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One-Way Analysis of Covariance. One-Way ANCOVA. ANCOVA. Allows you to compare mean differences in 1 or more groups with 2+ levels (just like a regular ANOVA), while removing variance from a 3 rd variable What does this mean?. ANCOVA. ANCOVA.

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ancova
ANCOVA
  • Allows you to compare mean differences in 1 or more groups with 2+ levels (just like a regular ANOVA), while removing variance from a 3rd variable
    • What does this mean?
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ANCOVA
  • Removing variance that is unrelated to the IV/intervention = removing error variance
    • Makes ANCOVA potentially a very powerful test (i.e. easier to find significant results than with ANOVA alone) by potentially reducing MSerror
    • Generally, the more strongly related are covariate and DV, and unrelated the covariate and IV, the more useful (statistically) the covariate will be in reducing MSerror
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ANCOVA
  • Why would this be useful?
    • Any longitudinal research design needs to control for T1 differences in the DV
      • I.e. If assessing change in symptoms of social anxiety over time between 2 groups, we need to control for group differences in T1 social anxiety
      • Even if random assignment is used, use of a covariate is a good idea – Random assignment doesn’t guarantee group equality
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ANCOVA
  • Why would this be useful?
    • Any DV’s with poor discriminant validity
      • I.e. SES and race are highly related – If we wanted to study the effects of SES, independent of race, on scholastic achievement we could use an ANCOVA using SES as the DV and race as a covariate
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ANCOVA
  • Why would this be useful?
    • If you’re using 2+ DV’s (MANOVA) and want to isolate the effects of one of them
      • ANCOVA with the DV of interest and all other DV’s used as covariates
      • Note: In this case we’re specifically predicting that IV’s and covariates are related, it’s not ideal, but what can you do?
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ANCOVA
  • However, ANCOVA should not be used as a substitute for good research design
    • If your groups are unequal on some 3rd variable, these differences are still a plausible rival hypothesis to your H1, with or without ANCOVA
    • Controlling ≠ Equalizing
    • Random assignment to groups still best way to ensure groups are equal on all variables
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ANCOVA
  • Also, covariates change the meaning of your DV
    • I.e. We studying the effects of a tutoring intervention for student athletes – We find out our Tx group is younger than our control group – (Using age as a covariate)  (DV = class performance – age)
    • What does this new DV mean??? Effects of Tx over and above age (???)
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ANCOVA
  • Also, covariates change the meaning of your DV
    • For this reason, DO NOT just add covariates thinking it will help you find sig. results
      • Adding a covariate highly correlated with a pre-existing covariate actually makes ANCOVA less powerful
        • df decreases slightly with each covariate
        • No increase in power since 2 covariates remove same variance due to high correlation
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ANCOVA
  • Assumptions:
    • Normality
    • Homoscedasticity
    • Independence of Observations
    • Relationship between covariate and DV
    • Relationship between IV and covariate is linear
    • Relationship between IV and covariate is equal across levels of IV
      • AKA Homogeniety of Regression Slopes
      • I.e. an interaction between IV and CV
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ANCOVA
  • Calculations
    • Don’t worry about them, in fact, you can skip pp. 577-585 in the text
    • Recall that in the one-way ANOVA we divided the total variance (SStotal) into variance attributable to our IV (SStreat) and not attributable to our IV (SSerror)
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ANCOVA
  • In ANCOVA, we just divide the variance once more (for the covariate)
    • IV: Inferences are made re: its effects on the DV by systematically separating its variance from everything else
    • Covariate: Inferences are made by separating its variance from everything else, however this separated variance is not investigated in-and-of itself