Manuals/calci/WILCOXONSIGNEDRANKTEST

WILCOXONSIGNEDRANKTEST(xRange,yRange,Confidencelevel,Logicalvalue)

• $xRange$ is the array of x values.
• $yRange$ is the array of y values.
• $Confidencelevel$ is the value between 0 and 1.
• $Logicalvalue$ is either TRUE or FALSE.

Description

• This function gives the summary values of Wilcoxon signed rank test.
• This test is the one of the Non Parametric test.
• Nonparametric test is also called Distribution Free Test.
• So Wilcoxon Rank test is not depending on the parameters.
• This test is designed to test a hypothesis about the median of a population distribution.
• It often involves the use of matched pairs, for example, before and after data, in which case it tests for a median difference of zero.
• Also this test does not require the assumption that the population is normally distributed.
• This test is the alternative of the Student's T-test.
• Normally this test is the version of the dependent samples t-test that can be performed on the ranked data.
• i.e., When the requirements for the t-test for two paired samples are not satisfied, the Wilcoxon Signed-Rank Test for Paired Samples non-parametric test can often be used.
• It is a more powerful alternative to the sign test, but does assume that the population probability distribution is symmetric.
• For this test let us consider the n subjects from the population with two observations ${x_{1},x_{2},....x_{n}}$ and ${y_{1},y_{2},....y_{n}}$.
• The requirements for the Wilcoxon Signed-Rank Tests for Paired Samples where Failed to parse (syntax error): {\displaystyle z_i = y_i–x_i} for all i = 1, … ,n, are as follows:

     1.the $z_{i}$ are independent. 
     2.$x_{i}$ and $y_{i}$ are interval data. 

• This test containing the following steps:
• 1.Define Null and Alternative hypothesis:
  • Null hypothesis $H_{0}$ = There is no difference between the two observations.
  • Alternative hypothesis $H_{1}$ = There is a difference between the two observations.
• 2.State alpha(Confidence level): alpha value is from 0 to 1.
• 3.State Decision Rule: Fix the hypothesis value according to the z table.
• 4.Calculate Test Statistic:$z={\frac {\frac {T-n(n+1)}{4}}{\sqrt {\frac {(n(n+1)(2n+1)}{24}}}}$

where T is the smallest rank value and n is the number of observations.

• 5.State Result: In this step we have to take a decision of null hypothesis either accept or reject depending on the z value.
• 6.State Conclusion: How far the value of the test before and after the grouping.
• The Wilcoxon signed Rank test result is contains the following values in the table:Difference of the ach observation, Absolute value of the difference,Rank value,Signed Rank and the test value.

Examples

WILCOXON SIGNED RANK TEST FOR PAIRED SAMPLES

X1	X2
188	186
177	171
176	177
169	168
196	191
172	172
165	177
190	191
165	170
180	171
181	188
172	187

1. =WILCOXONSIGNEDRANKTEST(A1:A12,B1:B12,0.05,2,TRUE) [For Two Tailed Test]

or

1. =WILCOXONSIGNEDRANKTEST(A1:A12,B1:B12,0.05,1,TRUE) [For One Tailed Test]

Wilcoxon Test

SUMMARY OUTPUT

X1	X2	Abs(Diff)	Sign	Rank	SignedRank
188	186	2	-1	4	-4
177	171	6	-1	7	-7
176	177	1	1	2	2
169	168	1	-1	2	-2
196	191	5	-1	5.5	-5.5
172	172	0	0	0	0
165	177	12	1	10	10
190	191	1	1	2	2
165	170	5	1	5.5	5.5
180	171	9	-1	9	-9
181	188	7	1	8	8
172	187	15	1	11	11

RANKS

	N	Mean Rank	SUM of Ranks
Negative Ranks	5	-5.5	-27.5
Positive Ranks	6	6.416666666666667	38.5

ANALYSIS

Nr	11
W	11
Standard Deviation	22.4944
Mean	33
Z-value	0.489
p-value(Two Tail)	0.6248

WILCOXON SIGNED RANK TEST FOR MEDIAN (SINGLE SAMPLE)

X
-2
-3
8
4
6
-4
-11
-2
-12
-6

1. MEDIAN = 0

1. =WILCOXONSIGNEDRANKTEST(A1:A9,0,0.05,2,TRUE) [For Two Tailed Test]

or

1. =WILCOXONSIGNEDRANKTEST(A1:A9,0,0.05,1,TRUE) [For One Tailed Test]

Wilcoxon Test SUMMARY OUTPUT

X1	X2	Abs(Diff)	Sign	Rank	SignedRank
-2	_	2	1	1.5	1.5
-3	_	3	1	3	3
8	_	8	-1	8	-8
4	_	4	-1	4.5	-4.5
6	_	6	-1	6.5	-6.5
-4	_	4	1	4.5	4.5
-11	_	11	1	9	9
-2	_	2	1	1.5	1.5
-12	_	12	1	10	10
-6	_	6	1	6.5	6.5

RANKS

	N	Mean Rank	SUM of Ranks
Negative Ranks	3	-6.333333333333333	-19
Positive Ranks	7	5.142857142857143	36

ANALYSIS

Nr	10
W	17
Standard Deviation	19.6214
Mean	27.5
Z-value	0.8664
p-value(Two Tail)	0.3863

WILCOXON SIGNED RANK TEST FOR SMALL SAMPLES

The lower and upper critical values,W, of the Wilcoxon Signed Rank test can be found from the table for samples size less than 10.

For a two-tail test, you reject the null hypothesis, if the computed W test statistic equals or is greater than the upper critical value or is equal to or less than the lower critical value.

For a one-tail test in the lower tail, you reject the null hypothesis if the computed W test statistic is less than or equal to the lower critical value.

For a onetail test in the upper tail, the decision rule is to reject the null hypothesis if the computed W test statistic equals or is greater than the upper critical value

X1	X2
85	75
70	50
40	50
65	40
80	20
75	65
55	40
20	25

1. =WILCOXONSIGNEDRANKTEST(A1:A9,B1:B9,0.05,2,TRUE) [For Two Tailed Test]

or

1. =WILCOXONSIGNEDRANKTEST(A1:A9,B1:B9,0.05,1,TRUE) [For One Tailed Test]

Wilcoxon Test SUMMARY OUTPUT

X1	X2	Abs(Diff)	Sign	Rank	SignedRank
85	75	10	-1	3	-3
70	50	20	-1	6	-6
40	50	10	1	3	3
65	40	25	-1	7	-7
80	20	60	-1	8	-8
75	65	10	-1	3	-3
55	40	15	-1	5	-5
20	25	5	1	1	1

RANKS

	N	Mean Rank	SUM of Ranks
Negative Ranks	6	-5.333333333333333	-32
Positive Ranks	2	2	4

ANALYSIS

Nr	8
W-Calculated	-28
W-LowerTail	5
W-UpperTail	31
Standard Deviation	14.2829
Mean	18
Z-value	-1.9604
p-value(Two Tail)	0.05

See Also

• KRUSKALWALLISTEST
• FRIEDMANTEST

References

• Wilcoxon Signed Rank test documentation on wikipedia
• Wilcoxon Signed Rank test for small samples
• Wilcoxon Signed Rank test for large samples in excel
• Wilcoxon Signed Rank test W critical value table