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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
Resources available at zstatistics.com including other videos and podcasts.
Encouragement to subscribe to the YouTube channel.
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