1. Overview of non parametric methods

2. Parametric methods.
Normality

3. Parametric methods.
Normality

4. Parametric methods.
Sample size.

5. Qualitative or Quantitative
Parametric methods.
Problems in measurement (scale of measurement)
Stanley Smith Stevens' typology
#  
Scale type  
Logical/math operations   allowed
  Examples:      Variable name                            (data values)
Measure of central tendency
Qualitative or Quantitative 1
Nominal
=/≠
Dichotomous:     Gender                              (male vs. female) Non-dichotomous: Nationality                              (American/Chinese/etc)
Mode
Qualitative 2
Ordinal
=/≠ ; </>
Dichotomous:     Health (healthy vs. sick),                          Truth (true vs. false),                          Beauty (beautiful vs. ugly) Non-dichotomous: Opinion ('completely agree'/ 'mostly agree'/ 'mostly disagree'/ 'completely disagree')
Median 3
Interval
=/≠ ; </> ; +/−
  Date (from 9999 BC to 2013 AD)   Latitude (from +90° to −90°)
Arithmetic Mean
Quantitative 4
Ratio
=/≠ ; </> ; +/− ; ×/÷
  Age (from 0 to 99 years)
Geometric Mean

6. «Loщ-quality» data Non-normal distribution Small sample size
Low level of measurement

7. nonparametric methods
and
nonparametric methods
(parameter-free methods or distribution-free methods.)

8. Nonparametric methods
Descriptive statistics;
Tests of differences between groups (independent samples);
Tests of differences between variables (dependent samples);
Tests of relationships between variables.

9. Nonparametric methods
Descriptive statistics
G = (x1 *x2 *...*xn )1/n - geometric mean
log(G) = {Σ[log(xi)]}/n

10. Nonparametric methods
Differences between dependent groups.
2 variables
Sign test
Wilcoxon's matched pairs test.
More than 2 vars
Cochran's Q test

11. Nonparametric methods
Sign test
Peanut Butter Taste Test
As part of a market research study, a sample of 36 consumers were asked to taste two brands of peanut
butter and indicate a preference.
Do the data shown
below indicate a significant difference in the consumer
preferences for the two brands?
18 preferred Hoppy Peanut Butter
12 preferred Pokey Peanut Butter
6 had no preference
The analysis is based on a sample size of = 30.

12. Nonparametric methods
Sign test
Peanut Butter Taste Test
p = Pr(X > Y)
H0: p = 0.50
W be the number of pairs for which yi − xi > 0
Reject H0 if z < or z > 1.96
z = ( )/2.74 = 3/2.74 = 1.095

13. Nonparametric methods
Wilcoxon Signed-Rank Test
District Office Overnight NiteFlite
Seattle hrs hrs.
Los Angeles 30 24
Boston 19 15
Cleveland 16 15
New York 15 13
Houston 18 15
Atlanta 14 15
St. Louis 10 8
Milwaukee 7 9
Denver 16 11

14. Nonparametric methods
Wilcoxon Signed-Rank Test
District Office Differ Diff. Rank Sign. Rank
Seattle
Los Angeles
Boston
Cleveland
New York
Houston
Atlanta
St. Louis
Milwaukee
Denver
+44

15. Nonparametric methods.
Nonparametric methods
Wilcoxon Signed-Rank Test
Compute the differences between the paired observations.
Discard any differences of zero.
Rank the absolute value of the differences from lowest to highest. Tied differences are assigned the average ranking of their positions.
Give the ranks the sign of the original difference in the data.
Sum the signed ranks.
. . . next we will determine whether the sum is significantly different from zero.

16. Nonparametric methods
Differences between independent groups.
2 variables
Mann-Whitney U test
Wald-Wolfowitz runs test
More than 2 vars
Kruskal-Wallis analysis of ranks

17. Nonparametric methods.
Nonparametric methods
Mann-Whitney U test
First, rank the combined data from the lowest to the highest values, with tied values being assigned the average of the tied rankings.
Then, compute U, the sum of the ranks for the each sample.
The smaller value of U1 and U2 is the one used when consulting significance tables.
If U < = Table value H1, else H0
U is just number of wins out of all pairwise contests

18. Nonparametric methods
Mann-Whitney U test, another way
T H H H H H T T T T T H

19. Nonparametric methods
Mann-Whitney U test

20. Nonparametric methods
Tests of relationships between variables.
Spearman, Kendall tau, Gamma correlation coefficients
2x2 Tables

21. Nonparametric methods
Spearman correllation
ui = rank of item i with respect to one variable
vi = rank of item i with respect to a second variable
di = ui - vi

22. Nonparametric methods
Spearman correllation
Car
Age
(months) Xi
Minimal Stopping dist at 40 kph (metres) Yi
Age Rank
(ui)
Stopping Rank
(vi)
Differences of the Ranks
(di = ui-vi) A 9
28.4 1 B 15
29.3 2 C 24
37.6 3 7 -4 D 30
36.2 4
4.5
-0.5 E 38
36.5 5 6 -1 F 46
35.3 G 53
2.5 H 60
44.1 8 I 64
44.8 J 76
47.2 10
d2=32.5

23. Nonparametric methods
Classification and clustering

24. Nonparametric methods
2x2 Tables
Suppose that you are considering whether to introduce a new formula for a successful soft drink. Before finally deciding on the new formula, you conduct a survey in which you ask male and female respondents to express their preference for either the old or new soft drink. Assume that out of 50 males, 41 prefer the new formula over the old formula; out of 50 females, only 27 prefer the new formula. 41 9 27 23
Chi-square (df=1)
9.01
p= .0027
V-square (df=1)
8.92
p= .0028
Yates corrected Chi-square
7.77
p= .0053
Phi-square
.09007
Fisher exact p, one-tailed
p= .0025
two-tailed
p= .0049
McNemar Chi-square (A/D)
4.52
p= .0336
Chi-square (B/C)
8.03
p= .0046