
19, 19 and 19 - which are ties. Since these are to get the ranks 7, 8 and 9, we
average the ranks and give each of the 19's the rank of 8. We then proceed
through the data until we finish by assigning rank 14 to the number 21. Note
how I maintained group identity.
The Wilcoxon Rank-Sum Test Statistic is T1, where T1 is
assigned to the group with the smallest sample size. If the groups have the
same sample size, as in our example, the assignment is arbitrary. I'll assign T1
to the first group, Brand A. T1 is the sum of the ranks in that
group, as shown in Worksheet 4.3.2. Note that T2 is the sum of the
ranks in the second group. Equation 4.3.2 shows a check on the rankings:
Eq. 4.3.2: T1 + T2 = n * (n + 1) / 2= 14 * (14 + 1) / 2 = 105
Since T1 + T2 equals 105 (41 + 64), our rankings check.
The Wilcoxon Rank Sum Test Statistic available in Excel is the Z for samples over size 10 (we are a little shy of that with samples of size 7, but I wanted to use a small size problem to illustrate the technique). To compute the Z for this test, we need the Wilcoxon Rank Sum formula for the mean and the standard deviation of T1.
Eq. 4.3.3:
MeanT1 = n1(n+1)/2 = 7(14+1)/2 = 52.5
Eq. 4.3.4: Std DevT1 = Sq Rt { [ (n1n2(n+1) ]
/ 12 } =
Sq Rt { [7*7(14+1)] 12 } = 7.826
Eq. 4.3.5: Z =(T1 - MeanT1) / Std DevT1 = (41-52.5) / 7.826 = -1.47
We can use the Excel Function
=NORMSDIST(-1.47) to get the p-value. The p-value returned by =NORMSDIST(-1.46)
in an Excel Cell is 0.07, which represents the cumulative probability of a
z-score less than or equal to -1.47. Since two times this p-value (2 * 0.07 =
0.14) is greater than 0.05, we fail to reject the null hypothesis and conclude
the populations, from a location point of view, are the same (no difference in
the medians). This is similar to our conclusion from the t-Test for Two Samples
in Module Notes 4.1. Note that I doubled the p-value because the alternative
hypothesis is two-tailed.
Let's do one more example, this time comparing Brand A against Brand C.
Worksheet 4.3.3 provides the data.
Worksheet 4.3.3
|
Brand A |
Brand C |
Brand A |
Brand C |
|
20 |
20 |
14 |
20 |
|
20 |
26 |
15 |
23 |
|
19 |
23 |
16 |
23 |
|
16 |
24 |
17 |
23 |
|
15 |
23 |
19 |
24 |
|
17 |
25 |
20 |
25 |
|
14 |
23 |
20 |
26 |
Worksheet 4.3.4 provides the rankings.
Worksheet 4.3.4
|
Brand A |
Brand C |
A Ranks |
C Ranks |
|
14 |
1 |
||
|
15 |
2 |
||
|
16 |
3 |
||
|
17 |
4 |
||
|
19 |
5 |
||
|
20 |
7 |
||
|
20 |
7 |
||
|
20 |
7 |
||
|
23 |
10 |
||
|
23 |
10 |
||
|
23 |
10 |
||
|
24 |
12 |
||
|
25 |
13 |
||
|
26 |
14 |
||
|
Sum of Ranks = |
T1 = 29 |
T2 = 76 |
Equations 4.3.1 - 4.3.4 are identical to the first example since we have the
same sample sizes in the two groups. The equation to compute the Z score is
repeated as Equation 4.3.6, below:
Eq. 4.3.6: Z = (T1 - MeanT1) / Std DevT1 = (29-52.5)/7.826 = -3.003
We
can use the Excel Function =NORMSDIST(-3.003) to get the p-value. The p-value
returned by =NORMSDIST(-3.003) in an Excel Cell is 0.0013. Since two times this
p-value (2 * 0.0013 = 0.0026) is less than 0.05, we reject the null hypothesis
and conclude the populations, from a location point of view, are different
(there is a significant difference in the medians). This is similar to our
conclusion from the t-Test for Two Samples in Module Notes 4.1.
This ends our work in comparing two independent samples for the purpose
of making inferences about the two populations from which they were drawn, and
our introduction to a nonparametric statistics test. There is a nonparametric
test for comparing two related samples, called the Wilcoxon Signed-Rank
Test. We cover that next.
Wilcoxon
Signed-Rank Test for Comparing Two Related Samples
Recall that in Module 4.2 Notes we introduced the situation in which we wanted
to compare two samples, but the observations in Sample 1 are related to Sample
2. The example used concerned automobiles that were run with two different
Brands of Gas. But rather than assign 7 cars randomly to Group 1, and a
different set of 7 cars randomly selected and assigned to Group 2, only 7 cars
are used. Car number 1 is run with Brand A gas and the mpg is recorded. Then
the tank and lines are purged and the same car is run with Brand B gas and the
mpg is recorded. The same happens for the other six cars in the sample.
In Module Notes 4.2 we used the Two Group t-Test for Paired Samples for the
analysis of the alternative hypothesis that the mean of the first group was not
equal to the mean of the second. That test requires that the distribution of
the differences follows a normal distribution. When that assumption is
seriously violated, it is better to use a nonparametric test that doesn't
require the normality assumption. For example, if the data being measured
involves preference scores on a 1 to 10 scale, with 10 being the highest, we
may find that answers are biased to the high point of the scale. In this case,
the Wilcoxon Signed-Rank Test would be better to use than the Two Group t-Test.
Recall the data shown in the Brand A and Brand B columns of Worksheet 4.3.5. It
comes from Worksheet 4.2.1 in Module 4.2 Notes. I have added some columns in
order to perform the Wilcoxon Sign-Rank Test procedure (Mason, 1999).
Worksheet 4.3.5
|
Car |
Brand A |
Brand B |
Difference |
Absolute Difference |
Rank |
Signed Rank R+ |
Signed Rank R- |
|
1 |
20 |
20 |
0 |
0 |
|||
|
2 |
20 |
20.5 |
-0.5 |
0.5 |
1.5 |
1.5 |
|
|
3 |
19 |
18.5 |
0.5 |
0.5 |
1.5 |
1.5 |
|
|
4 |
16 |
20 |
-4 |
4 |
5 |
5 |
|
|
5 |
15 |
19 |
-4 |
4 |
5 |
5 |
|
|
6 |
17 |
19 |
-2 |
2 |
3 |
3 |
|
|
7 |
14 |
18 |
-4 |
4 |
5 |
5 |
|
|
Total |
1.5 |
19.5 |
The hypothesis being tested in the Wilcoxon Signed-Rank Test nonparametric
method is:
H0: There is no difference in the mpg performance of the two brands
Ha: There is a difference in the mpg performance of the two brands
I
selected the two-tailed alternative hypothesis since I was simply trying to
determine if the brands of gas gave different mpg performance. If a special
additive was put into Brand B and we anticipated higher mpg performance from
Brand B, then I would use the one-tail alternative that Brand B mpg performance
is higher than Brand A.
The first step in the Wilcoxon Signed-Rant Test is to compute the difference
between each pairs of mpg scores as shown in the column labeled
"Difference" in Worksheet 4.3.5. We only are interested in positive
and negative differences, and samples with 0 differences are dropped from
further analysis. Next, the absolute value of the differences is stated in the
column "Absolute Difference." Ranks are then assigned using the
ranking procedure covered previously in the Wilcoxon Rank-Sum Test in the first
section of these Module 4.3 Notes. Note that higher ranks are assigned to
greater absolute differences. Next, each assigned rank is then given the same
sign from the "Difference" column and reported in the "Signed
Rank+" and "Signed Rank-" columns. Finally, the signed ranks are summed
and reported in the bottom "Total" row. The smaller of the two rank
sums is used as the test statistic and referred to as T.
The critical values for the Wilcoxon Rank-Sum Test are tabled in Worksheet
4.3.6 (Abridged from Mason, Appendix H, 1999).
Worksheet 4.3.5
|
n |
2 alpha à |
0.10 |
0.05 |
0.02 |
0.01 |
|
4 |
0 |
||||
|
5 |
1 |
0 |
|||
|
6 |
2 |
2 |
|||
|
7 |
4 |
3 |
0 |
||
|
8 |
7 |
5 |
1 |
0 |
|
|
9 |
9 |
8 |
5 |
1 |
|
|
10 |
12 |
10 |
3 |
3 |
|
|
11 |
16 |
13 |
7 |
5 |
|
|
12 |
19 |
17 |
9 |
7 |
|
|
13 |
24 |
21 |
12 |
9 |
|
|
14 |
28 |
25 |
15 |
12 |
|
|
15 |
33 |
30 |
19 |
15 |
|
|
16 |
39 |
35 |
23 |
19 |
|
|
17 |
45 |
41 |
27 |
23 |
|
|
18 |
51 |
47 |
32 |
27 |
|
|
19 |
58 |
53 |
37 |
32 |
|
|
20 |
65 |
60 |
43 |
37 |
Selecting an alpha value of 0.05, which gives 2 alpha of 0.10 for the two-tail test,
we go down that column to the intersection of the n = 7 row. The critical value
of the Wilcoxon Rank-Sum Test is shown to be 4. If the Test Statistic T
is less than the critical value, we fail to reject the null hypothesis; and
reject the null if T is greater than the critical value. Since T
= 1.5 for this sample, we fail to reject the null hypothesis, and conclude
there is no difference in mpg performance.
When this same data was analyzed with the parametric t-Test for Means for
Paired Samples, we rejected the null hypothesis. This should cause us to
recognize the importance of meeting assumptions. If we are certain that the
underlying distribution of the difference scores is normal, then the t-Test for
Paired Samples is appropriate. But if we are not sure about the underlying
distribution, especially when we have small samples and extreme values, the
more conservative and ethical approach would be to use the nonparametric
Wilcoxon Signed-Rank Test.
This finishes our coverage of
nonparametric tests for comparing two samples. The next topics concern
comparing more than two samples.
References:
Ken
Black. Business Statistics for Contemporary Decision Making. Fourth Edition,
Wiley. Chapter 10 & 11
D. Groebner, P. Shannon, P.
Fry & K. Smith. Business Statistics:
A Decision Making Approach, Fifth Edition, Prentice Hall, Chapter 10
Aczel, A. (1993). Complete
Business Statistics (2nd. ed.). Homewood, IL: Irwin.
Levine, D., Berenson,
M. & Stephan, D. (1999). Statistics for Managers Using Microsoft Excel (2nd.
ed.). Upper Saddle River, NJ: Prentice-Hall, Chapter 9.
Mason, R., Lind, D. & Marchal, W. (1999). Statistical Techniques in
Business and Economics (10th. ed.). Boston: Irwin McGraw Hill, Chapter
15.