Download to read offline
More Related Content
Manova
- 1.
- 2. Purpose To analyzewhether a combination of the DVs varies as a function of IVs Multiple DVs
- 3. Advantage of MANOVA Improve the chance of discovering what it is that changes as a result of different IVs and their interaction ↓type 1 error Reveal difference not show in separate ANOVA
- 4. Kinds of researchquestions The goal of MANOVA is to discover whether DVs is changed by manipulation Main effect of IVs Interaction among IVs Importance of DVs Parameter estimates (marginal, cell and adjusted mean) Specific comparisons and trend analysis Effect size Effects of covariates Repeated-measures analysis of variance
- 5. Limitations to factoranalysis Theoretical issues Causality Choice of variables generalizability Practical issues Unequal sample sizes, missing data and power (relations among DVs) Multivariate normality Absence of outliers Homogeneity of variance-covariance matrices (Box’s M) Linearity Homogeneity of regression (covariate; MANCOVA) Reliability of covariance Absence of multicollinearity and singularity
- 6. Fundamental equation forMANOVA ANOVA to MANOVA 𝑆𝑆𝑡𝑎𝑡𝑎𝑙 = 𝑆𝑆 𝑏𝑔 + 𝑆𝑆 𝑤𝑔 Y11mean level1 Y1∙1∙ 1 Y1∙1∙ 2 Y1∙1∙ 3 Y1∙1∙ 4 Y1∙1∙ 5 Y1∙1∙ 6 Y1∙1∙ 7 Y12mean level2 Y1∙2∙ 1 Y1∙2∙ 2 Y1∙2∙ 3 Y1∙2∙ 4 Y1∙2∙ 5 Y1∙2∙ 6 Y1∙2∙ 7 Y13mean level3 Y1∙3∙ 1 Y1∙3∙ 2 Y1∙3∙ 3 Y1∙3∙ 4 Y1∙3∙ 5 Y1∙3∙ 6 Y1∙3∙ 7 Y21mean level1 Y2∙1∙ 1 Y2∙1∙ 2 Y2∙1∙ 3 Y2∙1∙ 4 Y2∙1∙ 5 Y2∙1∙ 6 Y2∙1∙ 7 Y22mean level2 Y2∙2∙ 1 Y2∙2∙ 2 Y2∙2∙ 3 Y2∙2∙ 4 Y2∙2∙ 5 Y2∙2∙ 6 Y2∙2∙ 7 Y23mean level3 Y2∙3∙ 1 Y2∙3∙ 2 Y2∙3∙ 3 Y2∙3∙ 4 Y2∙3∙ 5 Y2∙3∙ 6 Y2∙3∙ 7 Y31mean level1 Y3∙1∙ 1 Y3∙1∙ 2 Y3∙1∙ 3 Y3∙1∙ 4 Y3∙1∙ 5 Y3∙1∙ 6 Y3∙1∙ 7 Y32mean level2 Y3∙2∙ 1 Y3∙2∙ 2 Y3∙2∙ 3 Y3∙2∙ 4 Y3∙2∙ 5 Y3∙2∙ 6 Y3∙2∙ 7 Y33mean level3 Y3∙3∙ 1 Y3∙3∙ 2 Y3∙3∙ 3 Y3∙3∙ 4 Y3∙3∙ 5 Y3∙3∙ 6 Y3∙3∙ 7
- 7. Fundamental equation forMANOVA Variance 𝑖 𝑗 𝑌𝑖𝑗 − 𝐺𝑀 2 = 𝑛 𝑗 𝑌𝑗 − 𝐺𝑀 2 + 𝑖 𝑗 𝑌𝑖𝑗 − 𝑌𝑗 2 𝑆𝑆𝑡𝑎𝑡𝑎𝑙 = 𝑆𝑆 𝑏𝑔 + 𝑆𝑆 𝑤𝑔
- 8. Fundamental equation forMANOVA > one IV 𝑛 𝑘𝑚 𝑘 𝑚 𝐷𝑇𝑘𝑚 − 𝐺𝑀 𝑘𝑚 2 = 𝑛 𝑘 𝑘 𝐷 𝑘 − 𝐺𝑀 2 + 𝑛 𝑚 𝑚 𝑇 𝑚 − 𝐺𝑀 2 + 𝑛 𝑘𝑚 𝑘 𝑚 𝐷𝑇𝑘𝑚 − 𝐺𝑀 2 − 𝑛 𝑘 𝑘 𝐷 𝑘 − 𝐺𝑀 2 − 𝑛 𝑚 𝑚 𝑇 𝑀 − 𝐺𝑀 2 𝑆𝑆 𝑏𝑔 = 𝑆𝑆 𝐷 + 𝑆𝑆 𝑇+𝑺𝑺 𝑫𝑻 Main effect Interaction
- 9. Fundamental equation forMANOVA Λ = 𝑆 𝑒𝑟𝑟𝑜𝑟 𝑆 𝑒𝑓𝑓𝑒𝑐𝑡 + 𝑆 𝑒𝑟𝑟𝑜𝑟 Approximate F = 𝑑𝑓1, 𝑑𝑓2 = 1−𝑦 𝑦 𝑑𝑓2 𝑑𝑓1 𝑦 = Λ 1 𝑆; 𝑠 = 𝑚𝑖𝑛 𝑝, 𝑑𝑓𝑒𝑓𝑓𝑒𝑐𝑡 𝑑𝑓1 = 𝑝 𝑑𝑓𝑒𝑓𝑓𝑒𝑐𝑡 𝑑𝑓2 = 𝑠 𝑑𝑓𝑒𝑓𝑓𝑐𝑡 − 𝑝 − 𝑑𝑓𝑒𝑓𝑓𝑒𝑐𝑡 + 1 2 − 𝑝 𝑑𝑓𝑒𝑓𝑓𝑒𝑐𝑡 − 2 2
- 10. Effect size 𝑝𝑎𝑟𝑡𝑖𝑎𝑙 η2 =1 − Λ 1 𝑠 η2 = 1 − Λ
- 11. Discriminant function Specialcase of canonical analysis χ1 χ2 X1 X2 X3 X4 X5 η1 η2 D1 D2 𝑟𝑐2 𝑟𝑐3 0 0
- 12. MANOVA Special caseof canonical analysis η1 η2 Y1 Y2 Y3 Y4 Y5 χ1 χ2 D1 D2 𝑟𝑐2 𝑟𝑐3 0 0
- 13. Some important issue MANOVA vs. ANOVA MANOVA rDVs = -.6 ANOVA Highly postively correlated DVs Uncorrelated DVs
- 14. Some important issue Criteria for statistical inference Wilk’s lambda Error variance / (effect variance + error variance) Hotelling’s trace Effect variance / error variance Pillai’s criterion Pooled effect variance (general effect size) Roy’s gcr (Effect variance)1∙23…n / error variance
- 15. Some important issue Assessing DVs Univariate F with Bonferroni correction (rDVs = 0) Roy-Bargmann stepdown analysis (rDVs ≠ 0) Discriminant analysis (structure loading) Specific comparisons and trend analysis Mathematical resolution of combining variables Design complexity Within-subjects and between-within design Unequal sample size
- 16. rDVs ≠ 0 Correlated DVs measure overlapping aspects of same behavior Inflation of type 1 error IV DV 1 DV 2
- 17. Roy-Bargmann stepdown analysis A series of ANCOVA Assessing DV1 DV1 as covariate, assessing DV2 DV1 , DV2 as covariate, assessing DV3 However, Order Homogeneity of regression
- 18. Homogeneity of regression 0 1 2 3 4 5 6 con exp Covariatevariable DV Heterogeneity → Johnson-Neyman correction
- 19. Procedure 1. Research question 2.Research design 3. Check the assumptions 4. Estimation of the MANOVA model and overall fit 5. Interpreting the effect of variables 6. Validating results
- 20.
- 21.
- 22.