3.5.6 1 Sample Wilcoxon

1 Sample Wilcoxon Concept and Purpose The 1 Sample Wilcoxon test (also called the Wilcoxon signed-rank test) is a nonparametric alternative to the 1-sample t-test. It is used when you want to test whether the median of a single population differs from a hypothesized value, and when the assumptions of the t-test (especially normality) are questionable. It is applied to: - A single set of numeric observations - Data that are at least ordinal (can be ranked) - Situations where you compare the population median to a benchmark or target The test uses the ranks of signed differences from the hypothesized value rather than using the raw data values. --- When to Use the 1 Sample Wilcoxon Appropriate Situations Use the 1 Sample Wilcoxon when: - You have one sample: A single group of observations from one process or condition. - You have a benchmark: A target or reference value to compare the median against. - Data are not normal: Skewed distributions or clear violation of normality. - You want to test the median, not the mean. - You have small samples: Especially when normality is hard to assess and t-test assumptions are doubtful. Typical examples: - Testing whether the median cycle time is lower than a customer requirement. - Checking whether the median defect count per batch differs from a specified value. - Evaluating whether the median satisfaction score meets a target. When Not to Use It It is not appropriate when: - Data are not at least ordinal (e.g., purely nominal categories). - There are many ties or zeros that prevent meaningful ranking. - You are interested specifically in the mean and the normality assumption is reasonably satisfied (then 1-sample t-test may be preferable). --- Assumptions and Data Requirements Key Assumptions The 1 Sample Wilcoxon test requires: - Random and independent observations: Each measurement is independent of the others. - Symmetry of differences around the median: The distribution of differences from the hypothesized value is symmetric. The test is robust but works best when this is roughly true. - Ordinal or better: Data must be at least rankable; interval or ratio data are fine. Data Requirements - Single sample of size n - Hypothesized median (M₀): A specific numeric benchmark - Reasonable sample size: The test can be used with very small samples, but power increases with n. When sample size is large (usually n ≥ 10–20, depending on the tool), a normal approximation is commonly used for the test statistic. --- Hypotheses and Test Direction Null and Alternative Hypotheses You are testing a statement about the population median (M). - Null hypothesis (H₀): The population median equals the hypothesized value: - H₀: M = M₀ - Alternative hypothesis (H₁) (choose form based on the question): - Two-sided: H₁: M ≠ M₀ - Lower one-sided: H₁: M < M₀ - Upper one-sided: H₁: M > M₀ Examples: - Two-sided: “Is the median cycle time different from 10 minutes?” - Lower: “Has the median defect rate been reduced below 2 per batch?” - Upper: “Is the median lead time greater than 5 days?” Significance Level and Decision Rule - Significance level (α): Commonly 0.05. - Decision rule: - If p-value ≤ α: Reject H₀ (evidence that the median differs as specified by H₁). - If p-value > α: Do not reject H₀ (insufficient evidence to conclude a difference). --- Mechanics of the 1 Sample Wilcoxon Test Step 1: Compute Differences from the Hypothesized Median Given observations ( X1, X2, ..., Xn ) and hypothesized median ( M0 ): - Compute differences: ( Di = Xi - M_0 ) - Ignore any zero differences (where ( D_i = 0 )); they do not enter the ranking. - Let ( n' ) be the number of nonzero differences. Step 2: Rank the Absolute Differences - Take absolute values: ( |D_i| ) - Rank the absolute differences from smallest to largest: - Smallest abs difference gets rank 1, next gets rank 2, etc. - Handle ties: - If two or more values of ( |D_i| ) are equal, assign them the average of the ranks they would occupy. Step 3: Apply Signs to the Ranks - For each nonzero difference: - If ( D_i > 0 ), assign the rank a positive sign. - If ( D_i < 0 ), assign the rank a negative sign. You now have signed ranks. Step 4: Compute the Test Statistic Two common equivalent forms are used: - Sum of positive ranks (T⁺): - ( T^+ = ) sum of all ranks where ( D_i > 0 ) - Sum of negative ranks (T⁻): - ( T^- = ) sum of all ranks where ( D_i < 0 ) Software may use one or both, but they carry the same information. Often the smaller of T⁺ and T⁻ is compared to distribution tables for exact p-values in small samples. For large samples, a normal approximation can be used by converting T⁺ (or T⁻) into a z-statistic. --- Interpreting Output and Results Typical Output Elements Statistical software for the 1 Sample Wilcoxon typically provides: - Sample size (n) and number of nonzero differences - Hypothesized median (M₀) - Test statistic: - Sum of positive ranks (T⁺) and/or negative ranks (T⁻) - Possibly a z-value (for normal approximation) - p-value: - Two-sided p-value for H₁: M ≠ M₀ - One-sided p-value for H₁: M < M₀ or M > M₀, as selected - Estimated median (often sample median) - Confidence interval for the median (typically a nonparametric CI) Practical Interpretation Connect the statistical result to the real question: - If p-value ≤ α: - There is statistically significant evidence that the population median differs from M₀ in the direction stated by H₁. - Example: If H₁: M < M₀, a small p-value supports that the median is below the target. - If p-value > α: - There is not enough evidence to conclude that the median differs from M₀. - You do not prove equality; you simply fail to reject H₀. - Use the confidence interval: - If the CI for the median does not contain M₀, this is consistent with rejecting H₀. - If the CI includes M₀, this is consistent with not rejecting H₀. Direction of Effect Interpret the pattern of signed ranks: - Many large positive signed ranks: - Data values tend to be greater than M₀. - Many large negative signed ranks: - Data values tend to be less than M₀. This aligns with one-sided hypothesis decisions. --- Comparison with the 1-Sample t-Test Similarities Both tests: - Compare a single sample to a numeric benchmark. - Use hypothesis testing with p-values and confidence intervals. - Require independent observations. Differences - Parameter tested: - 1-sample t-test: Mean (μ). - 1 Sample Wilcoxon: Median (M). - Assumptions: - t-test: Assumes approximate normality of data or at least of the sampling distribution of the mean. - 1 Sample Wilcoxon: Nonparametric, does not require normality, but assumes symmetry of differences. - Data basis: - t-test: Uses raw data values directly. - 1 Sample Wilcoxon: Uses ranks of signed differences from M₀. - Robustness to outliers: - 1 Sample Wilcoxon is generally more robust to outliers and heavy tails. When data are clearly non-normal, heavily skewed, or include extreme values, and when median is a more relevant measure of central tendency, the 1 Sample Wilcoxon is often preferable. --- Practical Considerations and Common Pitfalls Sample Size and Power - Small samples: - The test is valid but may have low power (less ability to detect real differences). - Exact p-values (based on the exact distribution of T⁺/T⁻) are often used. - Larger samples: - Normal approximation is common. - Power is generally reasonable if the effect size is not very small. Handling Ties and Zeros - Ties in |Dᵢ|: - Assign average ranks; software does this automatically. - Many ties can slightly change the distribution of the test statistic. - Zero differences (Xᵢ = M₀): - Exclude from ranking. - Large numbers of zeros reduce effective sample size, weakening the test. Checking Assumptions Informally Even though this is a nonparametric test, consider: - Symmetry: - If the differences ( Di = Xi - M_0 ) are extremely skewed, results may be harder to interpret as a median test. - Independence: - Do not use this test for time-series data with strong autocorrelation without appropriate adjustments. - Do not treat repeated measures on the same unit as independent observations. --- Reporting the 1 Sample Wilcoxon Test When summarizing results, include: - Purpose: - What median and benchmark were tested. - Data description: - Sample size and context (e.g., type of measurement). - Test details: - Test name (1 Sample Wilcoxon or Wilcoxon signed-rank). - Hypothesized median (M₀). - Direction of test (two-sided, less-than, greater-than). - Statistics: - Test statistic (T⁺ or T⁻ and, if applicable, z-value). - p-value. - Sample median and confidence interval for the median. - Conclusion in context: - State whether evidence supports a median different from M₀ and how (higher or lower). Example structure: - “A 1 Sample Wilcoxon test was used to evaluate whether the median cycle time differs from 10 minutes (H₀: M = 10). With n = 20, the sum of positive ranks was T⁺ = 180, p = 0.012 (two-sided). The sample median was 8.7 minutes (95% CI: 7.9 to 9.5). We conclude that the median cycle time is significantly different from 10 minutes and is lower in practice.” --- Summary The 1 Sample Wilcoxon test is a nonparametric method for testing whether the median of a single population equals a hypothesized value. It is based on the ranks of signed differences from that value and is well-suited for non-normal, ordinal, or skewed data. Effective use requires understanding: - When it is appropriate (single sample, benchmark, non-normal data). - How to set null and alternative hypotheses for the median. - How to compute and interpret the signed-rank test statistic. - How to read p-values and confidence intervals to draw practical conclusions. By focusing on medians and ranks instead of means and raw values, the 1 Sample Wilcoxon test provides a robust tool for assessing central tendency against a target under less restrictive assumptions than the 1-sample t-test.