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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

Analysis of Covariance

46-512: Statistics for Graduate Study in Psychology

learning outcomes
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?
first let s run it as an mra
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.









/METHOD=ENTER dc1 dc2 .


R2 = 204.056/1600.306 = .128

back to mra
Back to MRA

Enter our continuous variable (IQ)

Sans interaction term for the time being.

R2 = .527

what have we done
What have we done?
  • Analysis of Covariance
  • What does it tell us?
  • In general, why would we use this technique?
    • 1)
    • 2)
    • 3)
examples of different applications
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.
effects extensions
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
run through glm as ancova
Run through GLM as ANCOVA

Why is GPID now significant?

means and adjusted means
Means and Adjusted Means

Adjusted Means calculated as…

For Group 1…

parameter estimates from spss
Parameter Estimates from SPSS

Compare to those from our MRA

bryant paulson post hoc
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.

ancova intact groups
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.
assumptions of ancova
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
  • 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
things to consider about covariates
Things to consider about covariates
  • Number
  • Reliability
  • Pre-screening
  • Multicollinearity
  • Loss of df
more complicated designs
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