The Kendall rank correlation coefficient, based on the Kendall-
When we have a sample of observations of a given population it may be difficult to assume that they come from a certain distribution since we may not always have any type of information about the variable under study and when we do, it may not be enough to determine the type of distribution. In these cases, parametric inference is inappropriate. Moreover, this type of technique may be unsuitable should the observations not fulfill any of the basic assumptions on which they are based; normality and a large quantity of data.
Violation of the necessary assumptions in parametric statistics necessitates the use of non-parametric statistics. Non-parametric tests do not depend on the definition of a distribution function or statistical parameters such as mean, variance, etc. The use of non-parametric tests, despite being less powerful, is also adequate when there are not enough observations available, when data are non-normal data or when ordinal data are being analyzed.
Although the first steps in non-parametric statistics began earlier, it was not until the 1930s that a systematic study in this field appeared. (Fisher 1935) introduced the permutation test or randomization test as a simple way to compute the sampling distribution for any test statistic under the null hypothesis that does not establish any effect on all possible outcomes. Over the next two decades some of the main non-parametric tests emerged, (Pitman 1937; Kendall 1938; Kendall and Smith 1939; Friedman 1940; Wilcoxon 1945; Mann and Whitney 1947; Kruskal and Wallis 1952; Kruskal 1958), among others.
The main advantages of the non-parametric tests are: the data can be nonnumerical observations while they can be classified according to some criterion, they are usually easy to calculate and do not make any hypothesis about the distribution of the population from which the samples are taken. We can also cite two drawbacks: the non-parametric tests are less precise than other statistical models and they are based on the order of the elements in the sample and this order will likely stay the same even if the numerical data change.
There are many non-parametric tests in the literature, which can basically be classified into four categories depending on whether: it is a test to compare two or more than two related samples or a test for comparing related or unrelated samples. Examples of the most used non-parametric tests in the literature for each of these four situations are the following: the Wilcoxon signed-rank test (Wilcoxon 1945) for comparing two related samples, the Mann-Whitney (Wilcoxon) test (Mann and Whitney 1947) for comparing two unrelated samples, the Friedman test (Friedman 1940) for comparing three or more related samples, and the Kruskal-Wallis test (Kruskal and Wallis 1952) for comparing three or more unrelated samples. Several methods that exploit some characteristic of the samples have appeared in the literature in recent years, such as (Terpstra and Magel 2003; Alhakim and Hooper 2008).
It is also possible to measure the degree of association of two variables through a non-parametric approach, in that sense we can mention the Kendall rank correlation coefficient (Kendall 1938) and the Spearman rank correlation coefficient (Spearman 1904).
In (Aparicio et al. 2020), the authors introduce the Kendall-
The remainder of this paper is organized as follows. After a brief review in the next section of the main features of the Kendall rank correlation coefficient and the Kruskal-Wallis statistic, in the following two sections we present the coefficient we propose in this work and illustrate its use with our ConcordanceTest package. Specifically, in the third section we introduce the Concordance coefficient while in the fourth section the related statistical test is presented. The fifth section includes a comparison between the Kruskal-Wallis test in the stats package and that presented in this work. Some final remarks follow in the last section. Appendix A presents an example of the probability distribution of the Concordance coefficient and the Kruskal-Wallis statistic. Appendix B deals with a comparison between the probability density function of the Concordance coefficient and the Kruskal-Wallis statistic for several experiments. Finally, Appendix C presents some details of how the p-values for the Concordance coefficient have been calculated and shows some critical values and exact p-values.
This section presents the Kendall rank correlation coefficient (Kendall 1938), a coefficient to measure the relationship between two samples ordinally, and the Kruskal-Wallis statistical test (Kruskal and Wallis 1952), which is a rank-based statistical test to measure whether different samples come from the same distribution, without assuming a given distribution for the population.
Only these two non-parametric tests are presented in detail, since the
test proposed in this paper uses the Kendall-
The Kendall rank correlation coefficient is a non-parametric measure of
correlation. This measure is based on the Kendall-
The Kendall rank correlation coefficient between permutations
The Kruskal-Wallis test is a non-parametric statistical method to study whether different samples come from the same population. The test is the extension of the Mann-Whitney Test (Mann and Whitney 1947) when we have more than two samples or groups. The following example illustrates the Kruskal-Wallis test when comparing three samples.
Example 1. Let us assume that the effectiveness of three
different treatments (
Rank | 1 | 2 | 3 | 4 | 5 | 6 |
The Kruskal-Wallis statistic is determined by the difference between the
ranks of the individuals in each category with the average rank. In our
example, the average rank of the test is
0.00 | 0.06667 |
0.29 | 0.13333 |
0.86 | 0.13333 |
1.14 | 0.13333 |
2.00 | 0.13333 |
2.57 | 0.06667 |
3.43 | 0.13333 |
3.71 | 0.13333 |
4.57 | 0.06667 |
In (Aparicio et al. 2020), the authors introduce the Kendall-
This distance is also called the disorder of permutation
The authors (Aparicio et al. 2020) present the properties of the Kendall-
From (Aparicio et al. 2020), the maximum number of disagreements that may occur
in a permutation of
Definition 1. We define the Concordance coefficient (
The Concordance coefficient (
Example 1 (Cont.). Continuing with the data in
Example 1, the results of the experiment provide the following
order or permutation of the treatments
Given the order of individuals
where each element of the matrix
Then, the relative disorder of permutation
Table 3 shows the probability distribution of the
disorder and the Concordance coefficient for 3 treatments with 2
patients each. Appendix A presents the disorder and the Concordance
coefficient for all possible results in the experiment with sample sizes
6 | 0.0000 | 0.06667 |
5 | 0.1667 | 0.13333 |
4 | 0.3333 | 0.20000 |
3 | 0.5000 | 0.20000 |
2 | 0.6667 | 0.20000 |
1 | 0.8333 | 0.13333 |
0 | 1.0000 | 0.06667 |
The R package we have developed allows to calculate both the Concordance coefficient and the Kruskal-Wallis statistic in order to facilitate their comparison. Given the high combinatorial degree of the problem of ordering samples of populations, some of the functions implemented in the package can perform the calculations exactly, exploring the entire sample space or possibilities, or they can approximate the sample space or possibilities by simulation.
The ConcordanceTest package can be installed from CRAN:
install.packages("ConcordanceTest")
library("ConcordanceTest")
and its functions can perform the calculations related only to the
Concordance coefficient (default option, specified with the parameter
H
=0) or do them also for the Kruskal-Wallis statistic (H
=1),
allowing their comparison.
To obtain the probability distribution of the statistics, it is
necessary to have the set of all possible permutations that can occur in
the result of the experiment that we want to analyze (90=6!/2!2!2! in
Example 1). This can be obtained through the function
Permutations_With_Repetition()
, which has been developed and included
in the ConcordanceTest package.
The function CT_Distribution()
calculates the probability distribution
of the Concordance coefficient and the Kruskal-Wallis statistic. The set
of possibilities (sample space) grows very quickly with the number of
elements and with the number of sets and, in some cases, to calculate
the probability distribution in an exact way becomes unaffordable,
making it necessary to approximate calculations. Both an exact and an
approximate calculation (default option) can be done using the function
CT_Distribution()
. It is used as follows:
CT_Distribution(Sample_Sizes, Num_Sim = 10000, H = 0, verbose = TRUE)
where Sample_Sizes
is a numeric vector Num_Sim
is the number of simulations to be
performed in order to obtain the probability distribution of the
statistics (10,000 by default). If Num_Sim
is set to 0, the
probability distribution tables are obtained exactly using the function
Permutations_With_Repetition()
. H
is the parameter specifying
whether the calculations must also be performed for the Kruskal-Wallis
statistic, and verbose
is a logical parameter that indicates whether
some progress report of the simulations should be given.
Example 1 (Cont.). Using the function CT_Distribution()
with
Num_Sim
equal to 0, we could obtain the probability distribution of
the Kruskal-Wallis statistic and the Concordance coefficient in
Example 1 (Tables 2 and 3,
respectively) in an exact way. As shown in this example, we can also
approximate the probability distributions of Example 1 by
simulating, for example, 25,000 permutations of 3 treatments with 2
patients each. Note that, for reproducibility, we always initialize the
generator for pseudo-random numbers when the results rely on
simulation.
set.seed(12)
Sample_Sizes <- c(2,2,2)
CT_Distribution(Sample_Sizes, Num_Sim = 25000, H = 1)
$C_freq
disorder Concordance coefficient Frequency Probability
[1,] 6 0.00 6 0.0667
[2,] 5 0.17 12 0.1333
[3,] 4 0.33 18 0.2000
[4,] 3 0.50 18 0.2000
[5,] 2 0.67 18 0.2000
[6,] 1 0.83 12 0.1333
[7,] 0 1.00 6 0.0667
$H_freq
H Statistic Frequency Probability
[1,] 0.00 6 0.0667
[2,] 0.29 12 0.1333
[3,] 0.86 12 0.1333
[4,] 1.14 12 0.1333
[5,] 2.00 12 0.1333
[6,] 2.57 6 0.0667
[7,] 3.43 12 0.1333
[8,] 3.71 12 0.1333
[9,] 4.57 6 0.0667
The function CT_Distribution()
returns two elements. C_freq
is a
matrix with the probability distribution of the Concordance coefficient.
Each row in the matrix contains the disorder, the value of the
Concordance coefficient H_freq
(only returned if H
= 1) is a matrix with the probability
distribution of the Kruskal-Wallis statistic. Each row in the matrix
contains the value of the statistic CT_Distribution()
are the same as those previously shown in Table 3 and
Table 2 of Example 1.
In this section, we present the Concordance test in order to evaluate when different samples come from the same population distribution. The randomization test introduced by (Fisher 1935) establishes a framework for the statistical test based on permutations, see also (Box 1980; Stern 1990; Welch 1990).
If all the samples come from the same distribution, then all possible
ways to rank
There is no difference among the
At least one of the populations differs from the other populations.
The decision rule is to reject the null hypothesis if the disorder in
the permutation of observations is small, equivalently if the
Concordance coefficient
The following example illustrates the use of the Concordance test proposed in this work and compares it with the classical Kruskal-Wallis non-parametric test. The comparison will be made first considering that there are no ties and then modifying the data in the example so that ties appear.
Example 2.
Suppose we have applied three treatments to 18 patients, measuring the
number of hours it takes these patients to recover. The results are
shown in Table 4.
Hours | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Treatment A | 12 | 13 | 15 | 20 | 23 | 28 | 30 | 32 | 40 | 48 | ||
Treatment B | 29 | 31 | 49 | 52 | 54 | |||||||
Treatment C | 24 | 26 | 44 |
The experiment ranks the patients in the following ranking
If we perform the contrast using the disorder statistic or the
Concordance coefficient
The treatments A, B and C have average ranks of 7.3, 14.2 and 9,
respectively, and the sum of ranks are
The Kruskal-Wallis statistic is given by:
In (Meyer and Seaman 2015), exact values for the Kruskal-Wallis contrast at
different levels of significance are found. We can conclude by looking
at the tables that the p-value of the
Comparing both methods, the Concordance and Kruskal-Wallis tests provide similar results about the statistic but the conclusion differs.
Example 3.
Suppose we have the same experiment as in Example 2
but with ties. The results are shown in Table 5. Ties are
in bold.
Hours | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Treatment A | 12 | 13 | 15 | 20 | 24 | 29 | 30 | 32 | 40 | 49 | ||
Treatment B | 29 | 31 | 49 | 52 | 54 | |||||||
Treatment C | 24 | 26 | 44 |
The results of the experiment order the individuals according to the
sequence:
The order between treatments that maximizes the order between patients,
corresponds to the order
The treatments A, B and C have average ranks of 7.45, 14 and 8.83,
respectively, and the sum of ranks are
The Kruskal-Wallis statistic is given by:
If we make the adjustment in the statistic for ties, we get:
where
In this case, the Kruskal-Wallis test provides the same conclusion as the Concordance test; uncertainty being greater when we have ties.
The ConcordanceTest R-package allows to perform the hypothesis test
for testing whether samples originate from the same distribution with
the function CT_Hypothesis_Test()
, which carries out the calculations
by simulation. It is used as follows:
CT_Hypothesis_Test(Sample_List, Num_Sim = 10000, H = 0, verbose = TRUE)
where Sample_List
is a list of numeric data vectors with the elements
of each sample, Num_Sim
is the number of used simulations (10,000 by
default), H
specifies whether the Kruskal-Wallis test must also be
done, and verbose
is a logical parameter that indicates whether some
progress report of the simulations should be given.
Example 2 (Cont.). We use the ConcordanceTest* package to perform the Concordance and Kruskal-Wallis tests of Example 2. We use 25,000 simulations.*
set.seed(12)
A <- c(12,13,15,20,23,28,30,32,40,48)
B <- c(29,31,49,52,54)
C <- c(24,26,44)
Sample_List <- list(A, B, C)
CT_Hypothesis_Test(Sample_List, Num_Sim = 25000, H = 1)
$results
Statistic p-value
Concordance coefficient 0.574 0.04928
Kruskal Wallis 5.600 0.05292
$C_p_value
[1] 0.04928
$H_p_value
[1] 0.05292
The function CT_Hypothesis_Test()
provides the value of the statistics
together with the p-value associated with each of them. The result of
the Kruskal-Wallis test is only returned if H
= 1. Note that the
approximated p-values obtained by simulation are close to the exact
ones, 0.04927 and 0.05223 for the Concordance coefficient and the
Kruskal-Wallis statistic, respectively.
An alternative to the contrast performed with the function
CT_Hypothesis_Test()
is to obtain the critical values of our contrast.
This can be done with the ConcordanceTest package both in an exact or
approximate way, using the function CT_Critical_Values()
. It is used
as follows:
CT_Critical_Values(Sample_Sizes, Num_Sim = 10000, H = 0, verbose = TRUE)
where Sample_Sizes
is a numeric vector Num_Sim
is the number of simulations carried out in
order to obtain the probability distribution of the statistics (10,000
by default). If Num_Sim
is set to 0, the critical values are obtained
in an exact way. Otherwise they are obtained by simulation. H
is the
parameter specifying whether the critical values of the Kruskal-Wallis
test must be calculated and returned, and verbose
is a logical
parameter that indicates whether some progress report of the simulations
should be given.
The function returns a list with two elements. C_results
are the
critical values and p-values for a desired significance levels of 0.1,
.05 and .01 of the Concordance coefficient, and H_results
are the
critical values and p-values of the Kruskal-Wallis statistic (only
returned if H = 1).
Example 2 (Cont.). We show the results of the function
CT_Critical_Values()
with sample sizes
set.seed(12)
Sample_Sizes <- c(10,5,3)
CT_Critical_Values(Sample_Sizes, Num_Sim = 25000, H = 1)
$C_results
| disorder | Concordance coefficient | p-value
Sig level .10 23 0.51 0.0954
Sig level .05 20 0.57 0.0492
Sig level .01 14 0.70 0.0096
$H_results
| H Statistic | p-value
Sig level .10 4.55 0.0995
Sig level .05 5.72 0.0497
Sig level .01 7.78 0.0097
To obtain the Concordance coefficient and the Kruskal-Wallis statistic
from the result of an experiment, the ConcordanceTest package has the
function CT_Coefficient()
. This function is useful when we only want
to obtain the value of the statistic to check its significance using
statistical tables. The function CT_Coefficient()
is used as follows:
CT_Coefficient(Sample_List, H = 0)
where Sample_List
is a list of numeric data vectors with the elements
of each sample, and H
is defined as usual.
Example 2 (Cont.). We show the results of the function
CT_Coefficient()
for the data in Example 2.
A <- c(12,13,15,20,23,28,30,32,40,48)
B <- c(29,31,49,52,54)
C <- c(24,26,44)
Sample_List <- list(A, B, C)
CT_Coefficient(Sample_List, H = 1)
$Sample_Sizes
[1] 10 5 3
$order_elements
[1] 1 1 1 1 1 3 3 1 2 1 2 1 1 3 1 2 2 2
$disorder
[1] 20
$Concordance_Coefficient
[1] 0.5744681
$H_Statistic
[1] 5.6
The function CT_Coefficient()
returns a list with the following
elements: Sample_Sizes
is a numeric vector with the sample sizes,
order_elements
is a numeric vector containing the elements order,
disorder
is the disorder of the permutation given by order_elements
,
Concordance_Coefficient
is the value of the Concordance coefficient
order_elements
, and H_Statistic
is the Kruskal-Wallis
statistic (only returned if H
= 1).
Note that we can also solve problems with ties (as in Example 3) with the ConcordanceTest package.
The graphical visualization of the probability distributions of the
Concordance coefficient and the Kruskal-Wallis statistic can be done
with the function CT_Probability_Plot()
. It is used as follows:
CT_Probability_Plot(C_freq = NULL, H_freq = NULL)
Using the function CT_Density_Plot()
of the ConcordanceTest package,
we can make an approximate representation of the density functions of
the statistics, assuming that the probability distributions represent a
sample of a continuous variable. It is used as follows:
CT_Density_Plot(C_freq = NULL, H_freq = NULL)
In both functions, C_freq
is the probability distribution of the
Concordance coefficient and H_freq
is the probability distribution of
the Kruskal-Wallis statistic, obtained exactly or approximately with the
function CT_Distribution()
. The function CT_Probability_Plot()
can
represent both probability distributions or only one of them (if it only
receives the parameter C_freq
or H_freq
). Equivalently, the function
CT_Density_Plot()
can represent both density distributions or only one
of them. Appendix B presents the empirical density probability functions
for several experiments, where sample sizes vary form
Example 2 (Cont.). Graphical visualization of the probability
distributions and the density distributions of
Example 2 generated by simulation. The first row of
Figure 2 compares the probability distribution of the
Concordance coefficient and the Kruskal-Wallis statistic. The second row
of Figure 2 shows the probability density function of the
Concordance coefficient (continuous line) and the Kruskal-Wallis
statistic (dashed line). Note that the
set.seed(12)
Sample_Sizes <- c(10,5,3)
ProbDistr <- CT_Distribution(Sample_Sizes, Num_Sim = 25000, H = 1)
layout(matrix(c(1,3,2,3), ncol=2))
CT_Probability_Plot(C_freq = ProbDistr$C_freq, H_freq = ProbDistr$H_freq)
CT_Density_Plot(C_freq = ProbDistr$C_freq, H_freq = ProbDistr$H_freq)
As we mentioned in Figure 1, Figure 2 also shows that similar values of the Kruskal-Wallis statistic present very different probabilities, and this leads to a less smooth function than that presented by the Concordance coefficient. We can also see that the Concordance coefficient presents a more symmetrical distribution. This performance is generalized and, therefore, we consider that the Concordance coefficient is more reliable than the Kruskal-Wallis statistic.
The ConcordanceTest package also contains the function LOP()
, which
solves the Linear Ordering Problem from a square data matrix. This
function allows to calculate the disorder of a permutation of elements
from the preference matrix induced by that permutation and, therefore,
it is necessary for the calculation of the Concordance coefficient. The
function LOP()
is used by functions CT_Distribution()
,
CT_Hypothesis_Test()
and CT_Coefficient()
. It is used as follows:
LOP(mat_LOP)
where mat_LOP
is the preference matrix defining the Linear Ordering
Problem, a numeric square matrix for which we want to obtain the
permutation of rows/columns that maximizes the sum of the elements above
the main diagonal.
The function LOP()
returns a list with the following elements:
obj_val
is the optimal value of the solution of the Linear Ordering
Problem, that is, the sum of the elements above the main diagonal under
the permutation rows/columns solution, permutation
is the solution of
the Linear Ordering Problem, that is, the rows/columns permutation, and
permutation_matrix
is the optimal permutation matrix of the Linear
Ordering Problem.
Example 2 (Cont.). The matrix of preferences between treatments observed in Example 2 was:
LOP()
on this preference matrix we obtain the
following results:
mat_LOP <- matrix(c(0,7,11,43,0,13,19,2,0), nrow=3)
LOP(mat_LOP)
$obj_val
[1] 75
$permutation
[1] 1 3 2
$permutation_matrix
[,1] [,2] [,3]
[1,] 0 1 1
[2,] 0 0 0
[3,] 0 1 0
As we saw previously, the order between treatments that maximizes the
order between patients corresponds to the order permutation
= 1 3 2), satisfying obj_val
= 75 of the preferences
contained in the matrix.
kruskal.test()
function from stats packageThe well-known stats package contains, among many other functions, the
function kruskal.test()
that performs a Kruskal-Wallis rank sum test.
In this section, we compare the results obtained with the
ConcordanceTest package presented in this work and the function
kruskal.test()
, making use of the dataset from (Hollander and Wolfe 1973)
referenced in the kruskal.test()
examples.
Example 4. Comparison of kruskal.test()
(stats package)
and CT_Hypothesis_Test()
functions with 25,000 simulations
(ConcordanceTest package) using the dataset from (Hollander and Wolfe 1973).
## Hollander & Wolfe (1973), 116.
## Mucociliary efficiency from the rate of removal of dust in normal
## subjects, subjects with obstructive airway disease, and subjects
## with asbestosis.
x <- c(2.9, 3.0, 2.5, 2.6, 3.2) # normal subjects
y <- c(3.8, 2.7, 4.0, 2.4) # with obstructive airway disease
z <- c(2.8, 3.4, 3.7, 2.2, 2.0) # with asbestosis
Sample_List <- list(x, y, z)
kruskal.test(Sample_List)
Kruskal-Wallis rank sum test
data: Sample_List
Kruskal-Wallis chi-squared = 0.77143, df = 2, p-value = 0.68
set.seed(12)
CT_Hypothesis_Test(Sample_List, Num_Sim = 25000, H = 1)
results
Statistic p-value
Concordance coefficient 0.188 0.78408
Kruskal-Wallis 0.771 0.71080
$C_p_value
[1] 0.78408
$H_p_value
[1] 0.7108
As can be observed, the value of the Kruskal-Wallis statistic is the same with both functions (0.771). However, the p-values associated with the statistic differ.
The Kruskal-Wallis statistic follows approximately a kruskal.test()
uses a
pchisq()
). In the case of the function CT_Hypothesis_Test()
, it
calculates the p-values using the simulations performed (25,000 in this
example).
The function CT_Distribution()
of the ConcordanceTest package allows
the probability distribution tables of the Concordance coefficient and
Kruskal-Wallis statistic to be computed, and they can be obtained
exactly or by simulation. We can get the exact probability distribution
tables and, consequently, the exact p-values in Example 4 with
CT_Distribution(c(5,4,5), Num_Sim = 0, H = 1)
In Example 4, the exact p-value for the Kruskal-Wallis
statistic is 0.71077. Therefore, the difference between our p-value
obtained with 25,000 simulations (0.71080) and the exact one is 0.00003,
while the difference between the p-value approximated by the
It is worth noting that the function kruskal.test()
uses the
A new measure based on the Kendall-
This work aims to be an introduction of the new concordance measure between samples, but there still remains much to be done. There is a new problem and further challenges for researchers, for example: studying the asymptotic distribution of the Concordance coefficient, exploring the possibility of finding the exact distribution with the help of modern computing, or analyzing the power of the Concordance test presented in this work, among others.
The authors thank the grants PID2019-105952GB-I00 funded by Ministerio de Ciencia e Innovación/ Agencia Estatal de Investigación /10.13039/501100011033, Spain, and PROMETEO/2021/063 funded by the government of the Valencian Community, Spain.
Table 6 shows the Concordance coefficient (
a | a | b | b | c | c | 0 | 1.0000 | 4.57 | b | a | a | b | c | c | 2 | 0.6667 | 3.43 | c | a | a | b | b | c | 4 | 0.3333 | 1.14 | |||||||||||||||
a | a | b | c | b | c | 1 | 0.8333 | 3.71 | b | a | a | c | b | c | 3 | 0.5000 | 2.00 | c | a | a | b | c | b | 3 | 0.5000 | 2.00 | |||||||||||||||
a | a | b | c | c | b | 2 | 0.6667 | 3.43 | b | a | a | c | c | b | 4 | 0.3333 | 1.14 | c | a | a | c | b | b | 2 | 0.6667 | 3.43 | |||||||||||||||
a | a | c | b | b | c | 2 | 0.6667 | 3.43 | b | a | b | a | c | c | 1 | 0.8333 | 3.71 | c | a | b | a | b | c | 5 | 0.1667 | 0.29 | |||||||||||||||
a | a | c | b | c | b | 1 | 0.8333 | 3.71 | b | a | b | c | a | c | 2 | 0.6667 | 2.57 | c | a | b | a | c | b | 4 | 0.3333 | 0.86 | |||||||||||||||
a | a | c | c | b | b | 0 | 1.0000 | 4.57 | b | a | b | c | c | a | 3 | 0.5000 | 2.00 | c | a | b | b | a | c | 6 | 0.0000 | 0.00 | |||||||||||||||
a | b | a | b | c | c | 1 | 0.8333 | 3.71 | b | a | c | a | b | c | 4 | 0.3333 | 0.86 | c | a | b | b | c | a | 5 | 0.1667 | 0.29 | |||||||||||||||
a | b | a | c | b | c | 2 | 0.6667 | 2.57 | b | a | c | a | c | b | 5 | 0.1667 | 0.29 | c | a | b | c | a | b | 3 | 0.5000 | 1.14 | |||||||||||||||
a | b | a | c | c | b | 3 | 0.5000 | 2.00 | b | a | c | b | a | c | 3 | 0.5000 | 1.14 | c | a | b | c | b | a | 4 | 0.3333 | 0.86 | |||||||||||||||
a | b | b | a | c | c | 2 | 0.6667 | 3.43 | b | a | c | b | c | a | 4 | 0.3333 | 0.86 | c | a | c | a | b | b | 1 | 0.8333 | 3.71 | |||||||||||||||
a | b | b | c | a | c | 3 | 0.5000 | 2.00 | b | a | c | c | a | b | 6 | 0.0000 | 0.00 | c | a | c | b | a | b | 2 | 0.6667 | 2.57 | |||||||||||||||
a | b | b | c | c | a | 4 | 0.3333 | 1.14 | b | a | c | c | b | a | 5 | 0.1667 | 0.29 | c | a | c | b | b | a | 3 | 0.5000 | 2.00 | |||||||||||||||
a | b | c | a | b | c | 3 | 0.5000 | 1.14 | b | b | a | a | c | c | 0 | 1.0000 | 4.57 | c | b | a | a | b | c | 6 | 0.0000 | 0.00 | |||||||||||||||
a | b | c | a | c | b | 4 | 0.3333 | 0.86 | b | b | a | c | a | c | 1 | 0.8333 | 3.71 | c | b | a | a | c | b | 5 | 0.1667 | 0.29 | |||||||||||||||
a | b | c | b | a | c | 4 | 0.3333 | 0.86 | b | b | a | c | c | a | 2 | 0.6667 | 3.43 | c | b | a | b | a | c | 5 | 0.1667 | 0.29 | |||||||||||||||
a | b | c | b | c | a | 5 | 0.1667 | 0.29 | b | b | c | a | a | c | 2 | 0.6667 | 3.43 | c | b | a | b | c | a | 4 | 0.3333 | 0.86 | |||||||||||||||
a | b | c | c | a | b | 5 | 0.1667 | 0.29 | b | b | c | a | c | a | 1 | 0.8333 | 3.71 | c | b | a | c | a | b | 4 | 0.3333 | 0.86 | |||||||||||||||
a | b | c | c | b | a | 6 | 0.0000 | 0.00 | b | b | c | c | a | a | 0 | 1.0000 | 4.57 | c | b | a | c | b | a | 3 | 0.5000 | 1.14 | |||||||||||||||
a | c | a | b | b | c | 3 | 0.5000 | 2.00 | b | c | a | a | b | c | 5 | 0.1667 | 0.29 | c | b | b | a | a | c | 4 | 0.3333 | 1.14 | |||||||||||||||
a | c | a | b | c | b | 2 | 0.6667 | 2.57 | b | c | a | a | c | b | 6 | 0.0000 | 0.00 | c | b | b | a | c | a | 3 | 0.5000 | 2.00 | |||||||||||||||
a | c | a | c | b | b | 1 | 0.8333 | 3.71 | b | c | a | b | a | c | 4 | 0.3333 | 0.86 | c | b | b | c | a | a | 2 | 0.6667 | 3.43 | |||||||||||||||
a | c | b | a | b | c | 4 | 0.3333 | 0.86 | b | c | a | b | c | a | 3 | 0.5000 | 1.14 | c | b | c | a | a | b | 3 | 0.5000 | 2.00 | |||||||||||||||
a | c | b | a | c | b | 3 | 0.5000 | 1.14 | b | c | a | c | a | b | 5 | 0.1667 | 0.29 | c | b | c | a | b | a | 2 | 0.6667 | 2.57 | |||||||||||||||
a | c | b | b | a | c | 5 | 0.1667 | 0.29 | b | c | a | c | b | a | 4 | 0.3333 | 0.86 | c | b | c | b | a | a | 1 | 0.8333 | 3.71 | |||||||||||||||
a | c | b | b | c | a | 6 | 0.0000 | 0.00 | b | c | b | a | a | c | 3 | 0.5000 | 2.00 | c | c | a | a | b | b | 0 | 1.0000 | 4.57 | |||||||||||||||
a | c | b | c | a | b | 4 | 0.3333 | 0.86 | b | c | b | a | c | a | 2 | 0.6667 | 2.57 | c | c | a | b | a | b | 1 | 0.8333 | 3.71 | |||||||||||||||
a | c | b | c | b | a | 5 | 0.1667 | 0.29 | b | c | b | c | a | a | 1 | 0.8333 | 3.71 | c | c | a | b | b | a | 2 | 0.6667 | 3.43 | |||||||||||||||
a | c | c | a | b | b | 2 | 0.6667 | 3.43 | b | c | c | a | a | b | 4 | 0.3333 | 1.14 | c | c | b | a | a | b | 2 | 0.6667 | 3.43 | |||||||||||||||
a | c | c | b | a | b | 3 | 0.5000 | 2.00 | b | c | c | a | b | a | 3 | 0.5000 | 2.00 | c | c | b | a | b | a | 1 | 0.8333 | 3.71 | |||||||||||||||
a | c | c | b | b | a | 4 | 0.3333 | 1.14 | b | c | c | b | a | a | 2 | 0.6667 | 3.43 | c | c | b | b | a | a | 0 | 1.0000 | 4.57 |
Table 7 shows the probability density function of the
Concordance coefficient (continuous lines) and the Kruskal-Wallis
statistic (dashed lines) generated by simulation. Number of simulations
100,000. Note that the
Sample_Sizes=(4,4) | Sample_Sizes=(3,3,2) | Sample_Sizes=(2,2,2,2) |
Sample_Sizes=(5,4) | Sample_Sizes=(3,3,3) | Sample_Sizes=(3,2,2,2) |
Sample_Sizes=(5,5) | Sample_Sizes=(4,3,3) | Sample_Sizes=(3,3,2,2) |
Sample_Sizes=(7,6) | Sample_Sizes=(5,5,5) | Sample_Sizes=(4,4,4,3) |
Sample_Sizes=(10,10) | Sample_Sizes=(7,7,6) | Sample_Sizes=(5,5,5,5) |
Sample_Sizes=(15,15) | Sample_Sizes=(10,10,10) | Sample_Sizes=(8,8,7,7) |
Sample_Sizes=(6,6,6,6,6,6) | Sample_Sizes=(5,5,5,5,5,5) | Sample_Sizes=(5,5,4,4,4,4,4) |
In order to compute the probability distribution of the Concordance
coefficient, the enumeration of all the permutations of elements from an
order is required. Note for example that if we have 4 samples with 6
elements each,
Tables 8, 9 and 10 show the critical
values and exact p-values of the Concordance coefficient
Sample Sizes | p-value | p-value | p-value | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
.10 | .05 | .01 | ||||||||||||
4 | 1 | |||||||||||||
4 | 2 | |||||||||||||
4 | 3 | 0 | 1.000000 | 0.057143 | ||||||||||
4 | 4 | 1 | 0.875000 | 0.057143 | 0 | 1.000000 | 0.028571 | |||||||
5 | 1 | |||||||||||||
5 | 2 | 0 | 1.000000 | 0.095238 | ||||||||||
5 | 3 | 1 | 0.857143 | 0.071429 | 0 | 1.000000 | 0.035714 | |||||||
5 | 4 | 2 | 0.800000 | 0.063492 | 1 | 0.900000 | 0.031746 | |||||||
5 | 5 | 4 | 0.666667 | 0.095238 | 2 | 0.833333 | 0.031746 | 0 | 1.000000 | 0.007937 | ||||
6 | 1 | |||||||||||||
6 | 2 | 0 | 1.000000 | 0.071429 | ||||||||||
6 | 3 | 2 | 0.777778 | 0.095238 | 1 | 0.888889 | 0.047619 | |||||||
6 | 4 | 3 | 0.750000 | 0.066667 | 2 | 0.833333 | 0.038095 | 0 | 1.000000 | 0.009524 | ||||
6 | 5 | 5 | 0.666667 | 0.082251 | 3 | 0.800000 | 0.030303 | 1 | 0.933333 | 0.008658 | ||||
6 | 6 | 7 | 0.611111 | 0.093074 | 5 | 0.722222 | 0.041126 | 2 | 0.888889 | 0.008658 | ||||
7 | 1 | |||||||||||||
7 | 2 | 0 | 1.000000 | 0.055556 | ||||||||||
7 | 3 | 2 | 0.800000 | 0.066667 | 1 | 0.900000 | 0.033333 | |||||||
7 | 4 | 4 | 0.714286 | 0.072727 | 3 | 0.785714 | 0.042424 | 0 | 1.000000 | 0.006061 | ||||
7 | 5 | 6 | 0.647059 | 0.073232 | 5 | 0.705882 | 0.047980 | 1 | 0.941176 | 0.005051 | ||||
7 | 6 | 8 | 0.619048 | 0.073427 | 6 | 0.714286 | 0.034965 | 3 | 0.857143 | 0.008159 | ||||
7 | 7 | 11 | 0.541667 | 0.097319 | 8 | 0.666667 | 0.037879 | 4 | 0.833333 | 0.006993 | ||||
8 | 1 | |||||||||||||
8 | 2 | 1 | 0.875000 | 0.088889 | 0 | 1.000000 | 0.044444 | |||||||
8 | 3 | 3 | 0.750000 | 0.084848 | 2 | 0.833333 | 0.048485 | |||||||
8 | 4 | 5 | 0.687500 | 0.072727 | 4 | 0.750000 | 0.048485 | 1 | 0.937500 | 0.008081 | ||||
8 | 5 | 8 | 0.600000 | 0.093240 | 6 | 0.700000 | 0.045066 | 2 | 0.900000 | 0.006216 | ||||
8 | 6 | 10 | 0.583333 | 0.081252 | 8 | 0.666667 | 0.042624 | 4 | 0.833333 | 0.007992 | ||||
8 | 7 | 13 | 0.535714 | 0.093862 | 10 | 0.642857 | 0.040093 | 6 | 0.785714 | 0.009324 | ||||
8 | 8 | 15 | 0.531250 | 0.082984 | 13 | 0.593750 | 0.049883 | 7 | 0.781250 | 0.006993 | ||||
9 | 1 | |||||||||||||
9 | 2 | 1 | 0.888889 | 0.072727 | 0 | 1.000000 | 0.036364 | |||||||
9 | 3 | 3 | 0.769231 | 0.063636 | 2 | 0.846154 | 0.036364 | 0 | 1.000000 | 0.009091 | ||||
9 | 4 | 6 | 0.666667 | 0.075524 | 4 | 0.777778 | 0.033566 | 1 | 0.944444 | 0.005594 | ||||
9 | 5 | 9 | 0.590909 | 0.082917 | 7 | 0.681818 | 0.041958 | 3 | 0.863636 | 0.006993 | ||||
9 | 6 | 12 | 0.555556 | 0.087912 | 10 | 0.629630 | 0.049550 | 5 | 0.814815 | 0.007592 | ||||
9 | 7 | 15 | 0.516129 | 0.090734 | 12 | 0.612903 | 0.041783 | 7 | 0.774194 | 0.007867 | ||||
9 | 8 | 18 | 0.500000 | 0.092719 | 15 | 0.583333 | 0.046401 | 9 | 0.750000 | 0.007898 | ||||
9 | 9 | 21 | 0.475000 | 0.093912 | 17 | 0.575000 | 0.039984 | 11 | 0.725000 | 0.007775 | ||||
10 | 1 | |||||||||||||
10 | 2 | 1 | 0.900000 | 0.060606 | 0 | 1.000000 | 0.030303 | |||||||
10 | 3 | 4 | 0.733333 | 0.076923 | 3 | 0.800000 | 0.048951 | 0 | 1.000000 | 0.006993 | ||||
10 | 4 | 7 | 0.650000 | 0.075924 | 5 | 0.750000 | 0.035964 | 2 | 0.900000 | 0.007992 | ||||
10 | 5 | 11 | 0.560000 | 0.099234 | 8 | 0.680000 | 0.039960 | 4 | 0.840000 | 0.007992 | ||||
10 | 6 | 14 | 0.533333 | 0.093407 | 11 | 0.633333 | 0.041958 | 6 | 0.800000 | 0.007493 | ||||
10 | 7 | 17 | 0.514286 | 0.087824 | 14 | 0.600000 | 0.043089 | 9 | 0.742857 | 0.009667 | ||||
10 | 8 | 20 | 0.500000 | 0.083139 | 17 | 0.575000 | 0.043421 | 11 | 0.725000 | 0.008547 | ||||
10 | 9 | 24 | 0.466667 | 0.094720 | 20 | 0.555556 | 0.043474 | 13 | 0.711111 | 0.007621 | ||||
10 | 10 | 27 | 0.460000 | 0.089210 | 23 | 0.540000 | 0.043257 | 16 | 0.680000 | 0.008931 | ||||
11 | 1 | |||||||||||||
11 | 2 | 1 | 0.909091 | 0.051282 | 0 | 1.000000 | 0.025641 | |||||||
11 | 3 | 5 | 0.687500 | 0.087912 | 3 | 0.812500 | 0.038462 | 0 | 1.000000 | 0.005495 | ||||
11 | 4 | 8 | 0.636364 | 0.077656 | 6 | 0.727273 | 0.039560 | 2 | 0.909091 | 0.005861 | ||||
11 | 5 | 12 | 0.555556 | 0.089744 | 9 | 0.666667 | 0.038004 | 5 | 0.814815 | 0.008700 | ||||
11 | 6 | 16 | 0.515152 | 0.098255 | 13 | 0.606061 | 0.047673 | 7 | 0.787879 | 0.007111 | ||||
11 | 7 | 19 | 0.500000 | 0.085344 | 16 | 0.578947 | 0.044118 | 10 | 0.736842 | 0.008296 | ||||
11 | 8 | 23 | 0.477273 | 0.090842 | 19 | 0.568182 | 0.040883 | 13 | 0.704545 | 0.009103 | ||||
11 | 9 | 27 | 0.448980 | 0.095177 | 23 | 0.530612 | 0.046452 | 16 | 0.673469 | 0.009693 | ||||
11 | 10 | 31 | 0.436364 | 0.098618 | 26 | 0.527273 | 0.042964 | 18 | 0.672727 | 0.007950 | ||||
11 | 11 | 34 | 0.433333 | 0.087946 | 30 | 0.500000 | 0.047307 | 21 | 0.650000 | 0.008330 | ||||
12 | 1 | |||||||||||||
12 | 2 | 2 | 0.833333 | 0.087912 | 1 | 0.916667 | 0.043956 | |||||||
12 | 3 | 5 | 0.722222 | 0.070330 | 4 | 0.777778 | 0.048352 | 1 | 0.944444 | 0.008791 | ||||
12 | 4 | 9 | 0.625000 | 0.078022 | 7 | 0.708333 | 0.041758 | 3 | 0.875000 | 0.007692 | ||||
12 | 5 | 13 | 0.566667 | 0.081771 | 11 | 0.633333 | 0.048481 | 6 | 0.800000 | 0.009373 | ||||
12 | 6 | 17 | 0.527778 | 0.083064 | 14 | 0.611111 | 0.041478 | 9 | 0.750000 | 0.009696 | ||||
12 | 7 | 21 | 0.500000 | 0.083115 | 18 | 0.571429 | 0.044931 | 12 | 0.714286 | 0.009764 | ||||
12 | 8 | 26 | 0.458333 | 0.097880 | 22 | 0.541667 | 0.047345 | 15 | 0.687500 | 0.009558 | ||||
12 | 9 | 30 | 0.444444 | 0.095451 | 26 | 0.518519 | 0.049073 | 18 | 0.666667 | 0.009288 | ||||
12 | 10 | 34 | 0.433333 | 0.093090 | 29 | 0.516667 | 0.042570 | 21 | 0.650000 | 0.008957 | ||||
12 | 11 | 38 | 0.424242 | 0.090842 | 33 | 0.500000 | 0.043879 | 24 | 0.636364 | 0.008625 | ||||
12 | 12 | 42 | 0.416667 | 0.088734 | 37 | 0.486111 | 0.044902 | 27 | 0.625000 | 0.008293 | ||||
13 | 1 | |||||||||||||
13 | 2 | 2 | 0.846154 | 0.076190 | 1 | 0.923077 | 0.038095 | |||||||
13 | 3 | 6 | 0.684211 | 0.082143 | 4 | 0.789474 | 0.039286 | 1 | 0.947368 | 0.007143 | ||||
13 | 4 | 10 | 0.615385 | 0.078992 | 8 | 0.692308 | 0.044538 | 3 | 0.884615 | 0.005882 | ||||
13 | 5 | 15 | 0.531250 | 0.094538 | 12 | 0.625000 | 0.045985 | 7 | 0.781250 | 0.009804 | ||||
13 | 6 | 19 | 0.512820 | 0.087424 | 16 | 0.589744 | 0.046218 | 10 | 0.743590 | 0.009214 | ||||
13 | 7 | 24 | 0.466667 | 0.096801 | 20 | 0.555556 | 0.045562 | 13 | 0.711111 | 0.008462 | ||||
13 | 8 | 28 | 0.461538 | 0.089046 | 24 | 0.538462 | 0.044553 | 17 | 0.673077 | 0.009937 | ||||
13 | 9 | 33 | 0.431035 | 0.095557 | 28 | 0.517241 | 0.043376 | 20 | 0.655172 | 0.008910 | ||||
13 | 10 | 37 | 0.430769 | 0.088294 | 33 | 0.492308 | 0.049329 | 24 | 0.630769 | 0.009888 | ||||
13 | 11 | 42 | 0.408451 | 0.093307 | 37 | 0.478873 | 0.047448 | 27 | 0.619718 | 0.008848 | ||||
13 | 12 | 47 | 0.397436 | 0.097642 | 41 | 0.474359 | 0.045711 | 31 | 0.602564 | 0.009556 | ||||
13 | 13 | 51 | 0.392857 | 0.090847 | 45 | 0.464286 | 0.044117 | 34 | 0.595238 | 0.008601 | ||||
14 | 1 | |||||||||||||
14 | 2 | 2 | 0.857143 | 0.066667 | 1 | 0.928571 | 0.033333 | |||||||
14 | 3 | 7 | 0.666667 | 0.091176 | 5 | 0.761905 | 0.047059 | 1 | 0.952381 | 0.005882 | ||||
14 | 4 | 11 | 0.607143 | 0.079085 | 9 | 0.678571 | 0.046405 | 4 | 0.857143 | 0.007843 | ||||
14 | 5 | 16 | 0.542857 | 0.087031 | 13 | 0.628571 | 0.043688 | 7 | 0.800000 | 0.007224 | ||||
14 | 6 | 21 | 0.500000 | 0.091331 | 17 | 0.595238 | 0.040764 | 11 | 0.738095 | 0.008720 | ||||
14 | 7 | 26 | 0.469388 | 0.093774 | 22 | 0.551020 | 0.046096 | 15 | 0.693878 | 0.009684 | ||||
14 | 8 | 31 | 0.446429 | 0.095018 | 26 | 0.535714 | 0.042149 | 18 | 0.678571 | 0.008125 | ||||
14 | 9 | 36 | 0.428571 | 0.095574 | 31 | 0.507936 | 0.045585 | 22 | 0.650794 | 0.008568 | ||||
14 | 10 | 41 | 0.414286 | 0.095643 | 36 | 0.485714 | 0.048404 | 26 | 0.628571 | 0.008851 | ||||
14 | 11 | 46 | 0.402597 | 0.095427 | 40 | 0.480519 | 0.044228 | 30 | 0.610390 | 0.009022 | ||||
14 | 12 | 51 | 0.392857 | 0.095012 | 45 | 0.464286 | 0.046354 | 34 | 0.595238 | 0.009114 | ||||
14 | 13 | 56 | 0.384615 | 0.094479 | 50 | 0.450549 | 0.048173 | 38 | 0.582418 | 0.009150 | ||||
14 | 14 | 61 | 0.377551 | 0.093868 | 55 | 0.438776 | 0.049736 | 42 | 0.571429 | 0.009146 | ||||
15 | 1 | |||||||||||||
15 | 2 | 3 | 0.800000 | 0.088235 | 1 | 0.933333 | 0.029412 | |||||||
15 | 3 | 7 | 0.681818 | 0.075980 | 5 | 0.772727 | 0.039216 | 2 | 0.909091 | 0.009804 | ||||
15 | 4 | 12 | 0.600000 | 0.079979 | 10 | 0.666667 | 0.048504 | 5 | 0.833333 | 0.009288 | ||||
15 | 5 | 18 | 0.513514 | 0.098297 | 14 | 0.621622 | 0.041796 | 8 | 0.783784 | 0.007740 | ||||
15 | 6 | 23 | 0.488889 | 0.094833 | 19 | 0.577778 | 0.044855 | 12 | 0.733333 | 0.008367 | ||||
15 | 7 | 28 | 0.461538 | 0.091085 | 24 | 0.538462 | 0.046522 | 16 | 0.692308 | 0.008526 | ||||
15 | 8 | 33 | 0.450000 | 0.087332 | 29 | 0.516667 | 0.047304 | 20 | 0.666667 | 0.008456 | ||||
15 | 9 | 39 | 0.417910 | 0.095507 | 34 | 0.492537 | 0.047584 | 24 | 0.641791 | 0.008255 | ||||
15 | 10 | 44 | 0.413333 | 0.090971 | 39 | 0.480000 | 0.047524 | 29 | 0.613333 | 0.009616 | ||||
15 | 11 | 50 | 0.390244 | 0.097262 | 44 | 0.463415 | 0.047262 | 33 | 0.597561 | 0.009154 | ||||
15 | 12 | 55 | 0.388889 | 0.092610 | 49 | 0.455556 | 0.046866 | 37 | 0.588889 | 0.008710 | ||||
15 | 13 | 61 | 0.371134 | 0.097721 | 54 | 0.443299 | 0.046394 | 42 | 0.567010 | 0.009635 | ||||
15 | 14 | 66 | 0.371429 | 0.093216 | 59 | 0.438095 | 0.045875 | 46 | 0.561905 | 0.009115 | ||||
15 | 15 | 72 | 0.357143 | 0.097526 | 64 | 0.428571 | 0.045334 | 51 | 0.544643 | 0.009875 | ||||
16 | 1 | |||||||||||||
16 | 2 | 3 | 0.812500 | 0.078431 | 1 | 0.937500 | 0.026144 | |||||||
16 | 3 | 8 | 0.666667 | 0.084623 | 6 | 0.750000 | 0.047472 | 2 | 0.916667 | 0.008256 | ||||
16 | 4 | 14 | 0.562500 | 0.099484 | 11 | 0.656250 | 0.049948 | 5 | 0.843750 | 0.007430 | ||||
16 | 5 | 19 | 0.525000 | 0.091012 | 15 | 0.625000 | 0.040100 | 9 | 0.775000 | 0.008158 | ||||
16 | 6 | 25 | 0.479167 | 0.098026 | 21 | 0.562500 | 0.048731 | 13 | 0.729167 | 0.007988 | ||||
16 | 7 | 30 | 0.464286 | 0.088694 | 26 | 0.535714 | 0.046876 | 18 | 0.678571 | 0.009578 | ||||
16 | 8 | 36 | 0.437500 | 0.092602 | 31 | 0.515625 | 0.044823 | 22 | 0.656250 | 0.008748 | ||||
16 | 9 | 42 | 0.416667 | 0.095397 | 37 | 0.486111 | 0.049384 | 27 | 0.625000 | 0.009643 | ||||
16 | 10 | 48 | 0.400000 | 0.097414 | 42 | 0.475000 | 0.046707 | 31 | 0.612500 | 0.008685 | ||||
16 | 11 | 54 | 0.386364 | 0.098866 | 47 | 0.465909 | 0.044271 | 36 | 0.590909 | 0.009256 | ||||
16 | 12 | 60 | 0.375000 | 0.099904 | 53 | 0.447917 | 0.047276 | 41 | 0.572917 | 0.009707 | ||||
16 | 13 | 65 | 0.375000 | 0.091611 | 59 | 0.432692 | 0.049924 | 45 | 0.567308 | 0.008738 | ||||
16 | 14 | 71 | 0.366071 | 0.092540 | 64 | 0.428571 | 0.047205 | 50 | 0.553571 | 0.009064 | ||||
16 | 15 | 77 | 0.358333 | 0.093259 | 70 | 0.416667 | 0.049381 | 55 | 0.541667 | 0.009331 | ||||
16 | 16 | 83 | 0.351562 | 0.093812 | 75 | 0.414062 | 0.046815 | 60 | 0.531250 | 0.009551 | ||||
17 | 1 | |||||||||||||
17 | 2 | 3 | 0.823529 | 0.070175 | 2 | 0.882353 | 0.046784 | |||||||
17 | 3 | 9 | 0.640000 | 0.092982 | 6 | 0.760000 | 0.040351 | 2 | 0.920000 | 0.007018 | ||||
17 | 4 | 15 | 0.558824 | 0.098580 | 11 | 0.676471 | 0.040434 | 6 | 0.823529 | 0.009023 | ||||
17 | 5 | 20 | 0.523810 | 0.084909 | 17 | 0.595238 | 0.047695 | 10 | 0.761905 | 0.008582 | ||||
17 | 6 | 26 | 0.490196 | 0.086501 | 22 | 0.568627 | 0.043766 | 15 | 0.705882 | 0.009867 | ||||
17 | 7 | 33 | 0.440678 | 0.099490 | 28 | 0.525424 | 0.047171 | 19 | 0.677966 | 0.008518 | ||||
17 | 8 | 39 | 0.426471 | 0.097491 | 34 | 0.500000 | 0.049474 | 24 | 0.647059 | 0.009005 | ||||
17 | 9 | 45 | 0.407895 | 0.095246 | 39 | 0.486842 | 0.044566 | 29 | 0.618421 | 0.009248 | ||||
17 | 10 | 51 | 0.400000 | 0.092922 | 45 | 0.470588 | 0.045937 | 34 | 0.600000 | 0.009341 | ||||
17 | 11 | 57 | 0.387097 | 0.090623 | 51 | 0.451613 | 0.046916 | 39 | 0.580645 | 0.009331 | ||||
17 | 12 | 64 | 0.372549 | 0.097270 | 57 | 0.441176 | 0.047604 | 44 | 0.568627 | 0.009257 | ||||
17 | 13 | 70 | 0.363636 | 0.094466 | 63 | 0.427273 | 0.048075 | 49 | 0.554545 | 0.009141 | ||||
17 | 14 | 77 | 0.352941 | 0.099995 | 69 | 0.420168 | 0.048385 | 54 | 0.546219 | 0.008999 | ||||
17 | 15 | 83 | 0.346457 | 0.096996 | 75 | 0.409449 | 0.048571 | 60 | 0.527559 | 0.009973 | ||||
17 | 16 | 89 | 0.345588 | 0.094235 | 81 | 0.404412 | 0.048664 | 65 | 0.522059 | 0.009731 | ||||
17 | 17 | 96 | 0.333333 | 0.098687 | 87 | 0.395833 | 0.048686 | 70 | 0.513889 | 0.009494 | ||||
18 | 1 | |||||||||||||
18 | 2 | 4 | 0.777778 | 0.094737 | 2 | 0.888889 | 0.042105 | |||||||
18 | 3 | 9 | 0.666667 | 0.079699 | 7 | 0.740741 | 0.046617 | 2 | 0.925926 | 0.006015 | ||||
18 | 4 | 16 | 0.555556 | 0.098154 | 12 | 0.666667 | 0.042379 | 6 | 0.833333 | 0.007382 | ||||
18 | 5 | 22 | 0.511111 | 0.094327 | 18 | 0.600000 | 0.045707 | 11 | 0.755556 | 0.008916 | ||||
18 | 6 | 28 | 0.481481 | 0.089527 | 24 | 0.555556 | 0.047193 | 16 | 0.703704 | 0.009421 | ||||
18 | 7 | 35 | 0.444444 | 0.096701 | 30 | 0.523810 | 0.047418 | 21 | 0.666667 | 0.009445 | ||||
18 | 8 | 41 | 0.430556 | 0.090496 | 36 | 0.500000 | 0.046988 | 26 | 0.638889 | 0.009233 | ||||
18 | 9 | 48 | 0.407407 | 0.095074 | 42 | 0.481481 | 0.046198 | 31 | 0.617284 | 0.008893 | ||||
18 | 10 | 55 | 0.388889 | 0.098664 | 48 | 0.466667 | 0.045221 | 37 | 0.588889 | 0.009955 | ||||
18 | 11 | 61 | 0.383838 | 0.092197 | 55 | 0.444444 | 0.049392 | 42 | 0.575758 | 0.009388 | ||||
18 | 12 | 68 | 0.370370 | 0.094866 | 61 | 0.435185 | 0.047865 | 47 | 0.564815 | 0.008851 | ||||
18 | 13 | 75 | 0.358974 | 0.097070 | 67 | 0.427350 | 0.046401 | 53 | 0.547009 | 0.009505 | ||||
18 | 14 | 82 | 0.349206 | 0.098905 | 74 | 0.412698 | 0.049436 | 58 | 0.539683 | 0.008925 | ||||
18 | 15 | 88 | 0.348148 | 0.092994 | 80 | 0.407407 | 0.047795 | 64 | 0.525926 | 0.009432 | ||||
18 | 16 | 95 | 0.340278 | 0.094552 | 86 | 0.402778 | 0.046272 | 70 | 0.513889 | 0.009880 | ||||
18 | 17 | 102 | 0.333333 | 0.095895 | 93 | 0.392157 | 0.048652 | 75 | 0.509804 | 0.009265 | ||||
18 | 18 | 109 | 0.327160 | 0.097059 | 99 | 0.388889 | 0.047085 | 81 | 0.500000 | 0.009631 | ||||
19 | 1 | |||||||||||||
19 | 2 | 4 | 0.789474 | 0.085714 | 2 | 0.894737 | 0.038095 | 0 | 1.000000 | 0.009524 | ||||
19 | 3 | 10 | 0.642857 | 0.087013 | 7 | 0.750000 | 0.040260 | 3 | 0.892857 | 0.009091 | ||||
19 | 4 | 17 | 0.552632 | 0.097346 | 13 | 0.657895 | 0.043817 | 7 | 0.815789 | 0.008583 | ||||
19 | 5 | 23 | 0.510638 | 0.088368 | 19 | 0.595745 | 0.043902 | 12 | 0.744681 | 0.009270 | ||||
19 | 6 | 30 | 0.473684 | 0.092321 | 25 | 0.561404 | 0.042778 | 17 | 0.701754 | 0.009001 | ||||
19 | 7 | 37 | 0.439394 | 0.094199 | 32 | 0.515152 | 0.047622 | 22 | 0.666667 | 0.008489 | ||||
19 | 8 | 44 | 0.421053 | 0.094915 | 38 | 0.500000 | 0.044792 | 28 | 0.631579 | 0.009436 | ||||
19 | 9 | 51 | 0.400000 | 0.094882 | 45 | 0.470588 | 0.047700 | 33 | 0.611765 | 0.008572 | ||||
19 | 10 | 58 | 0.389474 | 0.094392 | 52 | 0.452632 | 0.049957 | 39 | 0.589474 | 0.009074 | ||||
19 | 11 | 65 | 0.375000 | 0.093614 | 58 | 0.442308 | 0.046502 | 45 | 0.567308 | 0.009429 | ||||
19 | 12 | 72 | 0.368421 | 0.092664 | 65 | 0.429825 | 0.048074 | 51 | 0.552632 | 0.009674 | ||||
19 | 13 | 80 | 0.349594 | 0.099454 | 72 | 0.414634 | 0.049346 | 57 | 0.536585 | 0.009835 | ||||
19 | 14 | 87 | 0.345865 | 0.097861 | 78 | 0.413534 | 0.046065 | 63 | 0.526316 | 0.009935 | ||||
19 | 15 | 94 | 0.338028 | 0.096301 | 85 | 0.401408 | 0.047054 | 69 | 0.514085 | 0.009986 | ||||
19 | 16 | 101 | 0.335526 | 0.094785 | 92 | 0.394737 | 0.047883 | 74 | 0.513158 | 0.009009 | ||||
19 | 17 | 109 | 0.322981 | 0.099827 | 99 | 0.385093 | 0.048578 | 81 | 0.496894 | 0.009991 | ||||
19 | 18 | 116 | 0.321637 | 0.098072 | 106 | 0.380117 | 0.049163 | 87 | 0.491228 | 0.009960 | ||||
19 | 19 | 123 | 0.316667 | 0.096409 | 113 | 0.372222 | 0.049656 | 93 | 0.483333 | 0.009914 | ||||
20 | 1 | 0 | 1.000000 | 0.095238 | ||||||||||
20 | 2 | 4 | 0.800000 | 0.077922 | 2 | 0.900000 | 0.034632 | 0 | 1.000000 | 0.008658 | ||||
20 | 3 | 11 | 0.633333 | 0.093732 | 8 | 0.733333 | 0.046302 | 3 | 0.900000 | 0.007905 | ||||
20 | 4 | 18 | 0.550000 | 0.096932 | 14 | 0.650000 | 0.045360 | 8 | 0.800000 | 0.009976 | ||||
20 | 5 | 25 | 0.500000 | 0.096970 | 20 | 0.600000 | 0.042349 | 13 | 0.740000 | 0.009561 | ||||
20 | 6 | 32 | 0.466667 | 0.094905 | 27 | 0.550000 | 0.045858 | 18 | 0.700000 | 0.008652 | ||||
20 | 7 | 39 | 0.442857 | 0.091932 | 34 | 0.514286 | 0.047798 | 24 | 0.657143 | 0.009315 | ||||
20 | 8 | 47 | 0.412500 | 0.099062 | 41 | 0.487500 | 0.048749 | 30 | 0.625000 | 0.009617 | ||||
20 | 9 | 54 | 0.400000 | 0.094682 | 48 | 0.466667 | 0.049091 | 36 | 0.600000 | 0.009687 | ||||
20 | 10 | 62 | 0.380000 | 0.099577 | 55 | 0.450000 | 0.049031 | 42 | 0.580000 | 0.009616 | ||||
20 | 11 | 69 | 0.372727 | 0.094896 | 62 | 0.436364 | 0.048718 | 48 | 0.563636 | 0.009458 | ||||
20 | 12 | 77 | 0.358333 | 0.098543 | 69 | 0.425000 | 0.048240 | 54 | 0.550000 | 0.009249 | ||||
20 | 13 | 84 | 0.353846 | 0.093978 | 76 | 0.415385 | 0.047661 | 60 | 0.538462 | 0.009012 | ||||
20 | 14 | 92 | 0.342857 | 0.096865 | 83 | 0.407143 | 0.047021 | 67 | 0.521429 | 0.009796 | ||||
20 | 15 | 100 | 0.333333 | 0.099377 | 90 | 0.400000 | 0.046348 | 73 | 0.513333 | 0.009462 | ||||
20 | 16 | 107 | 0.331250 | 0.094950 | 98 | 0.387500 | 0.049370 | 79 | 0.506250 | 0.009140 | ||||
20 | 17 | 115 | 0.323529 | 0.097069 | 105 | 0.382353 | 0.048471 | 86 | 0.494118 | 0.009721 | ||||
20 | 18 | 123 | 0.316667 | 0.098957 | 112 | 0.377778 | 0.047600 | 92 | 0.488889 | 0.009363 | ||||
20 | 19 | 130 | 0.315789 | 0.094835 | 119 | 0.373684 | 0.046761 | 99 | 0.478947 | 0.009856 | ||||
20 | 20 | 138 | 0.310000 | 0.096500 | 127 | 0.365000 | 0.049090 | 105 | 0.475000 | 0.009484 |
Sample Sizes | p-value | p-value | p-value | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
.10 | .05 | .01 | |||||||||||||
2 | 1 | 1 | |||||||||||||
2 | 2 | 1 | |||||||||||||
2 | 2 | 2 | 0 | 1.000000 | 0.066667 | ||||||||||
3 | 1 | 1 | |||||||||||||
3 | 2 | 1 | |||||||||||||
3 | 2 | 2 | 1 | 0.875000 | 0.085714 | 0 | 1.000000 | 0.028571 | |||||||
3 | 3 | 1 | 0 | 1.000000 | 0.042857 | 0 | 1.000000 | 0.042857 | |||||||
3 | 3 | 2 | 2 | 0.800000 | 0.085714 | 1 | 0.900000 | 0.032143 | |||||||
3 | 3 | 3 | 3 | 0.769231 | 0.064286 | 2 | 0.846154 | 0.028571 | 0 | 1.000000 | 0.003571 | ||||
4 | 1 | 1 | |||||||||||||
4 | 2 | 1 | 0 | 1.000000 | 0.057143 | ||||||||||
4 | 2 | 2 | 1 | 0.900000 | 0.042857 | 1 | 0.900000 | 0.042857 | |||||||
4 | 3 | 1 | 1 | 0.888889 | 0.064286 | 0 | 1.000000 | 0.021429 | |||||||
4 | 3 | 2 | 3 | 0.769231 | 0.077778 | 2 | 0.846154 | 0.038095 | 0 | 1.000000 | 0.004762 | ||||
4 | 3 | 3 | 5 | 0.687500 | 0.090000 | 3 | 0.812500 | 0.025714 | 1 | 0.937500 | 0.004286 | ||||
4 | 4 | 1 | 2 | 0.833333 | 0.060317 | 1 | 0.916667 | 0.028571 | 0 | 1.000000 | 0.009524 | ||||
4 | 4 | 2 | 4 | 0.750000 | 0.060952 | 3 | 0.812500 | 0.032381 | 1 | 0.937500 | 0.005714 | ||||
4 | 4 | 3 | 7 | 0.650000 | 0.095065 | 5 | 0.750000 | 0.035325 | 3 | 0.850000 | 0.009351 | ||||
4 | 4 | 4 | 9 | 0.625000 | 0.086580 | 7 | 0.708333 | 0.036883 | 4 | 0.833333 | 0.006580 | ||||
5 | 1 | 1 | |||||||||||||
5 | 2 | 1 | 1 | 0.875000 | 0.095238 | 0 | 1.000000 | 0.035714 | |||||||
5 | 2 | 2 | 2 | 0.833333 | 0.058201 | 1 | 0.916667 | 0.023810 | 0 | 1.000000 | 0.007937 | ||||
5 | 3 | 1 | 2 | 0.818182 | 0.075397 | 1 | 0.909091 | 0.035714 | |||||||
5 | 3 | 2 | 4 | 0.733333 | 0.072222 | 3 | 0.800000 | 0.038889 | 1 | 0.933333 | 0.007143 | ||||
5 | 3 | 3 | 6 | 0.684211 | 0.070130 | 5 | 0.736842 | 0.041558 | 2 | 0.894737 | 0.005195 | ||||
5 | 4 | 1 | 4 | 0.714286 | 0.098413 | 2 | 0.857143 | 0.031746 | 0 | 1.000000 | 0.004762 | ||||
5 | 4 | 2 | 6 | 0.684211 | 0.079654 | 5 | 0.736842 | 0.049062 | 2 | 0.894737 | 0.006926 | ||||
5 | 4 | 3 | 9 | 0.608696 | 0.098341 | 7 | 0.695652 | 0.042641 | 4 | 0.826087 | 0.008009 | ||||
5 | 4 | 4 | 11 | 0.607143 | 0.079343 | 9 | 0.678571 | 0.037163 | 6 | 0.785714 | 0.008658 | ||||
5 | 5 | 1 | 5 | 0.705882 | 0.077201 | 4 | 0.764706 | 0.047619 | 1 | 0.941176 | 0.006494 | ||||
5 | 5 | 2 | 8 | 0.636364 | 0.084416 | 6 | 0.727273 | 0.035714 | 3 | 0.863636 | 0.006133 | ||||
5 | 5 | 3 | 11 | 0.592593 | 0.089022 | 9 | 0.666667 | 0.042374 | 5 | 0.814815 | 0.005828 | ||||
5 | 5 | 4 | 14 | 0.562500 | 0.089498 | 12 | 0.625000 | 0.047072 | 8 | 0.750000 | 0.009039 | ||||
5 | 5 | 5 | 17 | 0.540541 | 0.088887 | 14 | 0.621622 | 0.036630 | 10 | 0.729730 | 0.008016 | ||||
6 | 1 | 1 | |||||||||||||
6 | 2 | 1 | 1 | 0.900000 | 0.063492 | 0 | 1.000000 | 0.023810 | |||||||
6 | 2 | 2 | 3 | 0.785714 | 0.066667 | 2 | 0.857143 | 0.034921 | 0 | 1.000000 | 0.004762 | ||||
6 | 3 | 1 | 3 | 0.769231 | 0.083333 | 2 | 0.846154 | 0.045238 | 0 | 1.000000 | 0.007143 | ||||
6 | 3 | 2 | 5 | 0.722222 | 0.067532 | 4 | 0.777778 | 0.039394 | 1 | 0.944444 | 0.003896 | ||||
6 | 3 | 3 | 8 | 0.636364 | 0.087554 | 6 | 0.727273 | 0.035390 | 3 | 0.863636 | 0.005844 | ||||
6 | 4 | 1 | 5 | 0.705882 | 0.089177 | 3 | 0.823529 | 0.032035 | 1 | 0.941176 | 0.007792 | ||||
6 | 4 | 2 | 8 | 0.636364 | 0.096537 | 6 | 0.727273 | 0.041414 | 3 | 0.863636 | 0.007359 | ||||
6 | 4 | 3 | 11 | 0.592593 | 0.099933 | 9 | 0.666667 | 0.048119 | 5 | 0.814815 | 0.006893 | ||||
6 | 4 | 4 | 14 | 0.562500 | 0.099310 | 11 | 0.656250 | 0.036934 | 7 | 0.781250 | 0.006394 | ||||
6 | 5 | 1 | 7 | 0.650000 | 0.094156 | 5 | 0.750000 | 0.040043 | 2 | 0.900000 | 0.007576 | ||||
6 | 5 | 2 | 10 | 0.615385 | 0.087468 | 8 | 0.692308 | 0.041570 | 4 | 0.846154 | 0.005661 | ||||
6 | 5 | 3 | 13 | 0.580645 | 0.081205 | 11 | 0.645161 | 0.041625 | 7 | 0.774194 | 0.007635 | ||||
6 | 5 | 4 | 17 | 0.540541 | 0.097296 | 14 | 0.621622 | 0.040721 | 10 | 0.729730 | 0.009238 | ||||
6 | 5 | 5 | 20 | 0.523810 | 0.087370 | 17 | 0.595238 | 0.039446 | 12 | 0.714286 | 0.007222 | ||||
6 | 6 | 1 | 9 | 0.625000 | 0.095571 | 7 | 0.708333 | 0.046287 | 3 | 0.875000 | 0.006660 | ||||
6 | 6 | 2 | 12 | 0.600000 | 0.080039 | 10 | 0.666667 | 0.041173 | 6 | 0.800000 | 0.007588 | ||||
6 | 6 | 3 | 16 | 0.555556 | 0.089258 | 13 | 0.638889 | 0.036473 | 9 | 0.750000 | 0.008044 | ||||
6 | 6 | 4 | 20 | 0.523810 | 0.094960 | 17 | 0.595238 | 0.043424 | 12 | 0.714286 | 0.008249 | ||||
6 | 6 | 5 | 24 | 0.500000 | 0.098268 | 21 | 0.562500 | 0.049106 | 15 | 0.687500 | 0.008321 | ||||
6 | 6 | 6 | 27 | 0.500000 | 0.082204 | 24 | 0.555556 | 0.042636 | 18 | 0.666667 | 0.008323 | ||||
7 | 1 | 1 | 0 | 1.000000 | 0.083333 | ||||||||||
7 | 2 | 1 | 2 | 0.818182 | 0.094444 | 1 | 0.909091 | 0.044444 | |||||||
7 | 2 | 2 | 4 | 0.750000 | 0.076768 | 3 | 0.812500 | 0.042424 | 1 | 0.937500 | 0.009091 | ||||
7 | 3 | 1 | 4 | 0.733333 | 0.092424 | 2 | 0.866667 | 0.028788 | 0 | 1.000000 | 0.004545 | ||||
7 | 3 | 2 | 7 | 0.650000 | 0.098737 | 5 | 0.750000 | 0.039394 | 2 | 0.900000 | 0.006061 | ||||
7 | 3 | 3 | 9 | 0.640000 | 0.071270 | 8 | 0.680000 | 0.047669 | 4 | 0.840000 | 0.006294 | ||||
7 | 4 | 1 | 6 | 0.684211 | 0.083333 | 4 | 0.789474 | 0.033333 | 1 | 0.947368 | 0.004545 | ||||
7 | 4 | 2 | 9 | 0.640000 | 0.078788 | 7 | 0.720000 | 0.035664 | 4 | 0.840000 | 0.007692 | ||||
7 | 4 | 3 | 12 | 0.600000 | 0.074026 | 10 | 0.666667 | 0.036680 | 6 | 0.800000 | 0.006061 | ||||
7 | 4 | 4 | 16 | 0.555556 | 0.090541 | 13 | 0.638889 | 0.036572 | 9 | 0.750000 | 0.007779 | ||||
7 | 5 | 1 | 8 | 0.652174 | 0.077506 | 6 | 0.739130 | 0.035354 | 3 | 0.869565 | 0.007770 | ||||
7 | 5 | 2 | 12 | 0.586207 | 0.089494 | 10 | 0.655172 | 0.046481 | 6 | 0.793103 | 0.008658 | ||||
7 | 5 | 3 | 16 | 0.542857 | 0.098957 | 13 | 0.628571 | 0.040904 | 9 | 0.742857 | 0.009108 | ||||
7 | 5 | 4 | 19 | 0.536585 | 0.081531 | 17 | 0.585366 | 0.048012 | 12 | 0.707317 | 0.009257 | ||||
7 | 5 | 5 | 23 | 0.510638 | 0.085929 | 20 | 0.574468 | 0.041698 | 15 | 0.680851 | 0.009291 | ||||
7 | 6 | 1 | 11 | 0.592593 | 0.096820 | 8 | 0.703704 | 0.035881 | 5 | 0.814815 | 0.009907 | ||||
7 | 6 | 2 | 15 | 0.558824 | 0.097303 | 12 | 0.647059 | 0.040593 | 8 | 0.764706 | 0.009135 | ||||
7 | 6 | 3 | 19 | 0.525000 | 0.096083 | 16 | 0.600000 | 0.043746 | 11 | 0.725000 | 0.008244 | ||||
7 | 6 | 4 | 23 | 0.510638 | 0.092608 | 20 | 0.574468 | 0.045458 | 14 | 0.702128 | 0.007419 | ||||
7 | 6 | 5 | 27 | 0.490566 | 0.088632 | 24 | 0.547170 | 0.046362 | 18 | 0.660377 | 0.009231 | ||||
7 | 6 | 6 | 31 | 0.483333 | 0.084562 | 28 | 0.533333 | 0.046592 | 21 | 0.650000 | 0.008248 | ||||
7 | 7 | 1 | 13 | 0.580645 | 0.088462 | 11 | 0.645161 | 0.049728 | 6 | 0.806452 | 0.007653 | ||||
7 | 7 | 2 | 17 | 0.552632 | 0.081371 | 15 | 0.605263 | 0.048067 | 10 | 0.736842 | 0.009368 | ||||
7 | 7 | 3 | 22 | 0.511111 | 0.093660 | 19 | 0.577778 | 0.045940 | 13 | 0.711111 | 0.007505 | ||||
7 | 7 | 4 | 26 | 0.500000 | 0.083679 | 23 | 0.557692 | 0.043172 | 17 | 0.673077 | 0.008389 | ||||
7 | 7 | 5 | 31 | 0.474576 | 0.090678 | 27 | 0.542373 | 0.040628 | 21 | 0.644068 | 0.009101 | ||||
7 | 7 | 6 | 36 | 0.454545 | 0.095828 | 32 | 0.515152 | 0.046525 | 25 | 0.621212 | 0.009656 | ||||
7 | 7 | 7 | 40 | 0.452055 | 0.085655 | 36 | 0.506849 | 0.043267 | 28 | 0.616438 | 0.007945 | ||||
8 | 1 | 1 | 0 | 1.000000 | 0.066667 | ||||||||||
8 | 2 | 1 | 2 | 0.846154 | 0.068687 | 1 | 0.923077 | 0.032323 | |||||||
8 | 2 | 2 | 5 | 0.722222 | 0.083502 | 3 | 0.833333 | 0.028283 | 1 | 0.944444 | 0.006061 | ||||
8 | 3 | 1 | 5 | 0.705882 | 0.097980 | 3 | 0.823529 | 0.036364 | 1 | 0.941176 | 0.009091 | ||||
8 | 3 | 2 | 8 | 0.652174 | 0.091064 | 6 | 0.739130 | 0.039627 | 3 | 0.869565 | 0.007615 | ||||
8 | 3 | 3 | 11 | 0.607143 | 0.084582 | 9 | 0.678571 | 0.041026 | 5 | 0.821429 | 0.006394 | ||||
8 | 4 | 1 | 7 | 0.681818 | 0.077389 | 5 | 0.772727 | 0.033877 | 2 | 0.909091 | 0.006216 | ||||
8 | 4 | 2 | 11 | 0.607143 | 0.091553 | 9 | 0.678571 | 0.046309 | 5 | 0.821429 | 0.007681 | ||||
8 | 4 | 3 | 14 | 0.588235 | 0.076546 | 12 | 0.647059 | 0.040884 | 8 | 0.764706 | 0.008560 | ||||
8 | 4 | 4 | 18 | 0.550000 | 0.083417 | 16 | 0.600000 | 0.048629 | 11 | 0.725000 | 0.008991 | ||||
8 | 5 | 1 | 10 | 0.615385 | 0.089355 | 8 | 0.692308 | 0.045732 | 4 | 0.846154 | 0.007881 | ||||
8 | 5 | 2 | 14 | 0.575758 | 0.090768 | 11 | 0.666667 | 0.036408 | 7 | 0.787879 | 0.007489 | ||||
8 | 5 | 3 | 18 | 0.538462 | 0.090415 | 15 | 0.615385 | 0.040041 | 10 | 0.743590 | 0.006990 | ||||
8 | 5 | 4 | 22 | 0.521739 | 0.087620 | 19 | 0.586957 | 0.042130 | 14 | 0.695652 | 0.009212 | ||||
8 | 5 | 5 | 26 | 0.500000 | 0.084260 | 23 | 0.557692 | 0.043404 | 17 | 0.673077 | 0.008280 | ||||
8 | 6 | 1 | 13 | 0.580645 | 0.096881 | 10 | 0.677419 | 0.040004 | 6 | 0.806452 | 0.008614 | ||||
8 | 6 | 2 | 17 | 0.552632 | 0.089066 | 14 | 0.631579 | 0.039832 | 9 | 0.763158 | 0.007065 | ||||
8 | 6 | 3 | 21 | 0.533333 | 0.081197 | 18 | 0.600000 | 0.038672 | 13 | 0.711111 | 0.008335 | ||||
8 | 6 | 4 | 26 | 0.500000 | 0.090348 | 23 | 0.557692 | 0.046990 | 17 | 0.673077 | 0.009265 | ||||
8 | 6 | 5 | 31 | 0.474576 | 0.097034 | 27 | 0.542373 | 0.043907 | 21 | 0.644068 | 0.009983 | ||||
8 | 6 | 6 | 35 | 0.469697 | 0.086285 | 32 | 0.515152 | 0.049960 | 24 | 0.636364 | 0.008137 | ||||
8 | 7 | 1 | 15 | 0.571429 | 0.081138 | 13 | 0.628571 | 0.047786 | 8 | 0.771429 | 0.009091 | ||||
8 | 7 | 2 | 20 | 0.534884 | 0.087307 | 17 | 0.604651 | 0.042356 | 12 | 0.720930 | 0.009450 | ||||
8 | 7 | 3 | 25 | 0.500000 | 0.091071 | 22 | 0.560000 | 0.047473 | 16 | 0.680000 | 0.009437 | ||||
8 | 7 | 4 | 30 | 0.482759 | 0.092169 | 26 | 0.551724 | 0.041134 | 20 | 0.655172 | 0.009202 | ||||
8 | 7 | 5 | 35 | 0.461538 | 0.091936 | 31 | 0.523077 | 0.044101 | 24 | 0.630769 | 0.008929 | ||||
8 | 7 | 6 | 40 | 0.452055 | 0.090824 | 36 | 0.506849 | 0.046234 | 28 | 0.616438 | 0.008639 | ||||
8 | 7 | 7 | 45 | 0.437500 | 0.089477 | 41 | 0.487500 | 0.047863 | 32 | 0.600000 | 0.008348 | ||||
8 | 8 | 1 | 18 | 0.550000 | 0.086668 | 15 | 0.625000 | 0.042022 | 10 | 0.750000 | 0.009297 | ||||
8 | 8 | 2 | 23 | 0.520833 | 0.085381 | 20 | 0.583333 | 0.044213 | 14 | 0.708333 | 0.008570 | ||||
8 | 8 | 3 | 29 | 0.482143 | 0.099335 | 25 | 0.553571 | 0.044954 | 18 | 0.678571 | 0.007749 | ||||
8 | 8 | 4 | 34 | 0.468750 | 0.093356 | 30 | 0.531250 | 0.044695 | 23 | 0.640625 | 0.009074 | ||||
8 | 8 | 5 | 39 | 0.458333 | 0.087277 | 35 | 0.513889 | 0.044004 | 27 | 0.625000 | 0.008054 | ||||
8 | 8 | 6 | 45 | 0.437500 | 0.094509 | 40 | 0.500000 | 0.043013 | 32 | 0.600000 | 0.009033 | ||||
8 | 8 | 7 | 50 | 0.431818 | 0.087903 | 46 | 0.477273 | 0.049055 | 37 | 0.579545 | 0.009894 | ||||
8 | 8 | 8 | 56 | 0.416667 | 0.093322 | 51 | 0.468750 | 0.047287 | 41 | 0.572917 | 0.008809 | ||||
9 | 1 | 1 | 0 | 1.000000 | 0.054545 | ||||||||||
9 | 2 | 1 | 3 | 0.785714 | 0.093939 | 1 | 0.928571 | 0.024242 | 0 | 1.000000 | 0.009091 | ||||
9 | 2 | 2 | 6 | 0.700000 | 0.090443 | 4 | 0.800000 | 0.035431 | 1 | 0.950000 | 0.004196 | ||||
9 | 3 | 1 | 5 | 0.736842 | 0.069231 | 4 | 0.789474 | 0.044056 | 1 | 0.947368 | 0.006294 | ||||
9 | 3 | 2 | 9 | 0.640000 | 0.084815 | 7 | 0.720000 | 0.039461 | 4 | 0.840000 | 0.009091 | ||||
9 | 3 | 3 | 13 | 0.580645 | 0.096883 | 10 | 0.677419 | 0.036064 | 6 | 0.806452 | 0.006533 | ||||
9 | 4 | 1 | 8 | 0.666667 | 0.073327 | 6 | 0.750000 | 0.034565 | 3 | 0.875000 | 0.007592 | ||||
9 | 4 | 2 | 12 | 0.612903 | 0.077416 | 10 | 0.677419 | 0.040573 | 6 | 0.806452 | 0.007752 | ||||
9 | 4 | 3 | 16 | 0.567568 | 0.078701 | 14 | 0.621622 | 0.044560 | 9 | 0.756757 | 0.007438 | ||||
9 | 4 | 4 | 21 | 0.522727 | 0.097718 | 18 | 0.590909 | 0.046871 | 12 | 0.727273 | 0.007022 | ||||
9 | 5 | 1 | 11 | 0.620690 | 0.075658 | 9 | 0.689655 | 0.040160 | 5 | 0.827586 | 0.007925 | ||||
9 | 5 | 2 | 16 | 0.555556 | 0.091767 | 13 | 0.638889 | 0.040152 | 8 | 0.777778 | 0.006618 | ||||
9 | 5 | 3 | 20 | 0.534884 | 0.083722 | 17 | 0.604651 | 0.039249 | 12 | 0.720930 | 0.008063 | ||||
9 | 5 | 4 | 25 | 0.500000 | 0.092920 | 22 | 0.560000 | 0.047915 | 16 | 0.680000 | 0.009130 | ||||
9 | 5 | 5 | 30 | 0.473684 | 0.099791 | 26 | 0.543860 | 0.044813 | 20 | 0.649123 | 0.009980 | ||||
9 | 6 | 1 | 15 | 0.558824 | 0.097278 | 12 | 0.647059 | 0.043681 | 7 | 0.794118 | 0.007617 | ||||
9 | 6 | 2 | 19 | 0.547619 | 0.082476 | 16 | 0.619048 | 0.039079 | 11 | 0.738095 | 0.008189 | ||||
9 | 6 | 3 | 24 | 0.510204 | 0.086697 | 21 | 0.571429 | 0.044388 | 15 | 0.693878 | 0.008372 | ||||
9 | 6 | 4 | 29 | 0.491228 | 0.088247 | 26 | 0.543860 | 0.048185 | 19 | 0.666667 | 0.008313 | ||||
9 | 6 | 5 | 34 | 0.468750 | 0.088358 | 30 | 0.531250 | 0.041791 | 23 | 0.640625 | 0.008178 | ||||
9 | 6 | 6 | 39 | 0.458333 | 0.087572 | 35 | 0.513889 | 0.044093 | 27 | 0.625000 | 0.008005 | ||||
9 | 7 | 1 | 18 | 0.538462 | 0.094755 | 15 | 0.615385 | 0.046318 | 9 | 0.769231 | 0.007240 | ||||
9 | 7 | 2 | 23 | 0.510638 | 0.092339 | 20 | 0.574468 | 0.048129 | 14 | 0.702128 | 0.009452 | ||||
9 | 7 | 3 | 28 | 0.490909 | 0.088931 | 25 | 0.545455 | 0.048734 | 18 | 0.672727 | 0.008511 | ||||
9 | 7 | 4 | 34 | 0.460317 | 0.099728 | 30 | 0.523810 | 0.048091 | 23 | 0.634921 | 0.009883 | ||||
9 | 7 | 5 | 39 | 0.450704 | 0.092938 | 35 | 0.507042 | 0.047167 | 27 | 0.619718 | 0.008744 | ||||
9 | 7 | 6 | 45 | 0.430380 | 0.099994 | 40 | 0.493671 | 0.045852 | 32 | 0.594937 | 0.009751 | ||||
9 | 7 | 7 | 50 | 0.425287 | 0.092778 | 45 | 0.482759 | 0.044526 | 36 | 0.586207 | 0.008675 | ||||
9 | 8 | 1 | 21 | 0.522727 | 0.091613 | 18 | 0.590909 | 0.047877 | 12 | 0.727273 | 0.009402 | ||||
9 | 8 | 2 | 26 | 0.509434 | 0.083606 | 23 | 0.566038 | 0.045645 | 16 | 0.698113 | 0.007832 | ||||
9 | 8 | 3 | 32 | 0.475410 | 0.090200 | 28 | 0.540984 | 0.042843 | 21 | 0.655738 | 0.008515 | ||||
9 | 8 | 4 | 38 | 0.457143 | 0.094142 | 34 | 0.514286 | 0.047762 | 26 | 0.628571 | 0.008914 | ||||
9 | 8 | 5 | 44 | 0.435897 | 0.096461 | 39 | 0.500000 | 0.043813 | 31 | 0.602564 | 0.009177 | ||||
9 | 8 | 6 | 50 | 0.425287 | 0.097552 | 45 | 0.482759 | 0.047178 | 36 | 0.586207 | 0.009342 | ||||
9 | 8 | 7 | 56 | 0.410526 | 0.098037 | 51 | 0.463158 | 0.049980 | 41 | 0.568421 | 0.009437 | ||||
9 | 8 | 8 | 62 | 0.403846 | 0.098020 | 56 | 0.461538 | 0.045624 | 46 | 0.557692 | 0.009473 | ||||
9 | 9 | 1 | 24 | 0.510204 | 0.089203 | 21 | 0.571429 | 0.049174 | 14 | 0.714286 | 0.008606 | ||||
9 | 9 | 2 | 30 | 0.482759 | 0.091114 | 26 | 0.551724 | 0.043471 | 19 | 0.672414 | 0.008661 | ||||
9 | 9 | 3 | 36 | 0.462687 | 0.091244 | 32 | 0.522388 | 0.046073 | 24 | 0.641791 | 0.008468 | ||||
9 | 9 | 4 | 42 | 0.447368 | 0.089294 | 38 | 0.500000 | 0.047313 | 29 | 0.618421 | 0.008104 | ||||
9 | 9 | 5 | 49 | 0.423529 | 0.099468 | 44 | 0.482353 | 0.048059 | 35 | 0.588235 | 0.009520 | ||||
9 | 9 | 6 | 55 | 0.414894 | 0.095225 | 50 | 0.468085 | 0.048205 | 40 | 0.574468 | 0.008960 | ||||
9 | 9 | 7 | 61 | 0.407767 | 0.091262 | 56 | 0.456311 | 0.048100 | 45 | 0.563107 | 0.008460 | ||||
9 | 9 | 8 | 68 | 0.392857 | 0.097772 | 62 | 0.446429 | 0.047751 | 51 | 0.544643 | 0.009469 | ||||
9 | 9 | 9 | 74 | 0.388430 | 0.093398 | 68 | 0.438017 | 0.047321 | 56 | 0.537190 | 0.008904 | ||||
10 | 1 | 1 | 0 | 1.000000 | 0.045455 | 0 | 1.000000 | 0.045455 | |||||||
10 | 2 | 1 | 3 | 0.812500 | 0.072261 | 2 | 0.875000 | 0.039627 | 0 | 1.000000 | 0.006993 | ||||
10 | 2 | 2 | 7 | 0.681818 | 0.095238 | 5 | 0.772727 | 0.041292 | 2 | 0.909091 | 0.007326 | ||||
10 | 3 | 1 | 6 | 0.714286 | 0.074925 | 5 | 0.761905 | 0.049950 | 2 | 0.904762 | 0.009491 | ||||
10 | 3 | 2 | 10 | 0.642857 | 0.079853 | 8 | 0.714286 | 0.039361 | 4 | 0.857143 | 0.006061 | ||||
10 | 3 | 3 | 14 | 0.588235 | 0.082105 | 12 | 0.647059 | 0.044843 | 7 | 0.794118 | 0.006581 | ||||
10 | 4 | 1 | 10 | 0.629630 | 0.094439 | 8 | 0.703704 | 0.049817 | 4 | 0.851852 | 0.009058 | ||||
10 | 4 | 2 | 14 | 0.588235 | 0.087796 | 12 | 0.647059 | 0.049534 | 7 | 0.794118 | 0.007709 | ||||
10 | 4 | 3 | 18 | 0.560976 | 0.080364 | 16 | 0.609756 | 0.047764 | 11 | 0.731707 | 0.009629 | ||||
10 | 4 | 4 | 23 | 0.520833 | 0.090451 | 20 | 0.583333 | 0.045339 | 14 | 0.708333 | 0.007948 | ||||
10 | 5 | 1 | 13 | 0.593750 | 0.085331 | 11 | 0.656250 | 0.048701 | 6 | 0.812500 | 0.007992 | ||||
10 | 5 | 2 | 18 | 0.550000 | 0.092437 | 15 | 0.625000 | 0.043398 | 10 | 0.750000 | 0.008731 | ||||
10 | 5 | 3 | 23 | 0.510638 | 0.096662 | 20 | 0.574468 | 0.049272 | 14 | 0.702128 | 0.009033 | ||||
10 | 5 | 4 | 28 | 0.490909 | 0.097473 | 24 | 0.563636 | 0.042664 | 18 | 0.672727 | 0.009027 | ||||
10 | 5 | 5 | 33 | 0.467742 | 0.096851 | 29 | 0.532258 | 0.045909 | 22 | 0.645161 | 0.008935 | ||||
10 | 6 | 1 | 17 | 0.552632 | 0.096918 | 14 | 0.631579 | 0.046615 | 9 | 0.763158 | 0.009887 | ||||
10 | 6 | 2 | 22 | 0.521739 | 0.095008 | 19 | 0.586957 | 0.048928 | 13 | 0.717391 | 0.009192 | ||||
10 | 6 | 3 | 27 | 0.500000 | 0.091298 | 24 | 0.555556 | 0.049602 | 17 | 0.685185 | 0.008356 | ||||
10 | 6 | 4 | 32 | 0.483871 | 0.086315 | 29 | 0.532258 | 0.049122 | 22 | 0.645161 | 0.009852 | ||||
10 | 6 | 5 | 38 | 0.457143 | 0.095262 | 34 | 0.514286 | 0.048127 | 26 | 0.628571 | 0.008761 | ||||
10 | 6 | 6 | 43 | 0.448718 | 0.088541 | 39 | 0.500000 | 0.046806 | 31 | 0.602564 | 0.009842 | ||||
10 | 7 | 1 | 20 | 0.534884 | 0.087281 | 17 | 0.604651 | 0.044752 | 11 | 0.744186 | 0.008261 | ||||
10 | 7 | 2 | 26 | 0.500000 | 0.096557 | 22 | 0.576923 | 0.042974 | 16 | 0.692308 | 0.009396 | ||||
10 | 7 | 3 | 31 | 0.483333 | 0.086789 | 28 | 0.533333 | 0.049625 | 20 | 0.666667 | 0.007738 | ||||
10 | 7 | 4 | 37 | 0.463768 | 0.090962 | 33 | 0.521739 | 0.045577 | 25 | 0.637681 | 0.008213 | ||||
10 | 7 | 5 | 43 | 0.441558 | 0.093529 | 39 | 0.493506 | 0.049803 | 30 | 0.610390 | 0.008552 | ||||
10 | 7 | 6 | 49 | 0.430233 | 0.094802 | 44 | 0.488372 | 0.045397 | 35 | 0.593023 | 0.008781 | ||||
10 | 7 | 7 | 55 | 0.414894 | 0.095476 | 50 | 0.468085 | 0.048288 | 40 | 0.574468 | 0.008936 | ||||
10 | 8 | 1 | 24 | 0.510204 | 0.095314 | 20 | 0.591837 | 0.042629 | 14 | 0.714286 | 0.009362 | ||||
10 | 8 | 2 | 30 | 0.482759 | 0.097293 | 26 | 0.551724 | 0.046727 | 19 | 0.672414 | 0.009415 | ||||
10 | 8 | 3 | 36 | 0.462687 | 0.096898 | 32 | 0.522388 | 0.049214 | 24 | 0.641791 | 0.009154 | ||||
10 | 8 | 4 | 42 | 0.447368 | 0.094647 | 37 | 0.513158 | 0.042505 | 29 | 0.618421 | 0.008738 | ||||
10 | 8 | 5 | 48 | 0.435294 | 0.091545 | 43 | 0.494118 | 0.043502 | 34 | 0.600000 | 0.008310 | ||||
10 | 8 | 6 | 54 | 0.425532 | 0.088149 | 49 | 0.478723 | 0.043991 | 40 | 0.574468 | 0.009585 | ||||
10 | 8 | 7 | 61 | 0.407767 | 0.095600 | 55 | 0.466019 | 0.044145 | 45 | 0.563107 | 0.009021 | ||||
10 | 8 | 8 | 67 | 0.401786 | 0.091405 | 61 | 0.455357 | 0.044080 | 50 | 0.553571 | 0.008511 | ||||
10 | 9 | 1 | 27 | 0.500000 | 0.086557 | 24 | 0.555556 | 0.049882 | 16 | 0.703704 | 0.007882 | ||||
10 | 9 | 2 | 34 | 0.468750 | 0.097666 | 30 | 0.531250 | 0.049878 | 22 | 0.656250 | 0.009353 | ||||
10 | 9 | 3 | 40 | 0.452055 | 0.091779 | 36 | 0.506849 | 0.048772 | 27 | 0.630137 | 0.008369 | ||||
10 | 9 | 4 | 47 | 0.433735 | 0.097637 | 42 | 0.493976 | 0.046800 | 33 | 0.602410 | 0.009150 | ||||
10 | 9 | 5 | 53 | 0.423913 | 0.089682 | 48 | 0.478261 | 0.044758 | 39 | 0.576087 | 0.009781 | ||||
10 | 9 | 6 | 60 | 0.411765 | 0.092986 | 55 | 0.460784 | 0.048988 | 44 | 0.568627 | 0.008606 | ||||
10 | 9 | 7 | 67 | 0.396396 | 0.095498 | 61 | 0.450450 | 0.046335 | 50 | 0.549550 | 0.009061 | ||||
10 | 9 | 8 | 74 | 0.388430 | 0.097309 | 68 | 0.438017 | 0.049566 | 56 | 0.537190 | 0.009435 | ||||
10 | 9 | 9 | 81 | 0.376923 | 0.098701 | 74 | 0.430769 | 0.046792 | 62 | 0.523077 | 0.009745 | ||||
10 | 10 | 1 | 31 | 0.483333 | 0.092777 | 27 | 0.550000 | 0.047070 | 19 | 0.683333 | 0.008615 | ||||
10 | 10 | 2 | 38 | 0.457143 | 0.097689 | 33 | 0.528571 | 0.044350 | 25 | 0.642857 | 0.009229 | ||||
10 | 10 | 3 | 45 | 0.437500 | 0.099986 | 40 | 0.500000 | 0.048153 | 31 | 0.612500 | 0.009472 | ||||
10 | 10 | 4 | 51 | 0.433333 | 0.088023 | 46 | 0.488889 | 0.043698 | 37 | 0.588889 | 0.009472 | ||||
10 | 10 | 5 | 59 | 0.410000 | 0.098994 | 53 | 0.470000 | 0.045715 | 43 | 0.570000 | 0.009375 | ||||
10 | 10 | 6 | 66 | 0.400000 | 0.097217 | 60 | 0.454545 | 0.047170 | 49 | 0.554545 | 0.009231 | ||||
10 | 10 | 7 | 73 | 0.391667 | 0.095144 | 67 | 0.441667 | 0.048198 | 55 | 0.541667 | 0.009064 | ||||
10 | 10 | 8 | 80 | 0.384615 | 0.092953 | 74 | 0.430769 | 0.048911 | 61 | 0.530769 | 0.008881 | ||||
10 | 10 | 9 | 88 | 0.371429 | 0.099700 | 81 | 0.421429 | 0.049390 | 68 | 0.514286 | 0.009996 | ||||
10 | 10 | 10 | 95 | 0.366667 | 0.096934 | 88 | 0.413333 | 0.049695 | 74 | 0.506667 | 0.009709 |
Sample Sizes | p-value | p-value | p-value | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
.10 | .05 | .01 | ||||||||||||||
2 | 2 | 1 | 1 | |||||||||||||
2 | 2 | 2 | 1 | 0 | 1.000000 | 0.038095 | 0 | 1.000000 | 0.038095 | |||||||
2 | 2 | 2 | 2 | 2 | 0.833333 | 0.095238 | 1 | 0.916667 | 0.038095 | 0 | 1.000000 | 0.009524 | ||||
3 | 1 | 1 | 1 | |||||||||||||
3 | 2 | 1 | 1 | 0 | 1.000000 | 0.057143 | ||||||||||
3 | 2 | 2 | 1 | 1 | 0.909091 | 0.050000 | 0 | 1.000000 | 0.014286 | |||||||
3 | 2 | 2 | 2 | 3 | 0.800000 | 0.077778 | 2 | 0.866667 | 0.036508 | 0 | 1.000000 | 0.003175 | ||||
3 | 3 | 1 | 1 | 1 | 0.888889 | 0.075000 | 0 | 1.000000 | 0.021429 | |||||||
3 | 3 | 2 | 1 | 3 | 0.785714 | 0.097619 | 2 | 0.857143 | 0.046429 | 0 | 1.000000 | 0.004762 | ||||
3 | 3 | 2 | 2 | 5 | 0.722222 | 0.097143 | 3 | 0.833333 | 0.027619 | 1 | 0.944444 | 0.003810 | ||||
3 | 3 | 3 | 1 | 4 | 0.750000 | 0.069286 | 3 | 0.812500 | 0.034286 | 1 | 0.937500 | 0.005714 | ||||
3 | 3 | 3 | 2 | 7 | 0.681818 | 0.096883 | 5 | 0.772727 | 0.034805 | 3 | 0.863636 | 0.008442 | ||||
3 | 3 | 3 | 3 | 9 | 0.640000 | 0.084091 | 7 | 0.720000 | 0.034221 | 4 | 0.840000 | 0.005325 | ||||
4 | 1 | 1 | 1 | |||||||||||||
4 | 2 | 1 | 1 | 1 | 0.900000 | 0.085714 | 0 | 1.000000 | 0.028571 | |||||||
4 | 2 | 2 | 1 | 2 | 0.857143 | 0.055556 | 1 | 0.928571 | 0.022222 | 0 | 1.000000 | 0.006349 | ||||
4 | 2 | 2 | 2 | 4 | 0.777778 | 0.064444 | 3 | 0.833333 | 0.033016 | 1 | 0.944444 | 0.005079 | ||||
4 | 3 | 1 | 1 | 2 | 0.833333 | 0.076190 | 1 | 0.916667 | 0.033333 | 0 | 1.000000 | 0.009524 | ||||
4 | 3 | 2 | 1 | 4 | 0.764706 | 0.077619 | 3 | 0.823529 | 0.041429 | 1 | 0.941176 | 0.007143 | ||||
4 | 3 | 2 | 2 | 6 | 0.727273 | 0.068312 | 5 | 0.772727 | 0.040173 | 2 | 0.909091 | 0.004329 | ||||
4 | 3 | 3 | 1 | 6 | 0.714286 | 0.081299 | 5 | 0.761905 | 0.048571 | 2 | 0.904762 | 0.005584 | ||||
4 | 3 | 3 | 2 | 9 | 0.653846 | 0.092309 | 7 | 0.730769 | 0.038615 | 4 | 0.846154 | 0.006342 | ||||
4 | 3 | 3 | 3 | 11 | 0.645161 | 0.071618 | 10 | 0.677419 | 0.048871 | 6 | 0.806452 | 0.006678 | ||||
4 | 4 | 1 | 1 | 3 | 0.812500 | 0.060952 | 2 | 0.875000 | 0.032381 | 0 | 1.000000 | 0.003810 | ||||
4 | 4 | 2 | 1 | 6 | 0.714286 | 0.089004 | 4 | 0.809524 | 0.031169 | 2 | 0.904762 | 0.007100 | ||||
4 | 4 | 2 | 2 | 8 | 0.692308 | 0.068283 | 7 | 0.730769 | 0.043579 | 4 | 0.846154 | 0.007734 | ||||
4 | 4 | 3 | 1 | 8 | 0.680000 | 0.078672 | 6 | 0.760000 | 0.030996 | 4 | 0.840000 | 0.009264 | ||||
4 | 4 | 3 | 2 | 11 | 0.645161 | 0.078326 | 9 | 0.709677 | 0.035791 | 6 | 0.806452 | 0.007792 | ||||
4 | 4 | 3 | 3 | 14 | 0.611111 | 0.077325 | 12 | 0.666667 | 0.038965 | 8 | 0.777778 | 0.006591 | ||||
4 | 4 | 4 | 1 | 11 | 0.633333 | 0.095984 | 9 | 0.700000 | 0.045594 | 5 | 0.833333 | 0.005808 | ||||
4 | 4 | 4 | 2 | 14 | 0.611111 | 0.084038 | 12 | 0.666667 | 0.043035 | 8 | 0.777778 | 0.007570 | ||||
4 | 4 | 4 | 3 | 18 | 0.571429 | 0.097366 | 15 | 0.642857 | 0.040186 | 11 | 0.738095 | 0.008775 | ||||
4 | 4 | 4 | 4 | 21 | 0.562500 | 0.083959 | 19 | 0.604167 | 0.049523 | 14 | 0.708333 | 0.009514 | ||||
5 | 1 | 1 | 1 | 0 | 1.000000 | 0.071429 | ||||||||||
5 | 2 | 1 | 1 | 1 | 0.909091 | 0.047619 | 1 | 0.909091 | 0.047619 | |||||||
5 | 2 | 2 | 1 | 3 | 0.812500 | 0.058730 | 2 | 0.875000 | 0.028571 | 0 | 1.000000 | 0.003175 | ||||
5 | 2 | 2 | 2 | 6 | 0.714286 | 0.091631 | 4 | 0.809524 | 0.030592 | 2 | 0.904762 | 0.006638 | ||||
5 | 3 | 1 | 1 | 3 | 0.785714 | 0.076190 | 2 | 0.857143 | 0.040476 | 0 | 1.000000 | 0.004762 | ||||
5 | 3 | 2 | 1 | 5 | 0.750000 | 0.065584 | 4 | 0.800000 | 0.037662 | 2 | 0.900000 | 0.008874 | ||||
5 | 3 | 2 | 2 | 8 | 0.680000 | 0.079221 | 6 | 0.760000 | 0.031025 | 4 | 0.840000 | 0.009235 | ||||
5 | 3 | 3 | 1 | 8 | 0.652174 | 0.092388 | 6 | 0.739130 | 0.036905 | 3 | 0.869565 | 0.005519 | ||||
5 | 3 | 3 | 2 | 11 | 0.633333 | 0.089419 | 9 | 0.700000 | 0.041492 | 6 | 0.800000 | 0.009232 | ||||
5 | 3 | 3 | 3 | 14 | 0.588235 | 0.087484 | 12 | 0.647059 | 0.044634 | 8 | 0.764706 | 0.007746 | ||||
5 | 4 | 1 | 1 | 5 | 0.722222 | 0.087446 | 3 | 0.833333 | 0.030303 | 1 | 0.944444 | 0.006061 | ||||
5 | 4 | 2 | 1 | 8 | 0.666667 | 0.099279 | 6 | 0.750000 | 0.041631 | 3 | 0.875000 | 0.006854 | ||||
5 | 4 | 2 | 2 | 11 | 0.633333 | 0.096947 | 9 | 0.700000 | 0.045865 | 5 | 0.833333 | 0.005972 | ||||
5 | 4 | 3 | 1 | 10 | 0.655172 | 0.077312 | 8 | 0.724138 | 0.034466 | 5 | 0.827586 | 0.007126 | ||||
5 | 4 | 3 | 2 | 14 | 0.600000 | 0.093723 | 12 | 0.657143 | 0.048620 | 8 | 0.771429 | 0.008841 | ||||
5 | 4 | 3 | 3 | 17 | 0.585366 | 0.082249 | 15 | 0.634146 | 0.045133 | 10 | 0.756098 | 0.006435 | ||||
5 | 4 | 4 | 1 | 13 | 0.617647 | 0.082489 | 11 | 0.676471 | 0.041660 | 7 | 0.794118 | 0.006974 | ||||
5 | 4 | 4 | 2 | 17 | 0.585366 | 0.088025 | 15 | 0.634146 | 0.048942 | 10 | 0.756098 | 0.007269 | ||||
5 | 4 | 4 | 3 | 21 | 0.553191 | 0.091439 | 18 | 0.617021 | 0.040930 | 13 | 0.723404 | 0.007254 | ||||
5 | 4 | 4 | 4 | 25 | 0.537037 | 0.091748 | 22 | 0.592593 | 0.044712 | 17 | 0.685185 | 0.009977 | ||||
5 | 5 | 1 | 1 | 7 | 0.666667 | 0.098966 | 5 | 0.761905 | 0.041126 | 2 | 0.904762 | 0.006854 | ||||
5 | 5 | 2 | 1 | 10 | 0.642857 | 0.094933 | 8 | 0.714286 | 0.044483 | 4 | 0.857143 | 0.005606 | ||||
5 | 5 | 2 | 2 | 13 | 0.617647 | 0.083004 | 11 | 0.676471 | 0.041903 | 7 | 0.794118 | 0.007191 | ||||
5 | 5 | 3 | 1 | 13 | 0.593750 | 0.093169 | 11 | 0.656250 | 0.047627 | 7 | 0.781250 | 0.008432 | ||||
5 | 5 | 3 | 2 | 17 | 0.575000 | 0.097342 | 14 | 0.650000 | 0.039675 | 10 | 0.750000 | 0.008422 | ||||
5 | 5 | 3 | 3 | 20 | 0.555556 | 0.078592 | 18 | 0.600000 | 0.045644 | 13 | 0.711111 | 0.008304 | ||||
5 | 5 | 4 | 1 | 16 | 0.589744 | 0.086815 | 14 | 0.641026 | 0.047844 | 9 | 0.769231 | 0.006849 | ||||
5 | 5 | 4 | 2 | 20 | 0.565217 | 0.083163 | 18 | 0.608696 | 0.048813 | 13 | 0.717391 | 0.009174 | ||||
5 | 5 | 4 | 3 | 25 | 0.528302 | 0.099365 | 22 | 0.584906 | 0.048982 | 16 | 0.698113 | 0.007882 | ||||
5 | 5 | 4 | 4 | 29 | 0.516667 | 0.091307 | 26 | 0.566667 | 0.047580 | 20 | 0.666667 | 0.009275 | ||||
5 | 5 | 5 | 1 | 19 | 0.558140 | 0.082239 | 17 | 0.604651 | 0.047987 | 12 | 0.720930 | 0.008799 | ||||
5 | 5 | 5 | 2 | 24 | 0.538462 | 0.091195 | 21 | 0.596154 | 0.044209 | 16 | 0.692308 | 0.009718 | ||||
5 | 5 | 5 | 3 | 29 | 0.500000 | 0.098583 | 25 | 0.568966 | 0.040931 | 19 | 0.672414 | 0.007511 | ||||
5 | 5 | 5 | 4 | 33 | 0.507463 | 0.084597 | 30 | 0.552239 | 0.046171 | 23 | 0.656716 | 0.007853 | ||||
5 | 5 | 5 | 5 | 38 | 0.479452 | 0.088106 | 34 | 0.534247 | 0.041674 | 27 | 0.630137 | 0.008096 | ||||
6 | 1 | 1 | 1 | 0 | 1.000000 | 0.047619 | 0 | 1.000000 | 0.047619 | |||||||
6 | 2 | 1 | 1 | 2 | 0.857143 | 0.066667 | 1 | 0.928571 | 0.028571 | 0 | 1.000000 | 0.009524 | ||||
6 | 2 | 2 | 1 | 4 | 0.789474 | 0.060462 | 3 | 0.842105 | 0.032468 | 1 | 0.947368 | 0.006061 | ||||
6 | 2 | 2 | 2 | 7 | 0.708333 | 0.077201 | 6 | 0.750000 | 0.048485 | 3 | 0.875000 | 0.007504 | ||||
6 | 3 | 1 | 1 | 4 | 0.764706 | 0.075325 | 3 | 0.823529 | 0.042857 | 1 | 0.941176 | 0.009091 | ||||
6 | 3 | 2 | 1 | 7 | 0.695652 | 0.088095 | 5 | 0.782609 | 0.033983 | 3 | 0.869565 | 0.009848 | ||||
6 | 3 | 2 | 2 | 10 | 0.655172 | 0.087568 | 8 | 0.724138 | 0.039494 | 5 | 0.827586 | 0.008225 | ||||
6 | 3 | 3 | 1 | 10 | 0.642857 | 0.099459 | 8 | 0.714286 | 0.045538 | 4 | 0.857143 | 0.005370 | ||||
6 | 3 | 3 | 2 | 13 | 0.617647 | 0.085707 | 11 | 0.676471 | 0.043064 | 7 | 0.794118 | 0.007064 | ||||
6 | 3 | 3 | 3 | 16 | 0.600000 | 0.075737 | 14 | 0.650000 | 0.040621 | 10 | 0.750000 | 0.008509 | ||||
6 | 4 | 1 | 1 | 6 | 0.727273 | 0.073737 | 5 | 0.772727 | 0.046609 | 2 | 0.909091 | 0.007792 | ||||
6 | 4 | 2 | 1 | 9 | 0.678571 | 0.074026 | 7 | 0.750000 | 0.032534 | 4 | 0.857143 | 0.006460 | ||||
6 | 4 | 2 | 2 | 13 | 0.617647 | 0.092254 | 11 | 0.676471 | 0.047099 | 7 | 0.794118 | 0.008249 | ||||
6 | 4 | 3 | 1 | 12 | 0.636364 | 0.074599 | 10 | 0.696970 | 0.036371 | 7 | 0.787879 | 0.009576 | ||||
6 | 4 | 3 | 2 | 16 | 0.600000 | 0.080891 | 14 | 0.650000 | 0.043967 | 10 | 0.750000 | 0.009549 | ||||
6 | 4 | 3 | 3 | 20 | 0.565217 | 0.085039 | 18 | 0.608696 | 0.049821 | 13 | 0.717391 | 0.009305 | ||||
6 | 4 | 4 | 1 | 16 | 0.589744 | 0.094578 | 13 | 0.666667 | 0.037925 | 9 | 0.769231 | 0.007777 | ||||
6 | 4 | 4 | 2 | 20 | 0.565217 | 0.090522 | 17 | 0.630435 | 0.040149 | 12 | 0.739130 | 0.006927 | ||||
6 | 4 | 4 | 3 | 24 | 0.547170 | 0.085709 | 21 | 0.603774 | 0.040995 | 16 | 0.698113 | 0.008781 | ||||
6 | 4 | 4 | 4 | 29 | 0.516667 | 0.097941 | 25 | 0.583333 | 0.040644 | 19 | 0.683333 | 0.007460 | ||||
6 | 5 | 1 | 1 | 8 | 0.680000 | 0.073704 | 7 | 0.720000 | 0.049728 | 3 | 0.880000 | 0.006494 | ||||
6 | 5 | 2 | 1 | 12 | 0.625000 | 0.089767 | 10 | 0.687500 | 0.045692 | 6 | 0.812500 | 0.007992 | ||||
6 | 5 | 2 | 2 | 16 | 0.589744 | 0.095524 | 13 | 0.666667 | 0.038420 | 9 | 0.769231 | 0.008052 | ||||
6 | 5 | 3 | 1 | 15 | 0.605263 | 0.079615 | 13 | 0.657895 | 0.042893 | 9 | 0.763158 | 0.009229 | ||||
6 | 5 | 3 | 2 | 20 | 0.555556 | 0.098821 | 17 | 0.622222 | 0.044475 | 12 | 0.733333 | 0.007945 | ||||
6 | 5 | 3 | 3 | 24 | 0.538462 | 0.093325 | 21 | 0.596154 | 0.045191 | 16 | 0.692308 | 0.009911 | ||||
6 | 5 | 4 | 1 | 19 | 0.568182 | 0.088703 | 16 | 0.636364 | 0.038969 | 12 | 0.727273 | 0.009856 | ||||
6 | 5 | 4 | 2 | 24 | 0.538462 | 0.098090 | 21 | 0.596154 | 0.048090 | 15 | 0.711538 | 0.007604 | ||||
6 | 5 | 4 | 3 | 28 | 0.525424 | 0.085898 | 25 | 0.576271 | 0.044131 | 19 | 0.677966 | 0.008293 | ||||
6 | 5 | 4 | 4 | 33 | 0.507463 | 0.090101 | 30 | 0.552239 | 0.049583 | 23 | 0.656716 | 0.008625 | ||||
6 | 5 | 5 | 1 | 23 | 0.540000 | 0.096573 | 20 | 0.600000 | 0.047042 | 14 | 0.720000 | 0.007284 | ||||
6 | 5 | 5 | 2 | 28 | 0.517241 | 0.097004 | 24 | 0.586207 | 0.040080 | 18 | 0.689655 | 0.007270 | ||||
6 | 5 | 5 | 3 | 33 | 0.500000 | 0.096429 | 29 | 0.560606 | 0.043157 | 23 | 0.651515 | 0.009514 | ||||
6 | 5 | 5 | 4 | 38 | 0.486486 | 0.093108 | 34 | 0.540541 | 0.044474 | 27 | 0.635135 | 0.008812 | ||||
6 | 5 | 5 | 5 | 43 | 0.475610 | 0.090015 | 39 | 0.524390 | 0.045390 | 31 | 0.621951 | 0.008195 | ||||
6 | 6 | 1 | 1 | 11 | 0.633333 | 0.097617 | 8 | 0.733333 | 0.034775 | 5 | 0.833333 | 0.009134 | ||||
6 | 6 | 2 | 1 | 14 | 0.621622 | 0.076902 | 12 | 0.675676 | 0.041463 | 8 | 0.783784 | 0.008910 | ||||
6 | 6 | 2 | 2 | 19 | 0.568182 | 0.097046 | 16 | 0.636364 | 0.043344 | 11 | 0.750000 | 0.007713 | ||||
6 | 6 | 3 | 1 | 18 | 0.581395 | 0.081966 | 16 | 0.627907 | 0.047471 | 11 | 0.744186 | 0.008665 | ||||
6 | 6 | 3 | 2 | 23 | 0.549020 | 0.091718 | 20 | 0.607843 | 0.044161 | 15 | 0.705882 | 0.009600 | ||||
6 | 6 | 3 | 3 | 28 | 0.517241 | 0.099031 | 24 | 0.586207 | 0.040890 | 18 | 0.689655 | 0.007417 | ||||
6 | 6 | 4 | 1 | 22 | 0.560000 | 0.082630 | 19 | 0.620000 | 0.038997 | 14 | 0.720000 | 0.008129 | ||||
6 | 6 | 4 | 2 | 27 | 0.534483 | 0.084812 | 24 | 0.586207 | 0.043401 | 18 | 0.689655 | 0.008070 | ||||
6 | 6 | 4 | 3 | 32 | 0.515152 | 0.085020 | 29 | 0.560606 | 0.046268 | 22 | 0.666667 | 0.007791 | ||||
6 | 6 | 4 | 4 | 38 | 0.486486 | 0.098717 | 34 | 0.540541 | 0.047595 | 27 | 0.635135 | 0.009613 | ||||
6 | 6 | 5 | 1 | 26 | 0.535714 | 0.082947 | 23 | 0.589286 | 0.042216 | 17 | 0.696429 | 0.007727 | ||||
6 | 6 | 5 | 2 | 32 | 0.507692 | 0.095002 | 28 | 0.569231 | 0.042368 | 22 | 0.661538 | 0.009272 | ||||
6 | 6 | 5 | 3 | 37 | 0.493151 | 0.088417 | 33 | 0.547945 | 0.041732 | 26 | 0.643836 | 0.008054 | ||||
6 | 6 | 5 | 4 | 43 | 0.475610 | 0.094851 | 39 | 0.524390 | 0.048227 | 31 | 0.621951 | 0.008874 | ||||
6 | 6 | 5 | 5 | 48 | 0.466667 | 0.086446 | 44 | 0.511111 | 0.045656 | 36 | 0.600000 | 0.009533 | ||||
6 | 6 | 6 | 1 | 31 | 0.507936 | 0.098636 | 27 | 0.571429 | 0.044191 | 21 | 0.666667 | 0.009750 | ||||
6 | 6 | 6 | 2 | 36 | 0.500000 | 0.087213 | 32 | 0.555556 | 0.041053 | 25 | 0.652778 | 0.007871 | ||||
6 | 6 | 6 | 3 | 42 | 0.481481 | 0.090321 | 38 | 0.530864 | 0.045479 | 30 | 0.629630 | 0.008168 | ||||
6 | 6 | 6 | 4 | 48 | 0.466667 | 0.090941 | 44 | 0.511111 | 0.048376 | 35 | 0.611111 | 0.008207 | ||||
6 | 6 | 6 | 5 | 54 | 0.454545 | 0.090669 | 49 | 0.505051 | 0.043100 | 40 | 0.595960 | 0.008145 | ||||
6 | 6 | 6 | 6 | 60 | 0.444444 | 0.089781 | 55 | 0.490741 | 0.044896 | 46 | 0.574074 | 0.009741 |
ConcordanceTest, Kendall, pspearman, PerMallows, rankdist, BayesMallows
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For attribution, please cite this work as
Alcaraz, et al., "The Concordance Test, an Alternative to Kruskal-Wallis Based on the Kendall-tau Distance: An R Package", The R Journal, 2022
BibTeX citation
@article{RJ-2022-039, author = {Alcaraz, Javier and Anton-Sanchez, Laura and Monge, Juan Francisco}, title = {The Concordance Test, an Alternative to Kruskal-Wallis Based on the Kendall-tau Distance: An R Package}, journal = {The R Journal}, year = {2022}, note = {https://rjournal.github.io/}, volume = {14}, issue = {2}, issn = {2073-4859}, pages = {26-53} }