Understanding Non-Parametric Statistical Tests

Oct 28, 2024

Non-Parametric Testing Lecture Notes

Introduction

  • Presenter: Justin Zeltser from zstatistics.com
  • Topic: Non-parametric testing in statistical inference
  • Focus on three non-parametric tests:
    • Sign test
    • Wilcoxon signed rank test
    • Mann-Whitney U test (Wilcoxon rank sum test)

Overview of Non-Parametric Methods

  • Definition: Non-parametric methods allow statistical inference without assuming a specific distribution for the sample (e.g., normal distribution).
  • Contrast with parametric methods that rely on means and standard deviations.
  • History:
    • John Arbuthnot (1710) used the sign test in his paper concerning the birth proportions of males and females.
    • First paper in inferential statistics.

Pathways for Non-Parametric Tests

  • Single Sample or Matched Pairs:
    • Sign Test or Wilcoxon Signed Rank Test
    • Example: Same subjects measured before and after an intervention.
  • Independent Samples:
    • Mann-Whitney U Test (Wilcoxon Rank Sum Test)
    • Example: Comparing males vs. females on some measure.

Non-Parametric Test Examples

1. Sign Test

  • Example: Hemoglobin levels from 10 female vegetarians to assess anemia prevalence.
    • Question: Is the median hemoglobin level < 13.0 g/dL?
    • Null Hypothesis (H0): Median = 13 g/dL
    • Alternate Hypothesis (H1): Median < 13 g/dL
  • Analysis:
    • Count of observations below 13 g/dL: 7 observations (more extreme than expected if H0 is true).
    • P-value Calculation: Binomial distribution with n=10, p=0.5;
    • P-value = 0.172; unable to reject H0 (not extreme enough).

2. Wilcoxon Signed Rank Test

  • Same Example as Sign Test: Hemoglobin levels for the same sample.
  • Incorporates Distance: Ranks differences from the median (13 g/dL).
    • Negative ranks for those below 13, positive ranks for those above 13.
  • Test Statistic Calculation:
    • Sum of positive ranks = 8. Compare with critical values for n=10 using the Wilcoxon signed ranks table.
    • Decision: Reject H0 at 5% significance level but not 1% (critical value for 1% is 5).

3. Mann-Whitney U Test

  • Example: Compare hemoglobin levels in 10 female vegetarians vs. 8 male vegetarians.
    • Question: Is there a difference in median hemoglobin levels?
  • Ranking: Rank all observations together, regardless of gender.
  • Test Statistic:
    • Sum of ranks for males = 85.
    • Compare with expected value for males (76).
    • Critical Values: Use Mann-Whitney U table to assess significance.
    • Result: Do not reject H0; not enough evidence to suggest a difference in medians.

Summary

  • Non-parametric methods provide a useful alternative to parametric tests when assumptions about distribution cannot be made.
  • Key tests: Sign Test, Wilcoxon Signed Rank Test, Mann-Whitney U Test (Wilcoxon Rank Sum Test).
  • Important to understand the different hypotheses, assumptions, and calculations involved.

Conclusion

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