psy 1950 ancova november 17, 2008
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PSY 1950 ANCOVA November 17, 2008. Analysis of Covariance (ANCOVA). Covariate. An independent variable that… you do not manipulate annoys you rather than interests you covaries/correlates with dependent variable may or may not relate to IVs of interest Cohen (1968) - PowerPoint PPT PresentationTRANSCRIPT
PSY 1950ANCOVA
November 17, 2008
Analysis of Covariance (ANCOVA)
Covariate• An independent variable that…
– you do not manipulate– annoys you rather than interests you– covaries/correlates with dependent variable
– may or may not relate to IVs of interest
• Cohen (1968)– “A covariate is, after all, nothing but an independent variable which, because of the logic dictated by the substantive issues of the research, assumes priority among the set of independent variables for Y variance.”
• e.g., pretest scores
ANCOVA• ANCOVA always does two things:
– Reduces unexplained variance•Almost always a good thing•Only exception is when the reduction is so small that it is offset by the loss of df
– Adjust the group DV means based upon group differences on covariate•Okay in experimental designs•Very questionable in non-experimental designs – Occasionally, an arguably okay thing– Oftentimes, a definitely bad thing
Reducing Error Variance
DV
IV Covariate
Total variancered + blue + green
Variance explained by covariateblue
Variance explained by IVgreen
Error variancered + blue
Adjusted error variancered
Adjusted total variance red + green
Analysis of Variance• Categorical IVs
– e.g., Color (black, red, green), SAT (low, high)
• Separable effects– Including “blocking” or nuisance factor reduces error term, increases F for effect of interest
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Black Red Green
Color
# Analogies Correct
low high
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White Red Green
Color
# Analoges Correct
ANCOVA• Categorical IV and quantitative covariate– e.g., Color (black, red, green), SAT score
• Separable effects– Removing the effect of a covariate reduces error term, increasing F for effect of interest
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White Red Green
Color
# Analoges Correct
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0 200 400 600 800 1000
SAT Score
# Analogies Correct
y = 0.0204x - 0.8929
y = 0.0204x - 2.0357
y = 0.0204x - 3.0357
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SAT Score
# Analogies Correct
-1.5
-1
-0.5
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0.5
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1.5
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2.5
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0 200 400 600 800 1000
SAT Score
Residuals (# Analogies Correct)
SAT W R G W R G W R G200 2 0.5 2.5 2.04 1.04 3.19 -0.04 -0.54 -0.68300 4.5 4 5.5 4.08 3.08 5.23 0.43 0.93 0.29400 5.5 4.5 6.5 6.12 5.12 7.27 -0.61 -0.61 -0.75500 8.5 7.5 12 8.16 7.16 9.31 0.36 0.36 2.71600 10 9.5 10.5 10.20 9.20 11.35 -0.18 0.32 -0.82700 12 10.5 12.5 12.24 11.24 13.39 -0.21 -0.71 -0.86800 14.5 13.5 15.5 14.28 13.28 15.43 0.25 0.25 0.11
116.9 118.4 125.9 <------ SS ------> 0.8 2.3 9.9
Actual scores Predicted scores Residual scores
Error Terms
-1.5
-1
-0.5
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0.5
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SAT Score
Residuals (# Analogies Correct)
ANOVA ANCOVA
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SAT Score
# Analogies Correct
Controlling Confounds
DV
IV Covariate
Total variancered + blue + green + yellow
DV variance explained by covariateblue + yellow
IV variance explained by covariate yellow + purple
DV variance explained by IVgreen + yellow
Corrected DV variance explained by IVgreen
Error variancered + blue
Adjusted error variancered
Adjusted total variance red + green
Adjusting Group Means
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Problem: Any group differences on covariate will bias group differences on DV
Solution: Equate groups on covariate, and use regression to adjust DV accordingly
Adjusting Group Means
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Group means on DV, adjusted for covariateGrand mean of covariate
Group means of covariate
Adjusting Group Means
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Otherwise significant group differences can become
insignificant
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Otherwise significant group differences can stay
significant
Otherwise significant group different can flip direction
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Interpreting ANCOVA Results• If subjects are randomly assigned to groups, then…– Any preexisting group differences on covariates are due to chance•If covariate is measured after treatment, you’re in trouble
– ANCOVA will reduce error term and remove any bias due to random variations in group assignment
Interpreting ANCOVA Results• If subjects are not randomly assigned to groups, then…– Any group differences on covariates may not be due to chance
– Lord (1967): “…there is simply no logical or statistical procedure that can be counted on to make proper allowances for uncontrolled prexisting differences between groups [not due to random assignment]”
ANCOVA Assumptions• All the usual, plus homogeneity of regression slopes– In other words, the relationship between the DV and the covariate is the same across groups
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