One-Way Analysis of Covariance

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One-Way Analysis of Covariance

• One-Way ANCOVA

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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
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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
  doesnt guarantee group equality

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ANCOVA

• Why would this be useful?
• Any DVs 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 youre using 2 DVs (MANOVA) and want to
  isolate the effects of one of them
• ANCOVA with the DV of interest and all other DVs
  used as covariates
• Note In this case were specifically predicting
  that IVs and covariates are related, its 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
• Dont 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