Analysis of Covariance KNNL Chapter 22 Analysis of

Analysis of Covariance KNNL – Chapter 22

Analysis of Covariance • Goal: To Compare treatments (1 -Factor or Multiple Factors) after Controlling for Numeric Predictor(s) that is (are) related to response • Makes use of Multiple Linear Regression Model with numeric and categorical predictors • Covariates (aka Concomitant Variables) can not be effected by the treatments assigned to units (often covariate is pre-treatment or baseline score) • Purpose is to reduce experimental error when it is large • Alternative to blocking: uses fewer degrees of freedom, and can be measured after trt assignment

Single Factor Model with 1 Covariate

Additive Model – Homogeneity of Slopes Analysis of Covariance - Additive Model 160 140 120 Y 100 mu_1 80 mu_2 mu_3 60 mu_4 40 20 0 0 2 4 6 8 10 X 12 14 16 18 20

Interaction Model – Heterogeneity of Slopes Analysis of Covariance - Interaction Model 140 120 100 80 Y mu_1 mu_2 60 mu_3 mu_4 40 20 0 0 2 4 6 8 10 X 12 14 16 18 20

Model Generalizations • Random Xij - Model is treated as conditional of observed values of X • Nonlinear relation between Response and Covariate – Include linear and quadratic centered X values • More than one covariate – No problem extending to multiple covariates • More than one treatment factor – No problem having multiple factors

Regression Model for 1 -Way ANCOVA

Comparing (Adjusted) Treatment Means

Testing for Common Slopes