Group Difference Methods

By Rama Krishna Kompella

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?

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

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

Assumptions• Homogeneity of Regression

– The relationship between each CV and the DV should be the same for each level of the IV

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

ANCOVA Model

Y = GMy + + [Bi(Ci – Mij) + …] + E

• Y is a continuous DV, GMy is grand mean of DV, 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 + + E

ANOVA and ANCOVA

• In analysis of variance the variability is divided into two components– Experimental effect– Error - experimental

and individual differences

•In ancova we partition variance into three basic components:

- Effect- Error- Covariate

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

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.

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.

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.

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.

Questions?