The Wilcoxon Signed-Rank test is a nonparametric test, it checks continuous or ordinal data for a significant difference between two dependent groups. The test finds the deltas of the two groups. Then, it sorts the pairs by the absolute value of the deltas.

The test usually comes as a failsafe option to a paired t-test when the test doesn't meet the normality assumption or contains many outliers.

A paired t-test compares the **means** of the two groups, while a Wilcoxon Signed-Rank test compares the entire **distributions**. If the two groups have a similar distribution curve, the test will also check if the **median** of the deltas is the expected value (usually zero).

When the two groups have a similar symmetrical distribution curve, the test will also check if the **mean** of the deltas is the expected value.

A paired t-test is slightly stronger than a Wilcoxon Signed-Rank Test. A Wilcoxon Signed-Rank Test has 95% efficiency in comparison to a paired t-test. If the population is similar to a normal distribution and reasonably symmetric, it is better to use the paired t-test.

**Not Normal**, the data is not normally distributes**Ordinal data**, but not interval scaled. You know the order but not the differences between the values. for example: unhappy, neutral, happy**Outliers**the test is more robust to outliers than t-test, but the process is similar - if you are sure that a value is an outlier, you should exclude it, otherwise, keep it.

**Dependent**pairs, but**independence**between the pairs**Ordinal / Continuous**the compared data consist of ordinal data or continuous data**Shape**the data is not necessarily normally distributed but should have a similar shape. If not you can compare the ranks but not the median of the differences

- Calculate the
**difference**between the pairs, for example after treatment value minus before treatment value - Exclude pairs with zero difference and sort by the absolute rank
- Sort the data from low absolute value to the high absolute value
- Rank the list, the lower absolute value get rank of 1, the second rank of 2, etc

When having ties group, identical value for several observations the rank will be the average of the ranks for the entire group

Give each rank a sign positive if the difference is positive and negative if the difference is negative. __Accumulate the ranks__

R_{i}- the absolute rank of the i rank difference.

n - number of pairs, where the difference is not zero. $$W_+=\sum_{i=1}^{n}{R_{i}}\enspace where \enspace sign\enspace is\enspace positive$$ $$W_-=\sum_{i=1}^{n}{R_{i}}\enspace where \enspace sign\enspace is\enspace negative$$ $$W_++W_-= \frac{n(1+n)}{2} \enspace sum \enspace of \enspace arithmetic \enspace progression$$ Since the distribution is symmetrical, usually w is the minimum between w_{+}and W_{-}. $$W=min(W_+ , W_-)$$ It is good for two tails test, but for one tail the test will always assume that H_{1}is the sample with bigger values is bigger than the sample with the smaller values. In this website we chose: statistics=W_{-}, in this way we can calculate the left-tailed test or the right-tailed test like any other test.- Check the critical W in the table, reject H
_{0}if W < W_{critical}.

When n is small, the tool will use the exact value from tables, the exact critical value is accurate (for common significant levels ), while the p-value is usually interpolated from two values in the table. The tool uses log interpolation which is more accurate with small p-values and less accurate with bigger p-values. (the common significant levels are small: 0.05, 0.01).

There is no consensus about what is a small n. When Method="automatic" the tool uses the Exact method when n ≤ 40, and normal approximation when n > 40.

When Method="z approximation" the tool uses only the normal approximation.

Calculated the critical W from a statistic table.

To get more accurate results, the tool uses continuity correction and ties correction. ties is a group of observations with the same value. $$ z = \frac { W_- - \mu_w + C_{continuity}} {\sigma_w}\\ \mu= \frac {n(n+1)} {4}\\ \sigma^2= \frac {n(n+1)(2n+1)} {24} - C_{ties}$$

When using continuous distribution to calculate discrete data it is better to use continuity correction.

P(X < a) => P(X < a - 0.5)

P(X > a) => P(X > a + 0.5)

As a result:**Right tail**, or two tails with positive Z, (W**-** > μ) , C_{continuity} = **-**0.5 .**Left tail**, or two tails with negative Z, (W**-** < μ) , C_{continuity} = 0.5 .

When we don't correct the data, C_{continuity} = 0.

$$C_{ties} = \sum_{i=1}^{t}{\frac{f_t^3-f_t}{48}}$$ t - group number of ties

f_t - number of values in group t

example: [1,2,2,2,3,4,5,5,5,5,6,7,8,8,8,8,8,8,9,10]

$$group_1: [2,2,2], \quad f_1=3\\ group_2: [5,5,5,5], \quad f_2=4\\ group_3: [8,8,8,8,8,8], \quad f_3=6$$

$$ f=\frac{2W}{n(1+n)}$$

In the following example, we check the number of questions answered correctly by the same subject, before and after performing a training.

The significant level (α) is 0.05;

- Calculate
**Difference**,**Absolute**and**Sign**:

Difference = After - Before, Absolute = Absolute(Difference), Sign = sign(Difference).Subject Before After Difference Absolute Sign A 1 5 4 4 + B 6 1 -5 5 - C 3 6 3 3 + D 4 4 0 0 E 10 13 3 3 + F 6 3 -3 3 - G 2 8 6 6 + H 3 16 13 13 + I 5 12 7 7 + J 2 10 8 8 + K 13 15 2 2 + L 6 7 1 1 + M 5 14 9 9 + - The sample size contains 13 pairs. Exclude D pair with zero difference

n = 13 - 1 = 12 - Sort by Absolute value
Subject Absolute Sign Simple Rank Rank W _{-}W _{+}L 1 + 1 1 1 K 2 + 2 2 2 C 3 + 3 4 4 E 3 + 4 4 4 F 3 **-**5 4 4 A 4 + 6 6 6 B 5 **-**7 7 7 G 6 + 8 8 8 I 7 + 9 9 9 J 8 + 10 10 10 M 9 + 11 11 11 H 13 + 12 12 12 **Total****11****67** - Simple Rank - rank by the Absolute value, the lower Absolute value get 1 rank, the second 2 etc.
- Rank - usually will be the same as Simple Rank. subjects C,E,F have identical Absolute value: 3. The rank is the average of the simple rank values

For subjects C,E,F

$$\frac{3+4+5}{3}=4$$ W- = 4 + 7 = 11

W+ = 1+ 2 + 4 + 4 + 6 + 8 + 9 + 10 + 11 + 12 = 67 W = min(11 , 67) = 11

**Critical Value**

Check the the two tails statistic table, for α = 0.05, n = 12.

Critical W is 13.**P-value**- For α=0.02, critical W is 9.

For α=0.05, critical W is 13.

Since 11 is between 9 and 13, the p-value will be between 0.02 and 0.05.

The tool will do a logarithmic extrapolation: p-value = 0.032 **Decision**

Since p-value < α (0.032 < 0.05 ) or alternatively since W < W_{critical}(11 < 17) we reject the H_{0}**Website**

The website uses**W-**instead of W.

Left critical W- = 13.

Right critical W- = n (1 + n) / 2 - Left critical W = 12 * 13 / 2 - 13 = 65.

Since W- (11) is in the following range: [13,65], accept H_{0}. When W- = 13 or 65 you still accept the H_{0}.

**Critical Value**

Check the the**two**tails statistic table, for α =**2*** 0.05 = 0.1, n = 12.

The critical W is 17.**P-value**

P-value = p-value(Two tailed) / 2 = 0.032 / 2 = 0.016**Decision**

Since p-value < α (0.016 < 0.05) or alternatively since W- < W_{critical}(11 < 17 ) reject H_{0}.

**Critical Value**

Check the the**two**tails statistic table, for α = 2 * 0.05 = 0.1, n = 12.

The in the table is is 17.

The critical W is n(1 + n) /2 - value from the table = 12 * 13 / 2 - 17 = 61.**P-value**

P-value =1 - p-value(Two tailed) / 2 = 1 - 0.016 = 0.984**Decision**

Since p-value < α (0.984 < 0.05) or alternatively since W- < W_{critical}(11 < 61), accept H_{0}

- There is only one tie group (t=1), that contains 3 values: f
_{1}=3

$$C_{ties} = \sum_{i=1}^{1}{\frac{3^3-3}{48}}=0.5$$ - Since W<μ , C
_{continuity}=0.5 $$ \mu_w= \frac {n(n+1)}{4}=\frac {12(12+1)}{4}=39 $$ $$ \sigma^2_w= \frac {n(n+1)(2n+1)} {24} - C_{ties}= \frac {12(12+1)(2*12+1)} {24} - 0.5=162.5 \quad => \sigma_w=12.7279 $$ $$ Z = \frac { W_- - \mu_w + C_{continuity}} {\sigma_w} = \frac { 11 - 39 + 0.5} {12.7279}=-2.16$$ - P(z≤Z) = P(z≤-2.16) = 0.01539.

- p-value = 2 * 0.01539 = 0.03078.
- Since 0.03078 < 0.05, reject H
_{0}.

- p-value = P(z≤-2.16) = 0.01539.
- Since 0.01539 < 0.05, reject H
_{0}.

- p-value =1 - P( z≤ -2.16) = 1 - 0.01539 = 0.98461.
- Since 0.98461 > 0.05, accept H
_{0}.