1 ANALYSIS OF COVARIANCE ANCOVA 2 ANCOVA Includes a continuous IV s in model w categorical IV s Eg May include math courses when testing sex differences in math ability math ability sex mathcourses The continuous variable is referred to as a covariate Can include as many covariates as desired 3 Potential Consequences of ANCOVA Increase power relative to ANOVA If 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 IV ANCOVA 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 4 Statistical Models of ANCOVA Full Yij j X ij ij Restriced Yij X ij ij 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 Model 5 Restriced Yij X ij ij Sex M M M F F F X 1 2 3 3 4 5 Y 4 9 8 12 11 16 Restricted model ignores SEX and predicts Y using only X Can solve for and with formulas to be discussed later and graph relation between Y and X Restricted Model Restriced Yij X ij ij Y 2 2 2 6 X i 6 Can graph predicted values of Y by plugging X scores into model 18 error Y Y 16 3rd male has X 3 Y 8 when X 3 Y 10 00 14 Y Error in model Y Y 12 10 8 6 4 2 Full Model 2 4 6 X 8 Full Yij j X ij ij Sex X Y M 1 4 2 7 M M F F F Full model using SEX Can solve for formulas to later and between Y males and 18 2 3 3 4 5 female 16 14 Y 12 male 10 8 9 8 12 11 16 predicts and X and with be discussed graph relation and X for females 6 Full Full Yij j X ij ij Y male 3 2 0 X i 4 Model 2 Y female 5 2 0 X i 2 4 6 X 8 8 Formulas for Slope and Y intercept Model Parameter slope y intercept 9 X j X Yij Y X Restricted j j ij X 2 i X Full ij i ij Y X X j Yij Y j i X j ij X j 2 i Y j X j 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 that j j F formula for the Treatment Effect 10 ER EF FA Error for each model EF a 1 N a 1 Y Y 2 dfdenominator loose 1 df for the covariate Power of ANCOVA vs ANOVA Model Full and Restricted Models for 1 factor ANOVA and ANCOVA ANOVA ANCOVA 11 Full Restricted Yij j ij Yij j X ij ij Yij ij Yij X ij ij 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 ANCOVA Error of ANCOVA vs ANOVA When covariate DV are related 12 1 factor ANOVA 1 factor ANCOVA 18 18 16 16 14 14 12 female Y10 12 Y 10 6 2 male 8 8 4 female 6 male 1 factor ANOVA 4 18 2 18 16 14 16 2 4 6 14 8 female X Lengt12 Y10 indicates amount of error 8 6 ANCOVA 12 2 4 6 Y10 8 X female h of vertical line 8 Error of vs ANOVA 6 male ANCOVA male When4 covariate DV are unrelated 2 2 4 6 8 X 4 2 2 4 6 8 X 13 Same SSerror however dferror ANCOVA N a 1 ANOVA N a ANCOVA less powerful Adjustment of Treatment Effect ANCOVA will adjust the treatment effect for group differences on the covariate 14 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 the covariate b Testing whether the group s differ in regard to the predicted i e adjusted means rather than the actual means Example of Adjustment Mean X Y as a Function of Sex Sex X Y 15 Male Female 2 4 7 13 Grand mean of X 3 If males and females had18the same X score i e 3 what female would be their predicted Y16scores Actual female mean Y male 3 2 0 X i 3 2 0 3 9 Y female 5 2 0 X i 5 2 0 3 11 14 ANCOVA tests whether the 12 adjusted means differ Predicted female mean Y male 10 Example8of Adjustment 6 4 2 X 2 4 X MX F 6 X 8 16 Adjustment of Treatment Effect Treatment Effect as a Function of the Covariate s Relation to the DV and IV Covariate Covariat DV e IV Effect on Magnitude of Treatment Effect unrelated unrelated related unrelated No effect No effect 17 related unrelated related related No effect decrease magnitude if group with larger score on DV has larger score on covariate increase magnitude if group with larger score on DV has smaller score on covariate increase magnitude if group with larger score on DV has larger score on covariate related related decrease magnitude if group with larger score on DV has smaller score on covariate Assumptions of ANOVA Assumptions can be phrased in regard to error ij Y scores for each population are normally distributed Y scores are independent 18 Population variances of Y are homogeneous Regression slope for the covariate is homogeneous across populations Problems 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 Effect 19 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 covariate Treatment Effect w Heterogeneous Slopes A Homogenous Slopes female B Heterogeneous Slopes female C Heterogeneous Slopes female male male male 20 Homogeneous slopes effect is constant at all values of X Heterogeneous slopes effect is different at all values of X ANCOVA in SAS Anxiety and depression are correlated Researcher assesses anxiety and randomly assigns Ps to 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