1-FACTOR ANOVASource DF SS MS F Pr > FModel 3 9.080 3.0267 12.55 0.00071ANALYSIS OF COVARIANCE(ANCOVA)2ANCOVA-Includes a continuous IV(s) in model w/ categorical IV(s)Eg.,May include # math courses when testing sex differences in math abilitymath_ability = sex mathcourses-The continuous variable is referred to as a covariate-Can include as many covariates as desired3Potential Consequences of ANCOVA-Increase power relative to ANOVAIf the covariate is related to the DV, inclusion of the covariate reduces SSerror and typically increases the power of the test of the categorical variable-Adjusts differences between groups on the DV for the relation between the covariate(s) and IVANCOVA asks “Would the groups differ on the DV if they were equivalent on the covariate?”Eg., Would males and females differ in math ability if they had the same experience with math (e.g., same number of math courses)?4Statistical Models of ANCOVA ijijjijXY :FullijijijXY :Restriced-Restricted Model -- (slope) indicates amount by which Y changes with change in X -- (y-intercept) indicates the point at which the line relating Y to X crosses the X-axis (i.e., value of Y when X = 0).--i error -Full Model--,-, & -i--j effect parameter indicating extent to which levels of categorical variable deviate from grand mean (y-intercept)Restricted Model5ijijijXY :RestricedSex X YM 1 4M 2 9M 3 8F 3 12F 4 11F 5 16-Restricted model ignores SEX and predicts Y using only XCan solve for - and - (with formulas to be discussed later) and graph relation between Y and X.Restricted ModelijijijXY :RestricediXY 6.22.2ˆ6-Can graph predicted values of Y by plugging X scores into modelFull ModelijijjijXY :FullSex X YM 1 4 22468 10 12 14 16 18 4 6 8YXYYerror ˆ3rd male has X =3 & Y = 8when X = 3 00.10ˆYError in model = 2)ˆ( YY7M 2 9M 3 8F 3 12F 4 11F 5 16-Full model predicts & using SEX and X Can solve for - and - (withformulas to be discussed later) and graph relationbetween Y and X for males and femalesFull ModelijijjijXY :FullimaleXY 0.23ˆifemaleXY 0.25ˆmale 22468 10 12 14 16 18 4 6 8YXfemale8Formulas for Slope and Y-interceptParameterModel - (slope) - (y-intercept)9Restrictedj iijj iijijXXYYXX2)())((XYˆFullj ijijj ijijjijXXYYXX2)())((jjXYˆ -Restricted model ignores grouping variable & treats as one sample-Full model-slope is a pooled (i.e., averages slopes for each group)-Y-intercept can be calculated for each group keeping in mind thatjjF-formula for the Treatment Effect10)1()1()(aNEaEEFFFRA-Error for each model: 2)ˆ( YY-dfdenominator loose 1 df for the covariatePower of ANCOVA vs ANOVAFull and Restricted Models for 1-factor ANOVA and ANCOVAModel ANOVA ANCOVA11FullijjijY ijijjijXY RestrictedijijY ijijijXY -ANOVA and ANCOVA differ in terms of the covariate-ANCOVA will be more powerful to the extent to which covariate & DV are related-when covariate & DV are related, ANCOVA will have smaller SSerror -if decrease in error outweighs increase in complexity (i.e., decreased dfdenominator) MSerror will be smaller for ANCOVAError of ANCOVA vs ANOVA-When covariate & DV are related22468 10 12 14 16 18 4 6 8YXmalefemale1-factor ANOVA 1-factor ANCOVAmale 22468 10 12 14 16 18 4 6 8YXfemale12-Lengt h of vertical line indicates amount of errorError of ANCOVA vs ANOVA-When covariate & DV are unrelated 22468 10 12 14 16 18 4 6 8YXmalefemale1-factor ANOVA 22468 10 12 14 16 18 4 6 8YXmalefemaleANCOVA13-Same SSerror-however, dferror ANCOVA= N-a-1 & ANOVA = N-a-ANCOVA less powerfulAdjustment of Treatment Effect-ANCOVA will adjust the treatment effect for group differences on the covariate14-”If groups were equal on covariate, would they differ on the DV?”-ANCOVA adjusts by:(a) Predicting the mean for each group on the DV at the grand mean of thecovariate(b) Testing whether the group’s differ in regard to the predicted (i.e., adjusted) means rather than the actual meansExample of AdjustmentMean X & Y as a Function of SexSex X Y15Male 2 7Female 4 13-Grand mean of X = 3-If males and females had the same X score (i.e., 3), what would be their predicted Y scores?9)3(0.230.23ˆimaleXY11)3(0.250.25ˆifemaleXY-ANCOVA tests whether the adjusted means differExample of Adjustmentmale 22468 10 12 14 16 18 4 6 8YXfemaleActual female meanPredicted female meanMXFXX16Adjustment of Treatment EffectTreatment Effect as a Function of the Covariate’s Relation to the DV and IVCovariate& DVCovariate & IV Effect on Magnitude of Treatment Effectunrelated related -No effectunrelated unrelated -No effect17related unrelated -No effect+ related related-decrease magnitude if group with larger score on DV has largerscore on covariate-increase magnitude if group with larger score on DV has smaller score on covariate- related related-increase magnitude if group with larger score on DV has larger score on covariate-decrease magnitude if group with larger score on DV has smaller score on covariateAssumptions of ANOVAAssumptions can be phrased in regard to error (ij)-Y-scores for each population are normally distributed-Y-scores are independent18-Population variances of Y are homogeneous-Regression slope for the covariate is homogeneous across populationsProblems via Heterogeneous Slopes-Estimating population slope from the full model -full model uses a pooled slope-when slopes in the population are heterogeneous it does not make sense to pool sample slopes to estimate the population slope.-Interpreting Adjusted Treatment Effect19-when slopes are homogeneous, the adjusted treatment effect is constant across values of the covariate-when slopes are heterogeneous, the adjusted treatment effect is different across values of the covariateTreatment Effect w/ Heterogeneous Slopesmalefemale(A) Homogenous Slopesmalefemale(B) Heterogeneous Slopesmalefemale(C) Heterogeneous Slopes20-Homogeneous slopes: effect is constant at all values of X-Heterogeneous slopes: effect is different at all values of XANCOVA in SAS-Anxiety and depression are
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