ANCOVAWhat is ANCOVA?Analysis of covarianceSlide 4Basic requirementsSlide 6Slide 7CovariateApplicationsSlide 10Slide 11Slide 12Slide 13AssumptionsUnequal sample sizes, missing data, and number of casesSlide 16Slide 17Slide 18Slide 19Slide 20Slide 21OutliersNo Multicollinearity/SingularityNormality of Sampling DistributionHomogeneity of VarianceSlide 26LinearityHomogeneity of RegressionSlide 29Reliability of CovariatesEquationsEquationsSlide 33Slide 34Slide 35Slide 36Slide 37Slide 38Slide 39Evaluating CovariatesAlternatives to ANCOVAAlternatives to ANCOVAANCOVAANCOVALecture 9Lecture 9Andrew AinsworthAndrew AinsworthWhat is ANCOVA?What is ANCOVA?Analysis of covariance Analysis of covariance ► an extension of ANOVA in which main an extension of ANOVA in which main effects and interactions are assessed effects and interactions are assessed on DV scores after the DV has been on DV scores after the DV has been adjusted for by the DV’s relationship adjusted for by the DV’s relationship with one or more Covariates (CVs)with one or more Covariates (CVs)ANCOVAANCOVA►In MR we saw this as the relationship In MR we saw this as the relationship between X1 and Y while controlling for between X1 and Y while controlling for other Xsother Xs1 1 2 2'j jy x x xb b b= + + +KBasic requirementsBasic requirements1 DV (I, R)1 DV (I, R)1 IV (N, O)1 IV (N, O)1 CV (I, R)1 CV (I, R)Basic requirementsBasic requirements►Minimum number of CVs that are Minimum number of CVs that are uncorrelated with each otheruncorrelated with each other►You want a lot of adjustment with minimum You want a lot of adjustment with minimum loss of degrees of freedomloss of degrees of freedom►The change in sums of squares needs to The change in sums of squares needs to greater than a change associated with a greater than a change associated with a single degree of freedom lost for the CVsingle degree of freedom lost for the CVBasic requirementsBasic requirements►CVs should also be uncorrelated with CVs should also be uncorrelated with the IVs (e.g. the CV should be the IVs (e.g. the CV should be collected before treatment is given) in collected before treatment is given) in order to avoid diminishing the order to avoid diminishing the relationship between the IV(s) and DV.relationship between the IV(s) and DV.CovariateCovariate►A covariate is a variable that is related A covariate is a variable that is related to the DV, which you can’t manipulate, to the DV, which you can’t manipulate, but you want to account for it in you but you want to account for it in you relationship.relationship.ApplicationsApplications►Three major applicationsThree major applicationsIncrease test sensitivity (main Increase test sensitivity (main effects and interactions) by effects and interactions) by using the CV(s) to account for using the CV(s) to account for more of the error variance more of the error variance therefore making the error term therefore making the error term smallersmallerApplicationsApplications►Adjust DV scores to what they would Adjust DV scores to what they would be if everyone scored the same on the be if everyone scored the same on the CV(s)CV(s)This second application is used often in This second application is used often in non-experimental situations where non-experimental situations where subjects cannot be randomly assignedsubjects cannot be randomly assignedApplicationsApplicationsSubjects cannot be made equal through Subjects cannot be made equal through random assignment so CVs are used to random assignment so CVs are used to adjust scores and make subjects more adjust scores and make subjects more similar than without the CVsimilar than without the CVThis second approach is often used as a This second approach is often used as a way to improve on poor research designs. way to improve on poor research designs. This should be seen as simple descriptive This should be seen as simple descriptive model building with no causalitymodel building with no causalityApplicationsApplicationsRealize that using CVs can adjust DV Realize that using CVs can adjust DV scores and show a larger effect or the CV scores and show a larger effect or the CV can eliminate the effectcan eliminate the effectApplicationsApplications►The third application will be addressed The third application will be addressed later in MANOVA, but is the adjustment later in MANOVA, but is the adjustment of a DV for other DVs taken as CVs.of a DV for other DVs taken as CVs.AssumptionsAssumptionsUnequal sample sizes, missing Unequal sample sizes, missing data, and number of casesdata, and number of cases►Missing data and unequal sample sizes Missing data and unequal sample sizes can be two different entities or seen as can be two different entities or seen as the same ideathe same ideaIf data was collected with equal samples If data was collected with equal samples sizes planned and there is data missing on sizes planned and there is data missing on the CV or DV then this can be seen as a the CV or DV then this can be seen as a missing data problemmissing data problemUnequal sample sizes, missing Unequal sample sizes, missing data, and number of casesdata, and number of cases►If data is unequal because of some If data is unequal because of some reason (e.g. larger population of reason (e.g. larger population of certain type of subject) then this isn’t certain type of subject) then this isn’t missing data and needs to be dealt missing data and needs to be dealt with appropriatelywith appropriatelyUnequal sample sizes, missing Unequal sample sizes, missing data, and number of casesdata, and number of cases►The problem here is that with unequal samples The problem here is that with unequal samples it is unclear how to calculate the marginal it is unclear how to calculate the marginal mean. mean. Is it the mean of the group means or the mean of the Is it the mean of the group means or the mean of the scores?scores?►Another problem is that the variances then Another problem is that the variances then start to overlap one another forcing the within start to overlap one another forcing the within plus between variances to be larger than the plus between variances to be larger than the total variance.total variance.Unequal sample sizes, missing Unequal sample sizes, missing data, and number of casesdata, and number of cases►Listwise deletion to balance cells is an Listwise deletion to balance cells is