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# Wilcoxon Rank-Sum test : A Non Parametric Alternative to two Sample T-Test

Statistics, a scientific approach to analyzing numerical data, is employed to discover relationships among the phenomena to describe, predict and control their occurrence.

Statistics helps the researcher to acquire precise, steadfast and dependable findings. Although there are several statistical tests such as ANOVA, independent t-test, etc. to arrive at the right result, one must choose the test according to the type of study.

For instance, if one wants to investigate if the means of two or more groups are different from each other, then he/she must use the ANOVA test. On the other hand, if a researcher wants to test the relationship between categorical variables, the Chi-square test is to be used.

Similarly, for the comparison of means of two independent groups, the two-sample t-test is used. However, if the t-test doesn’t satisfy the requirements for two independent samples, then Wilcoxon Rank-Sum is used as it can offer the two independent samples drawn from populations with an ordinal distribution. This test does not assume known distributions, does not deal with parameters, and hence it is considered as a non-parametric test.

Wilcoxon Rank-Sum test also known as Mann-Whitney U test makes two important assumptions. That is the assumption of independence and equal variance. These assumptions are sufficient for determining if the two populations are different. Additionally, if we assume that the two populations are identical (except for a difference in location), then Wilcoxon Rank-Sum can be utilized as a test of equal means or medians.

Power calculation for Wilcoxon Rank-Sum test

Power is nothing but the probability of rejecting the null hypothesis when it is false. The power calculation for the Wilcoxon Rank-Sum or Mann-Whitney U test is similar to that of the two sample equal-variance t-test except a few modifications are made to the sample size based on the assumed data distribution.

The sample size ni| is equal to ni|= ni/𝑊,

where 𝑊 is known as the Wilcoxon adjustment factor, which is based on the assumed data distribution.

In general, the valid range for the probability of accepting a false null hypothesis is 0 to 1. However, different domains have different standards for setting power.

Sample size conditions

While solving for sample size, the researcher must choose a condition that describes the constraints either on N1 or N2 or both.

1. Equal (N1 = N2) – This condition is utilized when a researcher has equal sample sizes in each group. Since both sample sizes are solved at once, no additional sample size parameters are required here.
2. Include N1, solve for N2 –  This condition is chosen to fix N1 at some value, and then solve only for N2. However, for some values of N1, N2 value that is large enough to acquire the desired power may be absent.
3. Enter N2, solve for N1–  In case a researcher wants to fix N2 at some value, and then solve only for N1, this condition is used. In this case, too, N1 that is large enough to get the desired power might be absent for some values of N2.
4. Enter R = N2/N1, solve for N1 & N2<span”> – To choose this condition, one must set a suitable value for the ratio of N2 to N1. This is followed by the determination of required N1 & N2 to obtain the desired power using PASS approach. An equivalent representation of R is
N2 = R * N1.
5. Include percentage in group 1, solve for N1 & N2 – Here, the researcher must set a definite value for the percentage of the total sample size in group1. Next, PASS determines the required N1 and N2 with the value of percentage entered to acquire the desired power.
6. N1 (sample size, group 1) – This condition is used if group allocation = “Enter N1, solve for N2.” Where N1 is the number of individuals sampled from the group 1 population and must be equal or greater than 2. Here a single or a series of values can be entered.
7. N2 (sample size, group 2) – If group allocation = “Enter N2, solve for N1,” this condition is utilized. Here N2 is the number of individuals sampled from the group 2 population and must be greater or equal to 2. A single or a series of values can be entered in this condition.

The Wilcoxon Rank-Sum test is less sensitive to outliers when compared to that of the two-sample t-test and valid for data from any distribution.

However, it reacts to other differences between the distributions such as differences in shape, especially if the focus is on the differences in location between the two distributions. This is considered as the major disadvantage of the Wilcoxon test. Also, when the assumptions of the two-sample t-test hold, this test is less likely to detect a location shift in comparison with the t-test.