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Chapter 14-4. Wilcoxon Rank Test

Recommended Article : 【Statistics】Lecture 14. Statistical Testing

1. Overview

2. Wilcoxon Rank Sum Test

3. Wilcoxon Signed Rank Test

1. Overview

⑴ Single Sample

① Parametric Test : Single Sample T-Test

② Non-Parametric Test : Sign Test, Wilcoxon Signed Rank Test

⑵ Two Samples (Paired Samples) : Essentially the same as single sample

① Parametric Test : Paired Sample T-Test

② Non-Parametric Test : Sign Test, Wilcoxon Signed Rank Test

⑶ Two Samples (Independent Samples)

① Parametric Test : Independent Sample T-Test

② Non-Parametric Test : Wilcoxon Rank Sum Test

⑷ Analysis of Variance

① Parametric Test : ANOVA

② Non-Parametric Test : Kruskal-Wallis Test

⑸ Randomness

① Non-Parametric Test : Run Test

⑹ Correlation Analysis

① Pearson Correlation Coefficient

② Spearman Rank Correlation Coefficient

2. Wilcoxon Rank Sum Test

⑴ Overview

① Also known as Mann-Whitney U statistics or Mann-Whitney-Wilcoxon Rank Sum Test

② A representative non-parametric test method for testing the median of two samples

③ A test method using the sum of ranks in a combined sample from two samples

④ Assumption 1. Data distribution is continuous and independent

⑤ Assumption 2. Symmetry assumption about the data distribution

⑵ 1^st. Null Hypothesis and Alternative Hypothesis

① Null Hypothesis H₀ : Δ = 0. That is, the medians of the two populations are the same

② Alternative Hypothesis H₁ : Δ ＞ 0 or Δ ＜ 0 or Δ ≠ 0

⑶ 2^nd. Setting the Significance Level

⑷ 3^rd. Calculation of Test Statistic

① 3-1. Combine the data from both samples and assign ranks

② 3-2. Sum the ranks of the group with the smaller sample size

○ If the sample sizes are equal, either can be chosen

○ In the case of data with the same values, assign a representative rank to those data collectively

③ 3-3. Calculate the following

○ n₁ : The sample size of the smaller group. Should be 10 or more

○ n₂ : The sample size of the larger group. Should be 10 or more

○ R : The sum of the ranks of the smaller group

⑸ 4^th. Statistical Testing

3. Wilcoxon Signed Rank Test

⑴ Overview

① Used for testing the median of a single sample or the difference in medians of two paired samples

② A test method that considers not only the sign of the difference but also its relative magnitude

③ Assumption 1. Data distribution is continuous and independent

④ Assumption 2. Symmetry assumption about the data distribution

⑵ 1^st. Assumptions

① N : Sample size

② Since there are N data in each of the two groups, there are a total of 2N data

③ Paired Comparison Assumption : x_1,i and x_2,i form a pair

④ H₀ : μ₁ - μ₂ = 0

⑤ H1 : μ₁ - μ₂ ≠ 0, μ₁ - μ₂ ＞ 0, or μ₁ - μ₂ ＜ 0

⑸ 4^th. Rearrange the absolute differences in increasing order and assign ranks

① In the case of data with the same values, assign a representative rank to those data collectively

⑹ 5^th. Calculation of Test Statistic

① Method 1.

② Method 2.

○ w⁺ : The sum of ranks of the data with positive differences

○ w^- : The sum of ranks of the data with negative differences

○ w_s = min(w⁺, w^-)

○ Calculation of the statistic

③ N* should be 10 or more

⑺ 6^th. Statistical Testing

Entered : 2021.05.10 09:13