Kendall's Tau Test

KENDALLSTAUTEST(Range1, Range2, alpha, NewTableFlag)

$Range1$ is the array of x values.
$Range2$ is the array of y values.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle alpha} is the value from 0 to 1.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle NewTableFlag} is either TRUE or FALSE. TRUE for getting results in a new cube. FALSE will display results in the same cube.

DESCRIPTION

It is a statistic test used to measure the ordinal association between two measured quantities.
It is a measure of rank correlation: the similarity of the orderings of the data when ranked by each of the quantities.
Kendall correlation between two variables will be high when observations have a similar rank.
It will be low when observations have a dissimilar rank between the two variables.

Let (x1, y1), (x2, y2), …, (xn, yn) be a set of observations of the joint random variables X and Y respectively, such that all the values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_i)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (y_i)} are unique.

concordant if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_i > x_j)}
 & Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (y_i > y_j)}
 or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_i < x_j)}
 & Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (y_i < y_j)}

discordant if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_i > x_j)}
 & Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (y_i < y_j)}
 or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_i < x_j)}
 & Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (y_i > y_j)}

neither if  $(x_{i}=x_{j})$  or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (y_i = y_j)}
 (i.e. ties are not counted).

The Kendall's Tau statistic is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau=1-\frac{4D}{n(n-1)}} .

C is the number of concordant pairs.
D is the number of discordant pairs.

RESULT

If number of values in a set is <15, critical tables are used to calculate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau} .
If number of values in a set is >=15, Normal approximation is used for calculation.

* If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau}
 > critical value from the Kendall's Tau Critical table, then reject the null hypothesis that there is no correlation.
* else if, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau}
 <critical value, correlation exists.

EXAMPLE

Spreadsheet
	A	B
1	80	5
2	78	23
3	60	25
4	53	48
5	85	17
6	84	8
7	73	4
8	79	26
9	81	11
10	75	19
11	68	14
12	72	35
13	58	29
14	92	3
15	65	24

=KENDALLSTAUTEST(A1:A15,B1:B15, 0.05, true)

KENDALL'S TAU TEST USING NORMAL APPROXIMATION
RANGE1 SORT	RANGE2 SORT	CONCORDANT	DISCORDANT
53	48	0	14
58	29	1	12
60	25	2	10
65	24	2	9
68	14	5	5
72	35	0	9
73	4	7	1
75	19	2	5
78	23	1	5
79	26	0	5
80	5	3	1
81	11	1	2
84	8	1	1
85	17	0	1
92	3	0	0

SUMMARY
VARIABLE	RESULT
COUNT	15
C	105
SUM CONCORDANT	25
SUM DISCORDANT	80
KENDALL'S TAU	-0.52381
STDERROR	0.19245
Z-VALUE	-2.7218
ZCRITICAL	1.95996
P-VALUE	0.00649
RESULT	REJECT NULL HYPOTHESIS, NO CORRELATION

CONCLUSION: REJECT NULL HYPOTHESIS, NO CORRELATION

Kendall's Tau Test

DESCRIPTION

RESULT

EXAMPLE

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	A	B
1	80	5
2	78	23
3	60	25
4	53	48
5	85	17
6	84	8
7	73	4
8	79	26
9	81	11
10	75	19
11	68	14
12	72	35
13	58	29
14	92	3
15	65	24

RANGE1 SORT	RANGE2 SORT	CONCORDANT	DISCORDANT
53	48	0	14
58	29	1	12
60	25	2	10
65	24	2	9
68	14	5	5
72	35	0	9
73	4	7	1
75	19	2	5
78	23	1	5
79	26	0	5
80	5	3	1
81	11	1	2
84	8	1	1
85	17	0	1
92	3	0	0

	A	B
1	80	5
2	78	23
3	60	25
4	53	48
5	85	17
6	84	8
7	73	4
8	79	26
9	81	11
10	75	19
11	68	14
12	72	35
13	58	29
14	92	3
15	65	24

RANGE1 SORT	RANGE2 SORT	CONCORDANT	DISCORDANT
53	48	0	14
58	29	1	12
60	25	2	10
65	24	2	9
68	14	5	5
72	35	0	9
73	4	7	1
75	19	2	5
78	23	1	5
79	26	0	5
80	5	3	1
81	11	1	2
84	8	1	1
85	17	0	1
92	3	0	0

	A	B
1	80	5
2	78	23
3	60	25
4	53	48
5	85	17
6	84	8
7	73	4
8	79	26
9	81	11
10	75	19
11	68	14
12	72	35
13	58	29
14	92	3
15	65	24

RANGE1 SORT	RANGE2 SORT	CONCORDANT	DISCORDANT
53	48	0	14
58	29	1	12
60	25	2	10
65	24	2	9
68	14	5	5
72	35	0	9
73	4	7	1
75	19	2	5
78	23	1	5
79	26	0	5
80	5	3	1
81	11	1	2
84	8	1	1
85	17	0	1
92	3	0	0