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) –“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 IVCovariate Total variance red + blue + green Variance explained by covariate blue Variance explained by IV green Error variance red + blue Adjusted error variance red 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

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

Error Terms ANOVAANCOVA

Controlling Confounds DV IVCovariate Total variance red + blue + green + yellow DV variance explained by covariate blue + yellow IV variance explained by covariate yellow + purple DV variance explained by IV green + yellow Corrected DV variance explained by IV green Error variance red + blue Adjusted error variance red Adjusted total variance red + green

Adjusting Group Means 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 Group means on DV, adjusted for covariate Grand mean of covariate Group means of covariate

Adjusting Group Means

Otherwise significant group differences can become insignificant

Otherwise significant group differences can stay significant

Otherwise significant group different can flip direction

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