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Analysis of Covariance

Analysis of Covariance. 46-512: Statistics for Graduate Study in Psychology. Learning Outcomes. What is an ANCOVA? How does it relate to what we have done already? When would we use it? What are the issues & assumptions? What are some limitations and alternatives?.

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Analysis of Covariance

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  1. Analysis of Covariance 46-512: Statistics for Graduate Study in Psychology

  2. Learning Outcomes • What is an ANCOVA? • How does it relate to what we have done already? • When would we use it? • What are the issues & assumptions? • What are some limitations and alternatives?

  3. Experiment: 3 instructional methods

  4. Treat group 3 as control: DC1 identifies Group 1 DC2 identifies Group 2 First, let’s run it as an MRA… compute dc1=0. compute dc2=0. if (gpid=1) dc1=1. if (gpid=2) dc2=1. execute. REGRESSION /DESCRIPTIVES MEAN STDDEV CORR SIG N /MISSING LISTWISE /STATISTICS COEFF OUTS CI R ANOVA /CRITERIA=PIN(.05) POUT(.10) /NOORIGIN /DEPENDENT y /METHOD=ENTER dc1 dc2 .

  5. Result… R2 = 204.056/1600.306 = .128

  6. Which agrees with GLM

  7. Back to MRA Enter our continuous variable (IQ) Sans interaction term for the time being. R2 = .527

  8. What have we done? • Analysis of Covariance • What does it tell us? • In general, why would we use this technique? • 1) • 2) • 3)

  9. Examples of different applications • Elimination of systematic bias • The relationship between questionnaire responses and business performance, controlling for pre-existing differences in business performance. • Reduce Error Variance • In a random assignment experiment, looking at vigilance and using age as a covariate • Step-down Analysis • Studying the effects of an educational intervention on performance & self-esteem.

  10. Effects & Extensions • Types of Effects • Significance of Covariate(s) • Main Effects • Interactions among Factors • Interactions between factors and covariate(s) = bad news. • Extensions • Can have multiple covariates • Factorial Designs • Mixed Randomized by Repeated Designs • Within Subjects Designs

  11. Back to our example (as one-way)

  12. Run through GLM as ANCOVA Why is GPID now significant?

  13. Means and Adjusted Means Adjusted Means calculated as… For Group 1…

  14. Parameter Estimates from SPSS Compare to those from our MRA

  15. Post-Hocs from SPSS

  16. Bryant-Paulson Post Hoc BPcrit = 3.55, Cell 3 is significantly higher than 1 & 2 Bryant-Paulson is an extension of Tukey’s Post-Hoc test, and more appropriate if X is random.

  17. ANCOVA & Intact Groups • Groups can still differ in unknown ways. • Question whether groups that are equivalent on the covariate ever exist – since ANCOVA adjusts for equivalence on the covariate. • Assumptions of linearity and homogeneity of regression slopes need to be satisfied. • Differential growth of subjects i.e., is difference due to treatment or differential growth? • Measurement error can produce spurious results.

  18. Assumptions of ANCOVA • Larger sample sizes (because of the regression of the DV on the CV) • Absence of Multicollinearity and Singularity • Normality of sampling distributions (of the means) • Homogeneity of Variance • Linearity – of relationship between covariate and dependent variable • Homogeneity of regression • Reliability of covariates

  19. Alternatives • In pre-post situations, using difference scores (assuming same metric) • Controversial and carries some risk • Incorporating pre-scores into a RM ANOVA design. • Residualize DV and run an ANOVA on the residualized scores. • Controversial, not a very popular approach • Blocking (rather than tackling!) • assigning/matching people based on pre-scores or creating appropriate IV categories of intact groups. • Utilizing the CV as a factor in the experiment, if it lends itself well to categorization. • This side-steps many issues, such as homogeneity of regression. • Johnson-Neyman technique • See Stevens (1999) for an alternative

  20. Things to consider about covariates • Number • Reliability • Pre-screening • Multicollinearity • Loss of df

  21. More complicated designs • More than one covariate • Factorial Designs • Repeated Measures Designs For now, we will suspend discussion of more complicated designs, but revisit when we cover MANOVA and MANCOVA

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