The document discusses the complexities of analyzing non-normal (ab-normal) data using statistical methods, highlighting that not all data fits traditional parametric assumptions. It emphasizes the importance of non-parametric statistics in fields such as psychology and biology, particularly for analyzing counts and ranks, and provides insights on reliability and reproducibility in data handling. The document also references various statistical tests and methodologies, including binomial and Poisson distributions, along with their application in real-world scenarios.

1. Para-normal Statistics:
Analyzing what doesn't add up.
Steven Lembark
Workhorse Computing
lembark@wrkhors.com

2. Normality
We expect data is normal.
It's what we are trained for.
Chi-Squared, F depend on it.
It's the guts of ANOVA.
Theory guarantees it, sort of.

3. What is "normal"?
Normal data is:
Parametric
Real
Symmetric
Unimodal

4. Ab-normal data
Not all data is parametric.

5. Ab-normal data
Not all data is parametric:
"Bold" + "Tide" / 2 ==

6. Ab-normal data
Not all data is parametric: Nominal Data
"Bold" + "Tide" / 2 == ??
"Bald" - "Harry" >= 0 ??

7. Ab-normal data
Not all data is parametric: Ordinal Data
"On a scale of 1 to 5 how would you rate..."
Is the average really 3?
Are differences between ranks unform?

8. Ab-normal data
Not all data is parametric: Ordinal Data
"On a scale of 1 to 5 how would you rate..."
Is the average really 3?
For different people?

9. Ab-normal data
Not all data is unimodal, symmetric.
Bi-modal data has higher sample variance.
Positive data is skewed.

10. Ab-normal data
Counts usually Binomial or Poission.
Binomial: Coin flips.
Poisson: Sample success/failure.

11. Power of Positive Thinking
Binomial: Count of
success from IID
experiments.
Mean = np
Variance = npq

12. Power of Positive Thinking
Poisson: Count of
occurrances in sample
size n.
Mean = np
Variance = np

13. Power of Positive Thinking
Curves all positive.
Right tailed.
Binomial has highest power if sample data is binomial.
Result: Smaller n for given Beta.

14. Kinda normal
Approximations work...

15. Kinda normal
Approximations work some of the time.
Rule: npq > 5 for binomial approximation.
Goal: Keep mean > 3σ so normal is all positive.
Q: How good an approximation?
A: It depends...

16. The middle way
Binomial:
n=20, p=0.5
Normal:
µ 10, σ = 2.23
Decent approximation.

17. Off to one side
Binomial:
n=20, p=0.3
Normal:
µ = 6, σ = 2.0
Drifting negative.

18. Life on the edge
Binomial:
n=20, p=0.1
Normal:
µ = 2, σ = 1.3
Significant negative.

19. Neverneverland
Binomial:
n=20, p=0.0013
Normal:
µ = 0.26
σ = 0.16
Heavily negative.

20. General rule: npq > 5
Small or large p is skewed.
Six-sigma range should be positive.
At that point n > 5 / pq.
For p = 0.0013, n = 3582.
Sample size around 4000?

21. When we assume we make...
Assuming normal data leaves a less robust conclusion.
Stronger, less robust:
Sensitive to individual datasets.
Not reproducable.

22. Non-parametric Statistics
Origins in Psychology, Biology, Marketing.
Analyze counts, ranks.
Tests based on discrete distributions.

23. Common in Quality
Frequency of failures.
QC with No-Go guages.
Variations between batch runs.
Customer feedback.

24. Example: Safety study
Q: Are departments equally "safe"?
Q: Is a new configuration any "safer"?
Compare sample populations.

25. What is "safe"?
Fewer reported injurys?
What is P( injury ) per operation?

26. What is "safe"?
Fewer reported injurys?
What is P( injury ) per operation?
0.5?
0.3?

27. What is "safe"?
Fewer reported injurys?
What is P( injury ) per operation?
0.5?
0.1?
A whole lot less?

28. What is "safe"?
Fewer reported injurys?
What is P( injury ) per operation?
0.5?
0.1?
A whole lot less?
N(0.01, 0.01) is heavily negative.

29. Severe?
Parametric measure of injurys?

30. Severe?
Parametric ranking of injurys?
( Finger + Thumb ) / 2 == ?

31. Severe?
Parametric ranking of injurys?
( Finger + Thumb ) / 2 == ?
( Hand + Eye ) == Arm ?
( Hand + Hand ) == 2 * Hand ?

32. Ordinal Data
Ranked data, not scaled.

33. Ordinal Data
Ranked data, not scaled.
Hangnail < Finger Tip < Finger < Hand < Arm
"Fuzzy Buckets"
Have p( accident ) from history.

34. Kolomogrov-Smirnov
Got tonic?

35. Kolomogrov-Smirnov
Nope, not Vodka.
Like F or ANOVA: Populations are "different".

36. K-S Test
Compare cumulative
data (blue) vs. Expeted
(red).
Measure is largest
difference (arrow).

37. K-S for safety
Rank the injurys on relative scale.
Compare counts by bucket.
Cumulative distribution:
accomodates empty cells.
minor mis-catagorization.

38. A good datum is hard to find,
You always get the other kind.
Apologies to Bessie Smith
Sliding-scale questions:
"How would you rate..."
"How well did..."
"How likely are you to..."

39. A good datum is hard to find,
You always get the other kind.
Apologies to Bessie Smith
Reproducability:
Variable skill.
Variable methods.
Variable data handling.

40. A good datum is hard to find,
You always get the other kind.
Apologies to Bessie Smith
Big Data:
Multiple sources.
Multiple populations.
Multiple data standards.

41. Repeatable Analysis
Variety of NP tests for "messy" data.
Handle protocol, sampling variations.
Robust conclusions with real data.

42. Summary
Non-parametric data: counts, nominal, ordinal data.
Non-parametric analysis avoids NID assumptions.
Robust analysis of real data.
Even the para-normal.

43. Questions?

44. References: N-P
http://www.uta.edu/faculty/sawasthi/Statistics/stnonpar.html
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC153434/
Nice writeups.

45. References: K-S
http://itl.nist.gov/div898/handbook/eda/section3/eda35g.htm
Exploratory data analysis is worth exploring.
https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test
As always... really good writeup of the test definition, math.

46. References: Robust Analysis
https://en.wikipedia.org/wiki/Robust_statistics
https://en.wikipedia.org/wiki/Robust_regression
Decent introductions.
Also look up "robust statistics" at nist.gov or "robust statistical
analysis" at duckduckgo.

47. References: This talk
http://slideshare.net/lembark
Along with everything else I've done...