2.1 Summary Slides for Analysis of Covariance (ANCOVA)

Question 1

ANCOVA defined

Answer 1

tests for mean differences among the levels of one or more factors, while statistically controlling for the effects of other continuous (or dichotomous) variables that are not of primary interest, known as covariates (CV) or nuisance variables

Question 2

Analysis of covariance or ANCOVA is an extension of ANOVA, where…

Answer 2

The dependent variable must be continuous
The factor(s) must be discrete
The covariate(s) can be either continuous or discrete (i.e., with only two categories)

Question 3

ANCOVA applied ANOVA after adjusting for one or more covariates in a regression type analysis

Answer 3

The effect of each covariate is removed from the data yielding revised cell and marginal means

Question 4

ANCOVA provides statistical control by statistically subtracting the effects of nuisance or control variables
provide example

Answer 4

Example: rehoming educational differences (CV) among participants before examining the effects of a training program on performance

Question 5

So, ANCOVA applies ANOVA analysis after adjusting one or more covariates in a two-step process:

Answer 5

First it runs a regression analysis between the dependent variable and one or more covariates, removing any effects of the covariates from the data
Second, an ANOVA is performed on the revised data (residualized data) in usual fashion

Question 6

Primary Reasons to use ANCOVA

Answer 6

1. When a covariate is important to the question being investigated in a nuisance sort of way

2.To increase power of a test by removing variability from the error term
–ANCOVA can be error purifying, where it can increase the signal-to-noise ratio of an F test

3.Equating non-equivalent groups for non-experimental designs where participants cannot be randomly assigned to treatments
–This has been criticized because it is unlikely to fully equate groups due to lack of control of other variables than cannot be measured or are unknown

Question 7

Practical Issues in ANCOVA: Selection of Covariates
Covariates should be few in number, where adding more covariates results in the law of diminishing returns
Covariates should be:

Answer 7

Correlated with the dependent variable
Not highly correlated with each other

To assess these associations, compute a correlation matrix that includes the dependent variable and covariate(s)

Question 8

Major limitations associated with ANCOVA are:

Answer 8

The power gained by cleaning the error term sum of squares may be offset by loss of one degree of freedom from the error term

Interpretation of the adjusted means should be done with caution
The adjusted means do not correspond to the real-world data
Real-world differences among the individuals have been removed

Question 9

Practical Issues in ANCOVA: The Null Hypothesis

Answer 9

The null hypothesis for the covariate is:
H0: B weight = 0

Null (H0) for main effects of factor states that there are no differences between the population means for the levels of a factor

Null: after controlling for CV, H0: µ1=µ2=µ3=….

Alternative: HA: after controlling for CV, there are likely to be differences among the means

Question 10

Practical Issues in ANCOVA: Research Hypotheses

Answer 10

Research hypotheses pertain primarily to the main effects and interactions of the factors and their effect of the dependent variable scores
Be sure to include the covariate in the hypotheses

Example for main effect: “After controlling for depression scores, participants in the brain training condition will score higher on the memory test compared to participants in the control condition.”

Example for interaction: “After controlling for depression scores, older participants will have larger gains compared to younger adults in the brain training condition compared to the control condition.”

Question 11

Practical Issues in ANCOVA: Effect Size
Two main measures of effect size for ANOVA:

Answer 11

Eta-squared (n2), which is included by request in SPSS and given as partial eta-squared
Omega-squared (w2), which can be hand calculated from the source table using the formula shown

Question 12

Interpretation of effect sizes is in terms of the proportion variance that is accounted for by the treatment effect

Answer 12

Where: .01=small effect; .06= medium effect; .14= large effect

Question 13

Assumption 1

Answer 13

There should be no significant outliers

Question 14

Assumption 2

Answer 14

the dependent variable should be approximately normally distributed for each category of the independent variable

Question 15

Assumptions 3

Answer 15

homogeneity of variance

Question 16

Assumption 4

Answer 16

The covariate should be linearly related to the dependent variable at each level of the independent variable

Question 17

Assumption 5

Answer 17

homoscedasticity of covariate and dependent variable

Question 18

Assumption 6

Answer 18

There needs to be homogeneity of regression slopes, which means that there is no interaction between the covariate and the independent variable