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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?

Experiment: 3 instructional methods

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 .

Result…

R2 = 204.056/1600.306 = .128

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

- 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

- 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

Why is GPID now significant?

Parameter Estimates from SPSS

Compare to those from our MRA

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

- 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

- 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

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

Things to consider about covariates

- Number
- Reliability
- Pre-screening
- Multicollinearity
- Loss of df

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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