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UT Knoxville STAT 201 - 13) sld_repeated_mulitvar_2factor

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Follow-up for B/w-sub Main EffectFollow-up for Within-sub Main EffectFollow-up for the InteractionSimple Effect of B/w-SubSimple Effect of Within-Sub1Factorial Designs With Repeated Measures:The Multivariate Approach2Rationale 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 variability3Organization 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 SAS4Factorial Designs: 2 Within-Sub VariablesE.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 trackTrackIndoor OutdoorSneaker SneakerSubjectA B C A B C1 15 13 15 20 18 102 09 08 08 12 14 083 13 12 13 18 19 144 11 11 10 16 13 105 16 16 16 21 19 156 11 11 11 15 15 1136 observations from 6 subjects52(Track: indoor, outdoor) x 3(Shoe: A, B, C)Main Effects & InteractionsSneakerTrack A B C Marginal MeansIndoor 12.50 11.83 12.17 12.17Outdoor 17.00 16.33 11.33 14.88Marginal Means 14.75 14.08 11.75-Track main effect-averages across sneaker and subjects-Shoe main effect-averages across track and subjects-Track x Shoe interaction-averages across subjects6Multivariate 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 effect7D-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 subjectE.g. Sneaker Aindoor – Sneaker Bindoorpairwise contrast in which Aindoor and Bindoor are weighted +1 & -1, respectively, and all other running times are weighted 0.Indoor OutdoorSubjectA B C A B C1 (1) 15 (-1) 13 (0) 15 (0) 20 (0) 18 (0) 10- Calculates D-scores for each omnibus effecttrack main effectsneaker main effectSneaker x Track interaction8D-Score for Track Main Effect-Track main effect averages across Sneaker-Involves comparison of two groups: Indoor vs Outdoor-Need 1 d-scorecompares indoor vs outdoor averaging across sneakerD-Score for Track Main EffectSneakerTrack A B CIndoor 1 1 1Outdoor -1 -1 -1- Apply weights to each subject to yield 1 d-score per subject9Weights Applied to Each SubjectTrackIndoor OutdoorSneaker SneakerSub A B C A B C D-Score1 (1) 15 (1) 13 (1) 15 (-1) 20 (-1) 18 (-1) 10 -52 (1) 09 (1) 08 (1) 08 (-1) 12 (-1) 14 (-1) 08 -93 (1) 13 (1) 12 (1) 13 (-1) 18 (-1) 19 (-1) 14 -134 (1) 11 (1) 11 (1) 10 (-1) 16 (-1) 13 (-1) 10 -75 (1) 16 (1) 16 (1) 16 (-1) 21 (-1) 19 (-1) 15 -76 (1) 11 (1) 11 (1) 11 (-1) 15 (-1) 15 (-1) 11 -8Mean - - - - - - -8.16-Track main effect involves 1-Dscore per subject10D-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 D2 Sneaker Main EffectSneaker SneakerTrack A B C Track A B CIndoor 1 1 -2 Indoor 1 -1 0Outdoor 1 1 -2 Outdoor 1 -1 0-Applied separately to data of each sub to yield 2 D’s per sub11D-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 D2 InteractionSneaker SneakerTrack A B C Track A B CIndoor 1 1 -2 Indoor 1 -1 0Outdoor -1 -1 2 Outdoor -1 1 0-Applied separately to data of each sub to yield 2 D’s per sub12Data for Multivariate AnalysisTrack SneakerTrack xSneakerSubject D1D1D2D1D21 -5 16 4 -20 02 -9 11 -1 -9 33 -13 8 0 -10 24 -7 11 3 -7 -35 -7 10 2 -10 -26 -8 8 0 -8 0Mean -8.16 10.66 1.33 -10.66 0-Use procedures of one-factor multivariate ANOVA to analyze D-scores of each effect13Follow-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 testsfollow-up significant simple effect with d-scores among the levels of the simple effect-Interaction contrasts14“How to” in SAS-Easiest way is with the repeated statement of proc GLM-We used the repeated statement with the univariate approachand 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…15Data Structure for SASdata track;input sub ia ib ic oa ob oc;cards;1 15 13 15 20 18 102 9 8 8 12 14 83 13 12 13 18 19 144 11 11 10 16 13 105 16 16 16 21 19 156 11 11 11 15 15 11;16SAS Commands for Omnibus Effectsproc 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 significant17Follow-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;18Follow-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) mnames =ab_v_c; manova h=intercept m=(1 -1 0) mnames =a_v_b; run;*Shoe simple effect in Outdoor Track and contrasts;proc glm;model oa ob oc = /nouni;repeated shoe 3 / nouni;manova h=intercept m=(1 1 -2) mnames =ab_v_c;19manova h=intercept m=(1 -1 0) mnames =a_v_b;


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UT Knoxville STAT 201 - 13) sld_repeated_mulitvar_2factor

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