This document provides an overview of group difference methods, specifically analysis of covariance (ANCOVA). ANCOVA is a statistical technique used to control for the influence of covariates (variables related to but not manipulated in an experiment) when assessing the differences between groups on a dependent variable. The document outlines the basic ANCOVA model and assumptions, including that the relationship between covariates and dependent variable is linear and homogeneous across groups. It also discusses how ANCOVA partitions variability, its usage to control for non-randomized but measurable factors, and limitations such as small number of covariates and independence from treatment.

1. Group Difference Methods
By Rama Krishna Kompella

2. The basic ANCOVA situation
• Three variables: 1 Categorical (IV), 1 Continuous (IV) which is
a covariate, 1 Continuous (DV)
• Main Question: Do the (means of) the quantitative variables
depend on which group (given by categorical variable) the
individual is in, after accounting for the covariate?

3. Analysis of Covariance (ANCOVA)
• When examining the differences in the mean values
of the dependent variable related to the effect of the
controlled independent variables, it is often
necessary to take into account the influence of
uncontrolled independent variables or covariates
• A covariate is a variable that is related to the DV,
which you can’t manipulate, but you want to account
for it’s relationship with the DV

4. Assumptions
• Absence of Multicollinearity –
– Multicollinearity is the presence of high
correlations between the covariates.
– If there are more than one covariate and they
are highly correlated they will cancel each
other out of the equations
– How would this work?
– If the correlations nears 1, this is known as
singularity
– One of the CVs should be removed

5. Assumptions
• Homogeneity of Regression
– The relationship between each CV and the DV should
be the same for each level of the IV

6. Assumptions
• The relation between the DV and the
covariate is linear.
– The best fitting regression line is straight
– If the relation has significant non-linearity,
ANCOVA is not useful

7. ANCOVA Model
Y = GMy + τ + [Bi(Ci – Mij) + …] + E
• Y is a continuous DV, GMy is grand mean of DV, τ is treatment
effect, Bi is regression coefficient for ith covariate, Ci, M is mean of
ith covariate in jth IV group, and E is error
• ANCOVA is an ANOVA on Y scores in which the relationships
between the covariates and the DV are partialled out of the DV.
• Y – Bi (Ci – Mij) = GMy + τ + E

8. ANOVA and ANCOVA
• In analysis of variance the •In ancova we partition
variability is divided into variance into three basic
two components components:
– Experimental effect - Effect
– Error - experimental - Error
and individual - Covariate
differences

9. ANCOVA
• When covariate scores are available we have
information about differences between treatment
groups that existed before the experiment was
performed
• Ancova uses linear regression to estimate the size of
treatment effects given the covariate information
• The adjustment for group differences can either
increase or decrease depending on the dependent
variables relationship with the covariate

10. Usage of ANCOVA
• In experimental designs, to control for factors
which cannot be randomized but which can be
measured on an interval scale
• In observational designs, to remove the
effects of variables which modify the
relationship of the categorical independents
to the interval dependent.

11. ANCOVA
• In most experiments the scores on the covariate are
collected before the experimental treatment. eg.
pretest scores, exam scores, IQ etc
• In some experiments the scores on the covariate are
collected after the experimental treatment.
e.g.anxiety, motivation, depression etc.
• It is important to be able to justify the decision to
collect the covariate after the experimental
treatment since it is assumed that the treatment and
covariate are independent.

12. Limitations of ANCOVA
• As a general rule a very small number of
covariates is best
– Correlated with the DV
– Not correlated with each other (multi-collinearity)
• Covariates must be independent of treatment
– Data on covariates be gathered before treatment
is administered
– Failure to do this often means that some portion
of the effect of the IV is removed from the DV
when the covariate adjustment is calculated.

13. EXAMPLES
• In determining how different groups exposed to different
commercials evaluate a brand, it may be necessary to control
for prior knowledge.
• In determining how different price levels will affect a
household's cereal consumption, it may be essential to take
household size into account.

14. Questions?