Group Difference Methods By Rama Krishna Kompella
The basic ANCOVA situation• Three variables: 1 Categorical (IV), 1 Continuous (IV) which isa covariate, 1 Continuous (DV)• Main Question: Do the (means of) the quantitative variablesdepend on which group (given by categorical variable) theindividual 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 + τ + [Bi(Ci – Mij) + …] + E• Y is a continuous DV, GMy is grand mean of DV, τ is treatmenteffect, Bi is regression coefficient for ith covariate, Ci, M is mean ofith covariate in jth IV group, and E is error• ANCOVA is an ANOVA on Y scores in which the relationshipsbetween the covariates and the DV are partialled out of the DV. • Y – Bi (Ci – Mij) = GMy + τ + E
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
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 households cereal consumption, it may be essential to take household size into account.