14.1 Sign Test and Wilcoxon Signed-Rank Test

Nonparametric tests for medians offer robust alternatives when data doesn't meet traditional assumptions. The Sign Test and Wilcoxon Signed-Rank Test allow us to analyze medians without relying on normal distributions, making them versatile tools for various business scenarios.

These tests shine when dealing with skewed data or outliers. By focusing on ranks or signs rather than exact values, they provide reliable insights into central tendencies, helping businesses make informed decisions even with challenging datasets.

Nonparametric Tests for Medians

Assumptions of nonparametric tests

Top images from around the web for Assumptions of nonparametric tests

• 

  Spearman's rank correlation coefficient - Wikipedia View original

  Is this image relevant?

• 

  File:Comparison mean median mode.svg - Wikimedia Commons View original

  Is this image relevant?

• 

  hypothesis testing - When To Use Negative Test Statistic over Positive Test Statistic in ... View original

  Is this image relevant?

• 

  Spearman's rank correlation coefficient - Wikipedia View original

  Is this image relevant?

• 

  File:Comparison mean median mode.svg - Wikimedia Commons View original

  Is this image relevant?

1 of 3

Top images from around the web for Assumptions of nonparametric tests

• 

  Spearman's rank correlation coefficient - Wikipedia View original

  Is this image relevant?

• 

  File:Comparison mean median mode.svg - Wikimedia Commons View original

  Is this image relevant?

• 

  hypothesis testing - When To Use Negative Test Statistic over Positive Test Statistic in ... View original

  Is this image relevant?

• 

  Spearman's rank correlation coefficient - Wikipedia View original

  Is this image relevant?

• 

  File:Comparison mean median mode.svg - Wikimedia Commons View original

  Is this image relevant?

1 of 3

• Sign Test assumes:
  • Data comes from a continuous distribution where the median is a meaningful measure of central tendency
  • Sample is randomly selected from the population to ensure representativeness
  • Observations within the sample are independent of each other (no relationship or influence between data points)
  • Primary interest lies in estimating or comparing the population median rather than the mean
• Wilcoxon Signed-Rank Test assumes:
  • Paired samples or matched data where each observation in one sample has a corresponding observation in the other sample (before and after measurements, matched subjects)
  • Differences between paired observations follow a symmetric distribution around the median difference
  • Differences between paired observations are mutually independent (knowing the difference for one pair does not provide information about the differences for other pairs)
• Sign Test is appropriate when:
  • Comparing a sample median to a specific hypothesized value (testing if the median salary in a company is equal to the industry median)
  • Data violates the assumptions of a t-test (non-normal distribution, presence of outliers)
• Wilcoxon Signed-Rank Test is suitable when:
  • Comparing the median difference between paired samples (assessing the median change in blood pressure before and after a treatment)
  • Data violates the assumptions of a paired t-test (non-normal distribution of differences, presence of outliers in the differences)

Application of Sign Test

• Formulate hypotheses:
  • $H_0$: The population median is equal to the hypothesized value ($M = M_0$)
  • $H_1$: The population median is not equal to the hypothesized value ($M \neq M_0$) for a two-tailed test, or $M < M_0$ or $M > M_0$ for one-tailed tests (testing if the median income is less than or greater than a specific value)
• Compute the test statistic:
  • Count the number of observations above (+) and below (-) the hypothesized median value (count the number of employees with salaries above and below the industry median)
  • Disregard observations exactly equal to the hypothesized median value (exclude employees with salaries exactly equal to the industry median)
  • The test statistic is the smaller of the two counts (+ or -) (if there are 12 employees above and 8 below the industry median, the test statistic is 8)
• Calculate the p-value:
  1. For small samples (n ≤ 20), use the binomial distribution with $p = 0.5$ (probability of success = 0.5 for a two-tailed test)
  2. For large samples (n > 20), use the normal approximation to the binomial distribution with $\mu = \frac{n}{2}$ and $\sigma = \sqrt{\frac{n}{4}}$ (mean = half the sample size, standard deviation = square root of one-fourth the sample size)
• Compare the p-value to the chosen significance level (α) to decide whether to reject or fail to reject the null hypothesis

Wilcoxon test for paired samples

• State hypotheses:
  • $H_0$: The median difference between paired observations is zero ($M_d = 0$)
  • $H_1$: The median difference is not zero ($M_d \neq 0$) for a two-tailed test, or $M_d < 0$ or $M_d > 0$ for one-tailed tests (testing if the median difference in test scores before and after a training program is significantly different from zero)
• Compute the test statistic:
  1. Calculate the differences between paired observations (subtract the pre-training score from the post-training score for each participant)
  2. Rank the absolute values of the differences, assigning the average rank to ties (if two differences have the same absolute value, assign them the average of the ranks they would have occupied)
  3. Sum the ranks of the positive differences ($W+$) and the ranks of the negative differences ($W-$) (calculate the sum of ranks for differences greater than zero and less than zero separately)
  4. The test statistic is the smaller of $W+$ and $W-$ (if the sum of positive ranks is 50 and the sum of negative ranks is 30, the test statistic is 30)
• Determine the p-value:
  1. For small samples (n ≤ 20), use the exact distribution of the test statistic (look up the p-value in a Wilcoxon Signed-Rank Test table)
  2. For large samples (n > 20), use the normal approximation with $\mu = \frac{n(n+1)}{4}$ and $\sigma = \sqrt{\frac{n(n+1)(2n+1)}{24}}$ (mean = one-fourth of n(n+1), standard deviation = square root of n(n+1)(2n+1) divided by 24)
• Compare the p-value to the significance level (α) to make a decision about the null hypothesis

Interpretation of nonparametric results

• If the p-value is less than the significance level (α), reject the null hypothesis
  • For the Sign Test, conclude that the population median is significantly different from the hypothesized value (the median salary in the company is significantly different from the industry median)
  • For the Wilcoxon Signed-Rank Test, conclude that the median difference between paired samples is significantly different from zero (the median change in test scores before and after the training program is significantly different from zero)
• If the p-value is greater than the significance level (α), fail to reject the null hypothesis
  • There is insufficient evidence to conclude that the population median differs from the hypothesized value (Sign Test) or that the median difference between paired samples differs from zero (Wilcoxon Signed-Rank Test) (there is not enough evidence to claim that the median salary differs from the industry median or that the training program had a significant effect on test scores)
• Consider the practical significance of the results in the context of the problem, in addition to statistical significance (even if the median difference in test scores is statistically significant, assess whether the magnitude of the difference is meaningful in terms of educational outcomes)