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UT Knoxville STAT 201 - 13) sld_repeated_mulitvar_2factor

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1 Factorial Designs With Repeated Measures The Multivariate Approach 2 Rationale of Multivariate Approach Transforms data with a 1 difference scores Calculates and removes correlation from D scores matrix algebra calculate determinant for full restricted models of a 1 D scores Analyzes the total amount of unique variability 3 Organization of Lecture Factorial design with 2 within sub factors Factorial design with 1 within sub 1 between sub factors mathematics is exponentially more complicated we ll examine conceptual analysis and SAS 4 Factorial Designs 2 Within Sub Variables E g Track coach is interested in 3 sneakers A B C Measures time s for 6 persons to run 100 yds with each shoe on an indoor and outdoor track Track Subjec t 1 2 3 4 5 6 Indoor Sneaker A B C Outdoor Sneaker A B C 15 13 15 20 18 09 08 08 12 14 13 12 13 18 19 11 11 10 16 13 16 16 16 21 19 11 11 11 15 15 36 observations from 6 subjects 10 08 14 10 15 11 5 2 Track indoor outdoor x 3 Shoe A B C Main Effects Interactions Track Indoor Outdoor Marginal Means A 12 50 17 00 14 75 Sneaker B 11 83 16 33 14 08 Track main effect averages across sneaker and subjects Shoe main effect averages across track and subjects Track x Shoe interaction averages across subjects C 12 17 11 33 11 75 Marginal Means 12 17 14 88 6 Multivariate Approach Calculates d scores that represent each ombibus effect Each effect is analyzed separately SS CP from full restricted models of relevant D scores Calculate determinants of full restricted models Enter determinants into 1 factor ANOVA We won t review the details of the matrix algebra Let s examine howD scores are formed for each effect 7 D Scores Contrasts D scores is a contrast applied to the data of each subject Contrasts apply weights to sample means to test hypotheses D scores are weights applied to data of each subject E g Sneaker Aindoor Sneaker Bindoor pairwise contrast in which Aindoor and Bindoor are weighted 1 1 respectively and all other running times are weighted 0 Subjec t 1 A Indoor B C 1 15 1 13 0 15 A Outdoor B C 0 20 0 18 0 10 Calculates D scores for each omnibus effect track main effect sneaker main effect Sneaker x Track interaction 8 D Score for Track Main Effect Track main effect averages across Sneaker Involves comparison of two groups Indoor vs Outdoor Need 1 d score compares indoor vs outdoor averaging across sneaker D Score for Track Main Effect Sneaker Track A B C Indoor 1 1 1 Outdoor 1 1 1 Apply weights to each subject to yield 1 d score per subject 9 Weights Applied to Each Subject Track Sub 1 2 3 4 5 6 Mean A 1 15 1 09 1 13 1 11 1 16 1 11 Indoor Sneaker B 1 13 1 08 1 12 1 11 1 16 1 11 C 1 15 1 08 1 13 1 10 1 16 1 11 A 1 20 1 12 1 18 1 16 1 21 1 15 Outdoor Sneaker B 1 18 1 14 1 19 1 13 1 19 1 15 C D Score 1 10 5 1 08 9 1 14 13 1 10 7 1 15 7 1 11 8 8 16 Track main effect involves 1 Dscore per subject 10 D Score for Sneaker Main Effect Sneaker main effect averages across track Involves comparison of three groups A B C Need 2 d scores to account for differences among A B C C vs mean of AB averaging across track A vs B averaging across track D1 Sneaker Main Effect Sneaker Track A B C Indoor 1 1 2 Outdoor 1 1 2 D2 Sneaker Main Effect Sneaker Track A B C Indoor 1 1 0 Outdoor 1 1 0 Applied separately to data of each sub to yield 2 D s per sub 11 D Score for Interaction Involves 6 groups BUT not 5 D scores Interaction tests whether sneaker effect changes across track vice versa Need d scores that test whether sneaker changes across track Not simple effects tests effect of sneaker in levels of track Interaction contrasts Do 1 C vs AB and 2 A vs B change across levels of track D1 Interaction Sneaker Track A B C Indoor 1 1 2 Outdoor 1 1 2 D2 Interaction Sneaker Track A B C Indoor 1 1 0 Outdoor 1 1 0 Applied separately to data of each sub to yield 2 D s per sub 12 Data for Multivariate Analysis Track x Track Sneaker Sneaker Subject D1 D1 D2 D1 D2 1 5 16 4 20 0 2 9 11 1 9 3 3 13 8 0 10 2 4 7 11 3 7 3 5 7 10 2 10 2 6 8 8 0 8 0 Mean 8 16 10 66 1 33 10 66 0 Use procedures of one factor multivariate ANOVA to analyze D scores of each effect 13 Follow Up Tests Significant effects with more than 2 levels require more tests Main effects test separate D scores among means for main effect use separate error term Interaction Simple effect tests follow up significant simple effect with d scores among the levels of the simple effect Interaction contrasts 14 How to in SAS Easiest way is with the repeated statement of proc GLM We used the repeated statement with the univariate approach and specified nom to suppress the multivariate results For multivariate results and no univariate results simply specify nouni option in the repeated statement We ll analyze the sneaker data 15 Data Structure for SAS data track input sub ia ib ic oa ob oc cards 1 15 13 15 20 18 10 2 9 8 8 12 14 8 3 13 12 13 18 19 14 4 11 11 10 16 13 10 5 16 16 16 21 19 15 6 11 11 11 15 15 11 16 SAS Commands for Omnibus Effects proc glm model ia ib ic oa ob oc nouni repeated track 2 shoe 3 nouni mean run Factor whose levels change least rapidly is listed first in repeated statement All 3 effects are significant 17 Follow up Tests for Sneaker Main Effect Can test any set of contrasts Assume track coach decided a priori to test 1 C vs AB 2 A vs B Each is tested with a separate error term Use MANOVA statement as we did previously proc glm model ia ib ic oa ob oc nouni repeated track 2 shoe 3 nouni mean manova h intercept m 1 1 2 1 1 2 mnames ab v c main manova h intercept m 1 1 0 1 1 0 mnames a v b main run 18 Follow up Tests for Interaction Simple effect of sneaker Conduct separate ANOVAs in levels of track Shoe simple effect in Indoor Track and contrasts proc glm model ia ib ic nouni repeated shoe 3 nouni manova h intercept m 1 1 2 …


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UT Knoxville STAT 201 - 13) sld_repeated_mulitvar_2factor

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