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Analysis of Covariance

Analysis of Covariance. Combines linear regression and ANOVA Can be used to compare g treatments, after controlling for quantitative factor believed to be related to response (e.g. pre-treatment score)

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Analysis of Covariance

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  1. Analysis of Covariance • Combines linear regression and ANOVA • Can be used to compare g treatments, after controlling for quantitative factor believed to be related to response (e.g. pre-treatment score) • Can be used to compare regression equations among g groups (e.g. common slopes and/or intercepts) • Model: (X quantitative, Z1,...,Zg-1dummy variables)

  2. Tests for Additive Model • Relation for group i (i=1,...,g-1): E(Y)=a+bX+bi • Relation for group g: E(Y)=a+bX • H0: b1=...=bg-1=0 (Controlling for covariate, no differences among treatments)

  3. Interaction Model • Regression slopes between Y and X are allowed to vary among groups • Group i (i=1,...,g-1): E(Y)=a+bX+bi+giX=(a+ bi)+(b +gi)X • Group g: E(Y)=a+bX • No interaction means common slopes: g1=...=gg-1=0

  4. Inference in ANCOVA • Model: • Construct 3 “sets” of independent variables: • {X} , {Z1,Z2,...,Zg-1}, {XZ1,...,XZg-1} • Fit Complete model, containing all 3 sets. • Obtain SSEC (or, equivalently RC2) and dfC • Fit Reduced, model containing {X} , {Z1,Z2,...,Zg-1} • Obtain SSER (or, equivalently RR2) and dfR • H0:g1=...=gg-1=0 (No interaction). Test Statistic:

  5. Inference in ANCOVA • Test for Group Differences, controlling for covariate • Fit Complete, model containing {X} , {Z1,Z2,...,Zg-1} • Obtain SSEC (or, equivalently RC2) and dfC • Fit Reduced, model containing {X} • Obtain SSER (or, equivalently RR2) and dfR • H0: b1=...=bg-1=0 (No group differences) Test Statistic:

  6. Inference in ANCOVA • Test for Effect of Covariate controlling for qualitative variable • H0:b=0 (No covariate effect) Test Statistic:

  7. Adjusted Means • Goal: Compare the g group means, after controlling for the covariate • Unadjusted Means: • Adjusted Means: Obtained by evaluating regression equation at • Comparing adjusted means (based on regression equation):

  8. Multiple Comparisons of Adjusted Means • Comparisons of each group with group g: • Comparisons among the other g-1 groups: • Variances and covariances are obtained from computer software packages (SPSS, SAS)

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