Special Topic:
Analysis of Covariance
(ANCOVA)
•
Uses of ANCOVA
•
Assumptions
•
Calculations (later)
•
Efficiency (later)
Why use ANCOVA?
 Measure a concomitant variable (covariate) to
control experimental error
 Examples
– Initial status of experimental units before treatments
are applied
• Plant height
• Nutrient status of plants or plots
– Events beyond our control that introduce error into the
experiment
• Stand losses not related to the treatments applied
– Performance of repeated check(s) in the experiment
– Trends in residuals over time or space
How is ANCOVA applied?
 Employs regression analysis within an ANOVA
 Means are adjusted to a common level of the
covariate(s)
 Treatments can be compared fairly
Linear Model for ANCOVA (for an RBD)
Yij    i   j    Xij  X..  ij
 = mean
ρi = ith block effect
j = jth treatment effect
β Xij − X. . = covariate effect
ij = random error
Means are adjusted to remove the covariate effect from error
Yij    Xij  X..    i   j  ij
ANCOVA assumptions
 The covariate(s) (X’s) are fixed, measured
without error, and independent of the treatments
 The regression of Y on X is linear
 Effects in the model are additive; the regression
of Y on X is independent of the block and
treatment effects
 Residuals are normally and independently
distributed with a mean of zero and common
variance