ANCOVA: Analysis of Covariance

Last updated 10/31/01

-design w/ both qualitative +quantitative predictor variables

-combines ANOVA + regression techniques

y =response variable (quantitative or continuous)

x =covariate (quantitative or continuous)

trt’s=independent variables of interest (qualitative or discrete)

ANCOVA applications

1. ­ precision in randomized experiments (even though you randomized it is possible that individuals in one trt group are a little older, little larger, etc.  By using age or size as a covariate, you can adjust for this.  In so doing, error MS is reduced (variation is associated with covariate, not error) and test becomes more sensitive).

2. adjustment of trt means

-group means of y can be adjusted to common value of x

-produces equitable comparisons among groups

Other related approaches:

-Y/X ratio assumes Y=bX and y-intercept of 0 is assumed (not always the case).  Also, relationship may not be linear.

-Y-X assumes Y=a+1X (relationship is not always linear)

ANCOVA makes no assumption about the relationship of Y & X.

 

3. examine heterogeneity among slopes (are slopes the same?  are intercepts the same? is the relationship b/w y and x the same for both trt groups?)

4. examine other influential or confounding variables (are differences between groups linked to differences in another factor).

ANCOVA assumptions

-covariate measured w/out error +under the control of investigator

 

-homoscedasticity

 

-normally distributed residuals

 

-random sampling

 

-homogeneity of slopes

Why homogeneity of slopes?

Y_ij = u + t_i + beta (x_ij-x_j) +error _ij

Y_ij = value of y in ith trt group + jth level of covariate

u = grand mean

t_i = effect of trt group i

beta= slope (same for all groups)

(x_ij-x_I) = difference b/w the value of x for Y_ij and the avg value of x in the ith group

error= random deviation