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WINKS
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Chapter
4 Part 4
Non-Parametric
Procedures
Mann-Whitney
test of two independent groups
Non-Parametric
procedures are appropriate if you are comparing two or more independent groups,
but you cannot make the assumption that the observed data follow a normal
distribution or that the variances are equal. It is also useful if you do not
have exact data values for the observations but you do have order statistics,
that is, you don't know the exact response values but you know which is largest,
next largest, and so forth, to smallest. The samples must be randomly and
independently taken from populations that differ only with respect to location,
and the variable of interest should be continuous.
The
hypotheses being tested are:
Ho:
There is no difference in the medians of the groups.
Ha: There is a difference in the medians of the groups.
The
Mann-Whitney and Kruskal-Wallis non-parametric procedures differ from the
independent groups analysis in that the ranks, or order, of the data are used
for the analysis rather than the data values themselves.
The
fertilizer data used for this example it the same as the data used in the t-test
example. Follow these steps to do this example:
Step
1: Open the
database named FERTILIZ (or create it as described in the t-test example) and
choose the Non-Parametric Comparisons option from the Analyze menu.
Step
2: From the
Non-Parametrics Comparisons menu select "Ind. Grp - Mann-Whitney, Kruskal-Wallis."
Note:
Critical Values of the Mann-Whitney U Distribution at 0.05 (two tailed) are
found in the printed manual.
Step
3: Select
GROUP as the group field and OBS (Height) as the data field and choose Ok. The
results will be displayed in the viewer.
Step
4: The
results appear in the viewer: the Mann-Whitney U statistic, the rank sums,
sample sizes and mean ranks of the groups, a z statistic and an approximate
p-value. In this case, U'=24.00, U = 18, z=0.357 and p=0.721.
The
p-value of 0.721 is large so the null hypothesis of no difference in medians
between groups is not rejected. There is not sufficient evidence based on this
procedure to say that there is a difference between the median heights of plants
in the two groups grown using different fertilizers.
Kruskal-Wallis
Procedure
If
more than two independent groups are being compared using non-parametric
methods, WINKS uses the Kruskal-Wallis test. The data are ranked and summed as
in the Mann-Whitney procedure. The hypotheses being tested are:
Ho:
There is no difference in the medians of the groups.
Ha:
There is a difference in the medians of the groups.
The
Kruskal-Wallis procedure tests whether there is a difference among several
treatment groups, but does not identify where the difference lies. WINKS
performs a multiple comparison test to identify which groups are different from
which others. Note: If any n’s
are less than 5 and k is 3, Kruskal-Wallis tables of critical values (available
in most statistical analysis non-parametric texts) should be used. When k is 2,
the test reduces to the Mann-Whitney test described above.
The
data used in this example are weights of four groups of seven randomly assigned
animals, each group given a different feed treatment. Suppose you want to test
whether there is a difference among the effects of the different treatments. You
have reason to believe that the populations from which these samples are taken
is not normally distributed, so you choose to use a non-parametric procedure.
Data for Kruskal-Wallis Procedure
Group
1 Group 2 Group
3 Group 4
50.8
68.7
82.6 76.9
57.0
67.7
74.1 72.2
44.6
66.3
80.5 73.7
51.7
69.8
80.3 74.2
48.2
66.9
81.5 70.6
51.3
65.2
78.6 75.3
49.0
62.0
76.1 69.8
Follow
these steps to perform this analysis:
Step
1: Since the
groups are independent, the database will include two fields (TREATMENT and
WEIGHT) and 28 records (one for each animal). Create a database by using
pre-defined structure "Independent group t-test and ANOVA." This will
create a database with the fields GROUP and OBS. The GROUP variable will be used
for the Treatment type (1, 2, 3, or 4) and the OBS will be used for the Weight.
Of course, you can choose to create a custom database and enter a structure
containing the fields named TREATMENT and WEIGHT. This database is similar to
the one used in the independent group ANOVA previously described. Or, open the
KRUSKAL.DBF database and skip to step 3.
Step
2: The data
you will enter in the first record is 1 (for Group 1) and 50.8. Enter the data
for the 28 records. For Example:
GROUP OBS
(TREATMENT)
1 50.8
1
57.0
1
44.6
1
51.7
1
48.2
etc...
4 75.3
4 69.8
Step
3: From the
Non-Parametrics Comparisons module menu, select "Ind. Gps - Mann-Whitney,
Kruskal-Wallis"
Step
4: Choose GROUP (TREATMENT) as the Group variable and choose OBS
(WEIGHT) as the data (response) variable.
Step
5: WINKS
will display the Kruskal-Wallis H-statistic, the rank sums, sample sizes and
mean ranks of the groups, a chi-square statistic and an approximate p-value.
A
low p-value (less than the significance level (e.g., less than 0.05) is usually
taken to indicate rejection of the null hypothesis. In this case, p<0.001 so
the null hypotheses is rejected. That is, there is enough evidence to say that the groups have different
medians, i.e., the groups are not identical with respect to location. If you
reject the null hypothesis and conclude that the groups have different medians,
you may also wish to know
which
groups differ.
Results of Non-Parametric analysis:
Group variable = GROUP Observation variable = OBS
Kruskal-Wallis H = 24.48
P-value for H estimated by Chi-Square with 3 degrees of freedom.
Chi-Square = 24.5 with 3 D.F. p < 0.001
Rank sum group 1 = 28. N = 7 Mean Rank = 4.
Rank sum group 2 = 77.5 N = 7 Mean Rank = 11.07
Rank sum group 3 = 171. N = 7 Mean Rank = 24.43
Rank sum group 4 = 129.5 N = 7 Mean Rank = 18.5
Critical q
Tukey Multiple Comp. Difference Q (.05)
------------------------------------------------------------------------
Rank(3)-Rank(1) = 20.4286 4.647 2.639 *
(SE used = 4.3964)
Rank(3)-Rank(2) = 13.3571 3.038 2.639 *
(SE used = 4.3964)
Rank(3)-Rank(4) = 5.9286 1.349 2.639
(SE used = 4.3964)
Rank(4)-Rank(1) = 14.5 3.298 2.639 *
(SE used = 4.3964)
Rank(4)-Rank(2) = 7.4286 1.69 2.639
(SE used = 4.3964)
Rank(2)-Rank(1) = 7.0714 1.608 2.639
(SE used = 4.3964)
The multiple comparison procedure based on ranks performs a
test to find specific differences.
In
the multiple comparison table, comparisons marked with an asterisk “*” are
significantly different at the 0.05 significance level (alpha-level). A
graphical description of the multiple comparisons is given by:
Gp Gp Gp Gp
1 2 4 3
---------
---------
---------
The
groups are listed in increasing order of the value of their average ranks.
(Group 1 has the smallest average rank and group 3 has the largest.) Since there
are four "populations", one for each group, the conclusion is that the
median of each group is significantly different from every other median (at the
0.05 significance level).
Friedman's
Test - Non-Parametric Repeated Measures Analysis
When
repeated observations are taken on the same subject, and there is interest in
comparing the observations for each repeated measure (e.g., each type of
treatment), then a repeated measures analysis may be appropriate. If you cannot
make the assumption that the data that are being observed are normally
distributed with equal variances between repeated measures, then a
non-parametric analysis is appropriate. One method of performing a
non-parametric one-way analysis of variance (ANOVA) with repeated measures
(randomized complete block experimental design) is with the Friedman test.
(When
there are only two groups, this test is equivalent to the sign text.) The
hypotheses for the Friedman test are:
Ho:There
is no difference in mean ranks between repeated measures.
Ha:There
is a difference in mean ranks between repeated measures.
The
Friedman analysis differs from a standard (parametric repeated measures) ANOVA
in that the analysis is performed on the ranks of the data rather than on the
actual data. For example, the following data are the same data used in a
previous example for a standard repeated measures ANOVA:
Drug1
Drug2
Drug3
Drug 4
31
29
17
35
15
17
11
23
25
21
19
31
35
35
21
45
27
27
15
31
The
data presented here are repeated measures of reaction times of 5 persons after
being given 4 drugs in randomized order. (See Winer, page 301 for more details.)
The
database for this analysis will include four fields (the repeated measures) and
will have five records, with each record representing a subject. Follow these
steps to perform this analysis:
Step
1: Create a
custom database that includes four
fields named DRUG1, DRUG2, DRUG3 and DRUG4. (Or open the database named DRUG on
disk and skip to step 3.)
Step
2: For the first record, enter the data for the first person
31,29,17,35. The second record will contain 15,17,11,23 and so forth. After you
finish entering your data, the database should look similar to the data listing
above. (This is the same database used in the Repeated Measures ANOVA earlier.)
Step
3: From the
Analyze menu, choose Non-Parametric
Comparisons. The Non-Parametric module menu appears. Choose the "Repeated
Measures Analysis" option.
Step
4: Choose
the fields DRUG1, DRUG2, DRUG2 and DRUG4 (in that order) to use for the
analysis. and click Ok.
Step
5: WINKS
displays the results in the viewer. For this data set, a Chi-Square value of
14.13 and a small p-value (p<0.01) is reported. The small p-value means that
there is a statistically significant difference in the mean ranks of times for
the four drugs.
In
the multiple comparison table, comparisons marked with an asterisk “*” are
significantly different at the 0.05 significance level (alpha-level).
Friedman's Test for Repeated Measures
Number of repeated measures= 4 Number of subjects = 5
1 )DRUG1 Rank sum = 13.0 Mean rank = 2.6
2 )DRUG2 Rank sum = 12.0 Mean rank = 2.4
3 )DRUG3 Rank sum = 5.0 Mean rank = 1.0
4 )DRUG4 Rank sum = 20.0 Mean rank = 4.0
Ho:There is no difference in mean ranks for repeated measures.
Ha:A difference exists in the mean ranks for repeated measures.
Friedman's Chi-Square = 14.13 with d.f. = 3 p = 0.003
Kendall's coefficient of concordance = 0.942
When the p-value is low, there is evidence to reject Ho,
and conclude that there is a difference between mean ranks.
Error term used for comparisons = 2.89
Critical q
Tukey Multiple Comp. Difference Q (.05)
------------------------------------------------------------------------
Rank( 4)-Rank( 3) = 15.0 5.196 3.63 *
Rank( 4)-Rank( 2) = 8.0 2.771 3.63
Rank( 4)-Rank( 1) = 7.0 (Do not test)
Rank( 1)-Rank( 3) = 8.0 2.771 3.63
Rank( 1)-Rank( 2) = 1.0 (Do not test)
Rank( 2)-Rank( 3) = 7.0 (Do not test)
Homogeneous Populations, repeated measures ranked
Gp 1 refers to DRUG1
Gp 2 refers to DRUG2
Gp 3 refers to DRUG3
Gp 4 refers to DRUG4
Gp Gp Gp Gp
3 2 1 4
---------
---------
This
graph is interpreted in the following way: Any two groups underlined by the same
line are considered not different at the 0.05 level of significance. Therefore,
the result of this analysis is that the mean rank for DRUG 3 is less than the
mean rank for DRUG 4. There are no other statistically significant pairwise
differences among the four groups.
Cochran's
Q -Non-Parametric Dichotomous Data Analysis
Cochran's
Q procedure is a non-parametric procedure appropriate for use with dichotomous
data when the experiment involves repeated measures on blocks.
Often the blocks are subjects (people or animals). The response of the
subjects to the treatments is dichotomous if it is taken as one of only two
possible outcomes, often labeled "success" and "failure",
rather than as a measurement.
Cochrans'
Q is used to test three or more treatments, or groups, and is in fact an
extension of McNemar's test for two groups (see Crosstabulation procedures).
(Cochran's Q test is placed under the Non-Parametric procedures rather than in
the Crosstabulation procedure because its data entry requirements fit better in
this section.) Cochran's Q can also be seen as similar to Friedman's test when
data are dichotomous. The hypotheses being tested are:
Ho:
The proportion of successes is the same for all treatments.
Ha:
The proportion of successes is not the same for all treatments.
The
data can be organized in a table of r rows and c columns, where the rows are the
subjects and the columns are the treatments. Each row by column entry of 0
(failure) or 1 (success) represents the response of that row's subject to that
column's treatment. For example, in the data used below, you can see that Drug 4
failed on Person 2.
Consider
an experiment where the response is not a measurement of a reaction, but instead
is simply a "success" or "failure". That is, an experiment
is conducted in which six persons are each given five test drugs (say headache
remedies) in random order, and a response of "success" or
"failure" is recorded in each case. The data are as follows:
Dichotomous
data for Cochran's Q (headache study)
Drug
1 Drug 2
Drug 3 Drug 4
Drug 5
1
1
0
1
0
0 0
0
0
1
0 0
0
1
0
1 1
0
1
1
0 0
0
1
1
1 0
0 1
1
where
1 = success and 0 = failure. Follow these steps to perform an analysis:
Step
1: Create a
database containing five fields, Drug1 to Drug5, similar to the databases
created previously for the repeated measures analyses described earlier. (Or,
open the database named COCHRAN on disk and skip to step 2.)
Step
2: From the
Analyze menu choose Non-Parametric Comparisons, then choose "Dichotomous
Data - Cochran's Q."
Step
3: Select
the fields DRUG1, DRUG2, ... DRUG5 as the fields to use in the analysis. The
results of the calculations appears in the viewer and click Ok.
Step
4: WINKS
summarizes the test information and reports a chi-square statistic of 9.87 and a
p-value of 0.04 in this case. In this case, the p-value is small. Therefore, the
null hypothesis is rejected in favor of the alternative hypothesis that there is
difference among the headache remedies.
Continue to Chapter
4. Part 5. (Regression and Correlation.)
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