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KRUSKAL-WALLIS TEST

In statistics, the Kruskal–Wallis one-way analysis of variance by ranks (named after William Kruskal
and W. Allen Wallis) is a non-parametric method for testing whether samples originate from the
same distribution. It is used for comparing more than two samples that are independent, or not
related. The parametric equivalence of the Kruskal-Wallis test is the one-way analysis of variance
(ANOVA). The factual null hypothesis is that the populations from which the samples originate
have the same median. When the Kruskal-Wallis test leads to significant results, then at least one
of the samples is different from the other samples. The test does not identify where the
differences occur or how many differences actually occur. It is an extension of the Mann–Whitney
U test to 3 or more groups. The Mann-Whitney would help analyze the specific sample pairs for
significant differences.



Since it is a non-parametric method, the Kruskal–Wallis test does not assume a normal
distribution, unlike the analogous one-way analysis of variance. However, the test does assume an
identically shaped and scaled distribution for each group, except for any difference in medians.



Kruskal–Wallis is also used when the examined groups are of unequal size (different number of
participants).[1]

Method:

    1. Rank all data from all groups together; i.e., rank the data from 1 to N ignoring group
       membership. Assign any tied values the average of the ranks they would have
       received had they not been tied.
    2. The test statistic is given by:



                                                        where:

            o        is the number of observations in group
            o       is the rank (among all observations) of observation from group
            o       is the total number of observations across all groups
            o


            o




            o                       is the average of all the    .
3. Notice that the denominator of the expression for      is exactly

                                  and              .    Thus




   Notice that the last formula only contains the squares of the average ranks.



4. A correction for ties can be made by dividing        by                         , where
   G is the number of groupings of different tied ranks, and ti is the number of tied
   values within group i that are tied at a particular value. This correction usually
   makes little difference in the value of K unless there are a large number of ties.




5. Finally, the p-value is approximated by                        . If some values are
   small (i.e., less than 5) the probability distribution of K can be quite different from
   this chi-squared distribution. If a table of the chi-squared probability distribution is
   available, the critical value of chi-squared,        , can be found by entering the
   table at g − 1 degrees of freedom and looking under the desired significance or
   alpha level. The null hypothesis of equal population medians would then be rejected
   if              . Appropriate multiple comparisons would then be performed on
   the group medians.



6. If the statistic is not significant, then there is no evidence of differences between the
   samples. However, if the test is significant then a difference exists between at least
   two of the samples. Therefore, a researcher might use sample contrasts between
   individual sample pairs, or post hoc tests, to determine which of the sample pairs are
   significantly different. When performing multiple sample contrasts, the Type I error
   rate tends to become inflated.
Este material es de otra web.

The Kruskal-Wallis test evaluates whether the population medians on a dependent variable are
the same across all levels of a factor. To conduct the Kruskal-Wallis test, using the K independent
samples procedure, cases must have scores on an independent or grouping variable and on a
dependent variable. The independent or grouping variable divides individuals into two or more
groups, and the dependent variable assesses individuals on at least an ordinal scale. If the
independent variable has only two levels, no additional significance tests need to be conducted
beyond the Kruskal-Wallis test. However, if a factor has more than two levels and the overall test
is significant, follow-up tests are usually conducted. These follow-up tests most frequently involve
comparisons between pairs of group medians. For the Kruskal-Wallis, we could use the Mann-
Whitney U test to examine unique pairs.



ASSUMPTIONS UNDERLYING A MANN-WHITNEY U TEST

Because the analysis for the Kruskal-Wallis test is conducted on ranked scores, the population
distributions for the test variable (the scores that the ranks are based on) do not have to be of any
particular form (e.g., normal). However, these distributions should be continuous and have
identical form.

Assumption 1: The continuous distributions for the test variable are exactly the same (except their
medians) for the different populations.

Assumption 2: The cases represent random samples from the populations, and the scores on the
test variable are independent of each other.

Assumption 3: The chi-square statistic for the Kruskal-Wallis test is only approximate and becomes
more accurate with larger sample sizes.

The p value for the chi-square approximation test is fairly accurate if the number of cases is
greater than or equal to 30.

EFFECT SIZE STATISTICS FOR THE MANN-WHITNEY U TEST

SPSS does not report an effect size index for the Kruskal-Wallis test. However, simple indices

can be computed to communicate the size of the effect. For the Kruskal-Wallis test, the median
and the mean rank for each of the groups can be reported. Another possibility for the Kruskal-
Wallis test is to compute an index that is usually associated with a one-way ANOVA, such as eta
square (h2), except h2 in this case would be computed on the ranked data. To do so, transform the
scores to ranks, conduct an ANOVA, and compute an eta square on the ranked scores. Eta square
can also be computed directly from the reported chi-square value for the Kruskal-Wallis test with
the use of the following equation:
n2=X2/N-1Where N is the total number of cases


THE RESEARCH QUESTIONS

The research questions used in this example can be asked to reflect differences in medians
between groups or a relationship between two variables.

1. Differences between the medians: Do the medians for change in the number of days of cold
symptoms differ among those who take a placebo, those who take low doses of vitamin C, and
those who take high doses of vitamin C?

2. Relationship between two variables: Is there a relationship between the amount of vitamin C
taken and the change in the number of days that individuals show cold symptoms?

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Kruskal wallis test

  • 1. KRUSKAL-WALLIS TEST In statistics, the Kruskal–Wallis one-way analysis of variance by ranks (named after William Kruskal and W. Allen Wallis) is a non-parametric method for testing whether samples originate from the same distribution. It is used for comparing more than two samples that are independent, or not related. The parametric equivalence of the Kruskal-Wallis test is the one-way analysis of variance (ANOVA). The factual null hypothesis is that the populations from which the samples originate have the same median. When the Kruskal-Wallis test leads to significant results, then at least one of the samples is different from the other samples. The test does not identify where the differences occur or how many differences actually occur. It is an extension of the Mann–Whitney U test to 3 or more groups. The Mann-Whitney would help analyze the specific sample pairs for significant differences. Since it is a non-parametric method, the Kruskal–Wallis test does not assume a normal distribution, unlike the analogous one-way analysis of variance. However, the test does assume an identically shaped and scaled distribution for each group, except for any difference in medians. Kruskal–Wallis is also used when the examined groups are of unequal size (different number of participants).[1] Method: 1. Rank all data from all groups together; i.e., rank the data from 1 to N ignoring group membership. Assign any tied values the average of the ranks they would have received had they not been tied. 2. The test statistic is given by: where: o is the number of observations in group o is the rank (among all observations) of observation from group o is the total number of observations across all groups o o o is the average of all the .
  • 2. 3. Notice that the denominator of the expression for is exactly and . Thus Notice that the last formula only contains the squares of the average ranks. 4. A correction for ties can be made by dividing by , where G is the number of groupings of different tied ranks, and ti is the number of tied values within group i that are tied at a particular value. This correction usually makes little difference in the value of K unless there are a large number of ties. 5. Finally, the p-value is approximated by . If some values are small (i.e., less than 5) the probability distribution of K can be quite different from this chi-squared distribution. If a table of the chi-squared probability distribution is available, the critical value of chi-squared, , can be found by entering the table at g − 1 degrees of freedom and looking under the desired significance or alpha level. The null hypothesis of equal population medians would then be rejected if . Appropriate multiple comparisons would then be performed on the group medians. 6. If the statistic is not significant, then there is no evidence of differences between the samples. However, if the test is significant then a difference exists between at least two of the samples. Therefore, a researcher might use sample contrasts between individual sample pairs, or post hoc tests, to determine which of the sample pairs are significantly different. When performing multiple sample contrasts, the Type I error rate tends to become inflated.
  • 3. Este material es de otra web. The Kruskal-Wallis test evaluates whether the population medians on a dependent variable are the same across all levels of a factor. To conduct the Kruskal-Wallis test, using the K independent samples procedure, cases must have scores on an independent or grouping variable and on a dependent variable. The independent or grouping variable divides individuals into two or more groups, and the dependent variable assesses individuals on at least an ordinal scale. If the independent variable has only two levels, no additional significance tests need to be conducted beyond the Kruskal-Wallis test. However, if a factor has more than two levels and the overall test is significant, follow-up tests are usually conducted. These follow-up tests most frequently involve comparisons between pairs of group medians. For the Kruskal-Wallis, we could use the Mann- Whitney U test to examine unique pairs. ASSUMPTIONS UNDERLYING A MANN-WHITNEY U TEST Because the analysis for the Kruskal-Wallis test is conducted on ranked scores, the population distributions for the test variable (the scores that the ranks are based on) do not have to be of any particular form (e.g., normal). However, these distributions should be continuous and have identical form. Assumption 1: The continuous distributions for the test variable are exactly the same (except their medians) for the different populations. Assumption 2: The cases represent random samples from the populations, and the scores on the test variable are independent of each other. Assumption 3: The chi-square statistic for the Kruskal-Wallis test is only approximate and becomes more accurate with larger sample sizes. The p value for the chi-square approximation test is fairly accurate if the number of cases is greater than or equal to 30. EFFECT SIZE STATISTICS FOR THE MANN-WHITNEY U TEST SPSS does not report an effect size index for the Kruskal-Wallis test. However, simple indices can be computed to communicate the size of the effect. For the Kruskal-Wallis test, the median and the mean rank for each of the groups can be reported. Another possibility for the Kruskal- Wallis test is to compute an index that is usually associated with a one-way ANOVA, such as eta square (h2), except h2 in this case would be computed on the ranked data. To do so, transform the scores to ranks, conduct an ANOVA, and compute an eta square on the ranked scores. Eta square can also be computed directly from the reported chi-square value for the Kruskal-Wallis test with the use of the following equation:
  • 4. n2=X2/N-1Where N is the total number of cases THE RESEARCH QUESTIONS The research questions used in this example can be asked to reflect differences in medians between groups or a relationship between two variables. 1. Differences between the medians: Do the medians for change in the number of days of cold symptoms differ among those who take a placebo, those who take low doses of vitamin C, and those who take high doses of vitamin C? 2. Relationship between two variables: Is there a relationship between the amount of vitamin C taken and the change in the number of days that individuals show cold symptoms?