A generalized rankorder method for nonparametric analysis of daa from exercise science A tutorial

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Slide 1 :

Parametric or Non-Parametric Statistics: Using Rank-Order Procedures Jerry R. Thomas Professor and Chair Department of Kinesiology Iowa State University Ames, IA

Slide 2 :

Reference for paper: Thomas, J.R., Nelson, J.K., & Thomas, K.T. (1999). A generalized rank-order method for nonparametric analysis of data from exercise science: A tutorial. Research Quarterly for Exercise and Sport, 70, 11-23. Iowa State University

Slide 3 :

Outline of Presentation • Procedures for rank-order data Correlation Differences among groups Review Measurement issue GLM

Slide 4 :

Measurement Issues Numbers don’t know what they represent (next slide) Statistical programs don’t know where numbers come from Important assumptions in parametric statistics Are the data normally distributed Do data fit a straight line (GLM) Assumptions must be met when comparing calculated statistics to tables

Slide 5 :

GLM—Regression of Group Membership on dv M(gp1) = 26.2 SD(gp1) = 1.6 M(gp2) = 35.5 SD(gp2) = 1.6 t(18) = 12.99, p<.0001 t2 = F = 168.85 r2 = .90 F (1,18) = 168.85, p<.0001

Slide 6 :

GLM—Regression of Predictor on Criterion M(pred) = 57.7 SD(pred) = 10.4 M(crit) = 30.9 SD(crit) = 5.0 r2 = .90 F(1,18) = 168.41 p< .0001

Slide 7 :

Understanding the GLM If data are normally distributed, all significance tests use F or t. If data are non-normal, change to ranks and all significance tests use L.

Slide 8 :

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!

Slide 9 :

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

Slide 10 :

Puri and Sen Rank-Order General Linear Method • This method maintains good power and protects against type I errors. • Three steps are involved 1. Change data to ranks 2. Use any of the standard parametric procedures for ranked scores using SPSS, SAS, BIMED 3. Calculate the L statistic instead of F

Slide 11 :

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

Slide 12 :

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

Slide 13 :

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

Slide 14 :

Examples from Regression (Distribution) 8 1* 01233444 20 1. 55555677777788889999 15 2* 000022223333344 17 2. 55555667777888899 6 3* 000224 7 3. 5578899 4 4* 2223 2 4. 66 Stem width: 10.0 Each leaf: 1 case(s) N = 79 M (mm) = 24.97 SD = 8.80 Skewness = 0.67 Kurtosis = -0.28

Slide 15 :

Examples from Regression (Distribution) 1 Extreme 4 1. 6799 5 2* 01222 12 2. 666777888999 17 3* 01111122222344444 23 3. 55556666778888889999999 15 4* 000001122233334 2 4. 56 Stem width: 10.0 Each leaf: 1 case(s) N = 79 M (%) = 33.89 SD = 7.48 Skewness = -0.67 Kurtosis = -0.09

Slide 16 :

Multiple Regression Using Original Data 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* F(4,74) = 32.87, p < .001, for linear composite of predictors 1 Subscap SF .68 .46 .332 1 35.83* 2 Calf SF .77 .60 .602 2 20.29* 3 Abdom SF .80 .64 .321 3 7.85* 4 Thigh SF .82 .68 -.327 4 9.56* L(4) = 53.01, p < .001, for the linear composite of predictors Multiple Regression Using Ranked Data

Slide 17 :

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

Slide 18 :

Stem-and-Leaf, Mean, Standard Deviation, Skewness, and Kurto-sis for Push-Up Scores for Boys and Girls in Grades 4, 5, and 6 30 0* 001111111122222222233334444444 30 0 555556666666777777778888889999 32 1* 00000001111122222233333334444444 21 1 555556667777777889999 25 2* 0000000111112233333344444 25 2 5555555566666666677788999 8 3* 00023444 7 3 5566889 2 4* 02 Stem width: 10 Each leaf: 1 N = 180 M = 15.63 SD = 10.27 Skewness = 0.41 Kurtosis = -0.70

Slide 19 :

3 × 2 ANOVA Results for Original Data • Original data - Grade: F(2, 174) = 7.30, p < .001 - Sex: F(1, 174) = 17.48, p < .001 - Interaction: Not significant 3 × 2 ANOVA Results for Ranked Data • Ranked data - Grade: L(2) = 11.67, p < .005 - Sex: L(1) = 13.21, p < .001 - Interaction: Not significant

Slide :

Slide 22 :

Summary Tables of Repeated-Measures ANOVAs for Original and Ranked Data Original data Age .14 (r2 = SSBet/SSTot) 1, 57 9.98 .003 Speed .98 4, 54 589.37 .0001 Age × Speed .22 4, 54 3.70 .01 Huynh-Feldt Epsilon = .65 Age .14 (r2 = SSBet/SSTot) 1 8.12 <.01 Speed .98 5 56.84 <.001 Age × Speed .22 5 12.76 <.05 Huynh-Feldt Epsilon = .77

Slide 23 :

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

Slide 24 :

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.

Slide 25 :

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 - Interaction: Not significant

Slide 26 :

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), ANCOVA - Multivariate techniques: Discriminant analysis, MANOVA (including repeated measures), MANCOVA

Slide 27 :

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.

Slide 28 :

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