Covariance Analysis

Question 1

What is Ancova?

Answer 1

•Analysis of Covariance is used to achieve statistical control of error when experimental control of error is not possible.

Question 2

The Ancova adjusts the analysis in two ways:-

Answer 2

• reducing the estimates of experimental error

* adjusting treatment effects with respect to the covariate

Question 3

Analysis of covariance

Answer 3

• 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.

Question 4

In analysis of variance the variability is divided into two components

Answer 4

• Experimental effect
• Error - experimental and individual differences

In a pie chart: bigger % of effect analysed than error

Question 5

•In ancova we partition variance into three basic components:

Answer 5

• Effect
• Error
• Covariate

In a pie chart: (largest % to lowest)
Effect
Error
Covariate

Question 6

Estimating treatment effects

When covariate scores are available we have information about differences…

Answer 6

between treatment groups that existed before the experiment was performed

Question 7

Ancova uses linear…

Answer 7

regression to estimate the size of treatment effects given the covariate information

Question 8

The adjustment for group differences can either…

Answer 8

increase or decrease the difference depending on the dependent and independent variables’ relationships with the covariate.

Question 9

Error variability in ANOVA

Answer 9

• In between groups analysis of variance the error variability comes from the subject within group deviation from the mean of the group.
• It is calculated on the basis of the S/A sum of squares

Question 10

Error variability in ANCOVA

Answer 10

• In regression the residual sum of squares is based on the deviation of the score from the regression line.
• The residual sum of squares will be smaller than the S/A sum of squares
• This is how ANCOVA works

Question 11

There are a number of assumptions that underlie the analysis of covariance

All the assumptions that apply to between groups ANOVA

Answer 11

• normality of treatment levels
• independence of variance estimates
• homogeneity of variance
• random sampling

Question 12

There are a number of assumptions that underlie the analysis of covariance

Two assumptions specific to ANCOVA

Answer 12

• The assumption of linear regression

* The assumption of homogeneity of regression coefficients

Question 13

The assumption of linear regression

This states that the deviations from the regression equation across the different levels of the independent variable have:

Answer 13

• normal distributions with means of zero

* homoscedasticity.

Question 14

If linear regression is used when the true regression is curvilinear then

Answer 14

• the ANCOVA will be of little use.

* adjusting the means with respect to the linear equation will be pointless

Question 15

Homogeneity of regression coefficients

Answer 15

• The regression coefficients for each of the groups in the independent variable(s) should be the same.
• Glass et al (1972) have argued that this assumption is only important if the regression coefficients are significantly different
• We can test this assumption by looking at the interaction between the independent variable and the covariate

Question 16

An example Ancova

Answer 16

• A researcher is looking at performance on crossword clues.
• Subjects have been grouped into three vocabulary levels.
• An anova & tukeys on this data finds that the high group and low groups are different

Question 17

Limitations to analysis of covariance

Answer 17

• As a general rule a very small number of covariates is best (min N=16-20 per covariate)
• Correlated with the dv
• Not correlated with each other (multi-collinearity)
• Covariates must be independent of treatment
• Data on covariates should 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.