1. Review
I volunteer in my son’s 2nd grade class on library day. Each kid gets to check out one book. Here are the types of books they picked this week:
Astronauts Ninjas Ponies Birds Total
Boys
Girls
Total
Suppose we want to know whether gender and book type are independent. Which of the following is NOT a correct statement of the null hypothesis?
The distribution of book preferences is the same for boys and girls.
Boys like all book types equally, and so do girls.
Knowing whether a kid is male or female gives no information about his or her likely book preference.
Knowing a kid’s book preference gives no information about the kid’s gender.

2. Review
I volunteer in my son’s 2nd grade class on library day. Each kid gets to check out one book. Here are the types of books they picked this week:
Astronauts Ninjas Ponies Birds Total
Boys
Girls
Total
If book type is independent of gender, how many boys would be expected to pick pony books?
2.25
2.63
3.00
3.50

3. Review Here are the expected frequencies, in red:
Astronauts Ninjas Ponies Birds Total
Boys
Girls
Total
Calculate the c2 statistic for testing independence.
2.32
5.89
9.04
9.69

4. Non-Parametric Tests
12/4

5. Parametric vs. Non-parametric Statistics
Most common type of inferential statistics
r, t, F
Make strong assumptions about the population
Mathematically fully described, except for a few unknown parameters
Powerful, but limited to situations consistent with assumptions
When parametric statistics fail
Assumptions not met
Ordinal data: Assumptions not meaningful
Non-parametric statistics
Alternatives to parametric statistics
"Naive" approach: Far fewer assumptions about data
Work in wider variety of situations
Not as powerful as parametric statistics (when applicable)

6. Assumption Violations
Parametric statistics work only if data obey certain properties
Normality
Shape of population distribution
Determines shape of sampling distributions
Tells how likely extreme results should be; critical for correct p-values
More important with small sample sizes (Central Limit Theorem)
Homogeneity of variance
Variance of groups is equal (t-test or ANOVA)
Variance from regression line does not depend on values of predictors
Linear relationships
Pearson correlation cannot recognize nonlinear relationships

7. Assumption Violations
Parametric statistics only work if data obey certain properties
If these assumptions are true:
Population is almost fully described in advance
Goal is simply to estimate a few unknown parameters
If assumptions violated:
Parametric statistics will not give correct answer
Need more conservative and flexible approach
Normal(m1, s2)
Normal(m2, s2)

8. Ordinal Data
Some variables have ordered values but are not as well-defined as interval/ratio variables
Preferences
Rankings
Nonlinear measures, e.g. money as indicator of value
Can't do statistics based on differences of scores
Mean, variance, r, t, F
More structure than nominal data
Scores are ordered
Chi-square goodness of fit ignores this structure
Want to answer same types of questions as with interval data, but without parametric statistics
Are variables correlated?
Do central tendencies differ?

9. Non-parametric Tests
Can use without parametric assumptions and with ordinal data
Basic idea
Convert raw scores to ranks
Do statistics on the ranks
Answer similar questions as parametric tests
Your job: Understand what each is used for and in what situations
Parametric Test
Non-parametric Test
Pearson correlation
Spearman correlation
Independent-samples t-test
Mann-Whitney
Single- or paired-samples t-test
Wilcoxon
Simple ANOVA
Kruskal-Wallis
Repeated-measures ANOVA
Friedman

10. Spearman Correlation Alternative to Pearson correlation
Produces correlation between -1 and 1
Convert data on each variable to ranks
For each subject, find rank on X and rank on Y within sample
Compute Pearson correlation from ranks
Works for
Ordinal data
Monotonic nonlinear relationships (consistently increasing or decreasing) X: 65 72 69 75 66 62 68 Y:
157
194
185
148
163
127
173
RX: 2 6 5 7 3 1 4
RY: X Y RY RX

11. Mann-Whitney Test Alternative to independent-samples t-test
Do two groups differ?
Combine groups and rank-order all scores
If groups differ, high ranks should be mostly in one group and low ranks in the other
Test statistic (U) measures how well the groups' ranks are separated
Compare U to its sampling distribution
Is it smaller than expected by chance?
Compute p-value in usual way
Works for
Ordinal data
Non-normal populations and small sample sizes A B 17 11 23 22 9 15 18 16 20 21 14 13 19 A B 7 2 13 12 1 5 8 6 10 11 4 3 9
Perfect separation U
No separation

12. Wilcoxon Test Alternative to single- or paired-samples t-test
X – m0
Rank
107 7 4 92 -8 5
115 15 9
104 3 87
-13 8
102 2 1 97 -3
112 12 90
-10 6
Alternative to single- or paired-samples t-test
Does median differ from m0?
Does median difference score differ from 0?
Subtract m0 from all scores
Can skip this step for paired samples or if m0 = 0
Rank-order the absolute values
Sum the ranks separately for positive and negative difference scores
If m > m0, positive scores should be larger
If m < m0, negative scores should be larger
If sums of ranks are more different than likely by chance, reject H0
Works for
Ordinal data
Non-normal populations and small sample sizes

13. Kruskal-Wallis Test Alternative to simple ANOVA
Do groups differ?
Extends Mann-Whitney Test
Combine groups and rank-order all scores
Sum ranks in each group
If groups differ, then their sums of ranks should differ
Test statistic (H) essentially measures variance of sums of ranks
If H is larger than likely by chance, reject null hypothesis that populations are equal
Works for
Ordinal data
Non-normal populations and small sample sizes

14. Friedman Test Alternative to repeated-measures ANOVA
Do measurements differ?
Look at each subject separately and rank-order his/her scores
Best to worst for that subject, or favorite to least favorite
For each measurement, sum ranks from all subjects
If measurements differ, then their sums of ranks should differ
Test statistic (c2r) essentially measures variance of sums of ranks
If larger than likely by chance, reject H0
Works for
Ordinal data
Non-normal populations and small sample sizes
Subject
Morning
Noon
Night 1 27 31 35 2 54 41 48 3 62 59 61 4 37 33 5 25 21 23 6 56 50 47
Subject
Morning
Noon
Night 1 2 3 4 5 6
Total 14 10 12

15. Summary Test Replaces When Useful Approach Spearman correlation
Pearson correlation
Ordinal data
Nonlinear relationships
Rank each variable
Mann-Whitney
Independent-samples t-test
Ordinal Data
Non-normal + small samples
Rank all scores
Wilcoxon
Single- or paired-samples t-test
Rank absolute values
Kruskal-Wallis
Simple ANOVA
Friedman
Repeated-measures ANOVA
Rank all scores for each subject

16. Review
Comparing average score between two groups, on an ordinal-scale variable.
What test should you use?
Friedman
Mann-Whitney
Spearman
Kruskal-Wallis

17. Review
Five subjects each measured in three conditions, on an interval-scale variable with a non-normal distribution.
What test should you use?
Repeated-measures ANOVA
Kruskal-Wallis
Friedman
Spearman

18. Review
Comparing average scores among five groups, on an interval-scale variable with a non-normal distribution subjects per group.
What test should you use?
Kruskal-Wallis
Friedman
One-way ANOVA
Mann-Whitney