1. Nonparametric Techniques
chapter 10
Nonparametric Techniques

2. Chapter Outline Chi square: testing the observed versus the expected
Procedures for rank-order data
Correlation
Differences among groups

3. Analyzing Data Appropriately
Behavioral scientists believe most data are normally distributed: God loves the normal curve!
Is that true? Micceri (1989) said no for large data sets in psychology.
Do we as scientists look carefully at the distribution of our data? God may not love the normal curve!

4. Parametric Statistical Procedures
Are parametric statistical procedures sensitive to nonnormality?
Substantial evidence exists that parametric statistical procedures are not as robust to violations of the normality assumption as once thought.

5. Chi Square: Testing the Observed Versus the Expected
Formula for chi square X2 = (O – E)2 / E where O = observed frequency and E = expected frequency

6. Coach Rabbitfoot and His Tennis Courts
Known values Court number
Total
Observed losses O =
Expected losses E =
(O – E) –6 +4 –8 +10
(O – E)
(O – E)2/E
X2 = (O – E)2/E = = 7.19
df = # cells – 1 = # courts – 1 = 4 – 1 = 3
X2 (3) = 7.19, p > .05, not significant

7. Contingency Tables
Chi square with two or more categories and two or more groups
Athletes and nonathletes respond to an ethical statement about whether one should tell the umpire if they trap a fly ball in baseball.
The athletes respond on a Likert-type scale: 3 = Agree, 2 = No opinion, 1 = Disagree

8. Working Out the Answer Obs resp. Agree No Disagree Total opinion
Athletes
Nonathletes
Total
Expected (column total  row total)/N
Athletes
Nonathletes
X2 = (–342/64) + (–342/80) + (–102/56) + (102/70) +(442/80) + (–442/100) = 79.29
df = (r – 1) (c – 1) = (2 – 1)(3 – 1) = 2, p < .01

9. Puri and Sen Rank-Order General Linear Method
This method maintains good power.
This method protects against type I error.
Change data to ranks.
Use any of the standard parametric procedures for ranked scores using SPSS or SAS.

10. General Linear Model (GLM)
Basis for procedures of
regression: r, R, Rc
differences: t, ANOVA, MANOVA
Y = B  X + E Y = vector of scores on p dvs X = vector of scores on q Ivs B = p  q matrix of reg. Coefficients E = vector of errors

11. Calculating the Test Statistic for Ranked Data
Instead of the parametric test statistic (t or F), calculate L. L = (N – 1)r2 df = p  q

12. Example From Regression
Can skinfold measurements be used to predict percentage fat (determined by underwater weighing) in women grouped by ethnicity?
Data from K.T. Thomas et al., 1997.

13. Examples From Regression (Distribution)
Thigh
Frequency Stem & Leaf
8 1*
15 2*
6 3*
4 4* 2223
Stem width: Each leaf: 1 case(s)
N = M (mm) = SD = 8.80
Skewness = Kurtosis = 0.28

14. Examples From Regression (Distribution)
Percent fat from hydrostatic weighing
Frequency Stem & Leaf
1 Extreme
5 2* 01222
17 3*
15 4*
Stem width: Each leaf: 1 case(s)
N = M = SD = 7.48
Skewness = Kurtosis = 0.09

15. Multiple Regression on Original Data
Step
Variable R R2 b df
F-to-enter 1
Subscap SF
.67
.45
.297
1,77
62.98* 2
Calf SF
.75
.56
.569
2,76
19.78* 3
Abdom SF
.78
.61
.339
3,75
8.86* 4
Thigh SF
.80
.64
–.289
4,74
6.26*
*p < .05
F(4,74) = 32.87, p < .001, for linear composite of predictors

16. Multiple Regression Using Ranked Data
Step
Variable R R2 b df
F-to-enter 1
Subscap SF
.68
.46
.332
35.83* 2
Calf SF
.77
.60
.602
20.29* 3
Abdom SF
.80
.64
.321
7.85* 4
Thigh SF
.82
–.327
9.56*
*p < .05
L(4) = 53.01, p < .001, for linear composite of predictors

17. Example Using Factorial ANOVA
Do boys and girls differ in push-up scores in grades 4, 5, and 6?
Data from J.K. Nelson et al., 1991.

18. Stem-and-Leaf, Mean, Standard Deviation, Skewness, and Kurtosis for Push-Up Scores for Boys and Girls in Grades 4, 5, and 6
Frequency Stem & Leaf
30 0*
32 1*
25 2*
8 3*
2 4* 02
Stem width: Each leaf: 1
N = M = SD = 10.27
Skewness = Kurtosis = 0.70

19. 3  2 ANOVA Results for Original Data
Grade: F(2, 174) = 7.30, p < .001
Sex: F(1, 174) = 17.48, p < .001
Interaction: not significant

20. 3  2 ANOVA Results for Ranked Data
Grade: L(2) = 11.67, p < .005
Sex: L(1) = 13.21, p < .001
Interaction: not significant

21. Example Using Repeated-Measures ANOVA.
Does VO2 differ by walking speeds in older and younger participants?
Data from P.E. Martin, D.E., Rothstein, & D.D. Larish, 1992, “Effects of age and physical activity status on the speed-aerobic demand relationship of walking,” Journal of Applied Physiology, 73:

22. Characteristics of VO2 at Five Walking Speeds.
Miles per hour M SD
Median
Skewness
Kurtosis –0.13 – –

23. Summary Tables of Repeated-Measures ANOVAs for Original and Ranked Data
Source Pillai’s trace df F Signif.
Original data
Age .14 (r2 – SSBet/SSTot) 1,
Speed .98 4,
Age  Speed .22 4,
Huynh-Feldt Epsilon = .65
Ranked Data L
Age .14 (r2 – SSBet/SSTot) <.01
Speed <.001
Age  Speed <.05
Huynh-Feldt Epsilon = .77

24. Example Using Factorial MANOVA
Do four ethnic groups at two age levels differ on two skinfold measurements and hip-to-waist ratio?
Data from K.T. Thomas et al., 1997.

25. Using MANOVA on Original and Ranked Data
Data are for four ethnic groups (African American, European American, Mexican American, and Native American) at two age levels (20–30 and 40–50), include the previously reported data on abdomen and calf skinfolds, and add a third dependent variable, hip-to-waist ratio.

26. 4 (Ethnic Group)  2 (Age Level) MANOVA on Three Dependent Variables
Original data
Ethnic group: F(3, 152) = 5.64, p <.0001
Age level: F(3, 152) = 7.86, p <.0001
Interaction: not significant
Ranked data (Pillai’s trace = R2)
Ethnic group: L(3) = 22.54, p <.0001
Age level: L(9) = 41.86, p <.0001

27. Applications to GLM
These procedures are appropriate for all GLM models.
Regression: Pearson r, multiple R canonical (Rc)
ANOVA: t, simple and factorial ANOVA (including repeated measures), ANOCOVA
Multivariate techniques: Discriminant analysis, MANOVA (including repeated measures), MANCOVA

28. Summary Are data from physical activity normally distributed?
If not, changing data to ranks and using nonparametric procedures allows the researcher the alternative of using standard statistical packages while calculating only the L statistic.