Lecture 3 - ANCOVA Flashcards
What is ANCOVA?
(Analysis of Covariance)
It is similar to ANOVA: It compares means of different groups/conditions and if they differ significantly while statistically controlling for the effects of other continuous variables that are not of interest, called COVARIATES (Basically covariates are confounding factors, so an ANCOVA also takes this confounding variables into account when analyzing data)
ANCOVA
What does an ANCOVA test include?
- 1 DV
- 1 or more IV
- 1 or more covariates
ANCOVA
Why should we include covariates?
- They reduce within-group variance: With an ANOVA, we have variance explained by model (SSm) and variance not explained by model, but some other random factors (SSr). In ANCOVA, a proportion of SSr includes the variance created by the covariate(s). Therefore, if we attribute part of SSr to our covariates, we reduce SSr. This in turn allows us to assess the difference between group means with more precision
- Eliminate confounds
All these lead to a etter measure of the effect of the experimental manipulation
ANCOVA
How is the linear model in ANCOVA?
Take the puppy example from the ANOVA chapter. Researchers realiszed that love for dogs affects happiness, so love for dogs is a covariate:
happiness = bo(^) + b1(^)long(i) + b2(^)short(i) + b3(^) covariate(i) + e
ANCOVA
How are the b-values different in an ANCOVA compared to an ANOVA?
- In ANOVA, b-values represent the differences between the means of each group and the control group
- In ANCOVA, b-values represent the differences between the means of each group and the control group ADJUSTED for the COVARIATES.
In this case, the b-values are called adjusted means - Usually used for follow-up tests, e.g. post hoc tests or planned contrasts)
ANCOVA
How can we get the normal b-values from adjusted-means values?
1) In equation with adjusted means we replace the covariate with the mean value of the covariate (average value of the covariate among participants in all conditions)
~ Mean of covariate is stable, regardless of which condition is present
2) If we replace values in adjusted means equation we get the values for our DV in our different conditions whilst controlling for the mean of the covariate. If we subtract the mean of the covariate from each value of the DV we got, then we get the actual b-values
(This last sentence is a bit complex maybe, but no need to learn it 100%, only the last sentence. If you want to understand it better go to the ANCOVA chapter pages and read pages 4-6, it explains this last concept as well and this whole flashcard just in detailed steps).
Effect size for ANCOVA
Effect size for ANCOVA
What effect size is usually used in ANCOVA?
ω^2 -> Main effect size measure of choice in ANCOVA.
- because in ANCOVA we can calculate ω^2 for each level of the IV and the covariate, another effect size, partial ω^2 is preferred.
Effect size for ANCOVA
What does partial ω^2 indicate?
The proportion of variance that a variable explains which is not explained by other variables.
In the puppoy-treatment example: The proportion of variance in happiness that the dose of puppy therpay shares that is not attributable to love of puppies.
Effect size for ANCOVA
Formulas for ω^2 & partial ω^2
(See Equations 1)
Assumptions
Assumptions
Other than the 3 initial Assumptions for ANOVA, what other assumptions does ANCOVA entail?
- Independence of the covariate and treatment effect
- Homogeneityof regression slopes
Assumptions
Independence of the covariate and treatment effect
(See Slide 2)
If 2 variables share variance then we can’t seperate the 2 variances by defining one variable as a covariate. E.G. anxiety and depression share a lot of variance because they’re highly correlated. In an experiment comparing an anxiety levels, a high-anxiety condition will also have higher depression than a low-anxiety group. We can’t consider depression as a covariate though. If we do then we’re creating the third scenario in Slide 2 which is bad, since depression amounts for some of the variance produced by anxiety.
SOLUTION:
- Randomize participants to experimental groups
- Match experimental groups on the covariate (e.g. on the anxiety-depression example, make sure participants in both high and low levels of anxiety have similar levels of depression)
Assumptions
Homogeneity of regression slopes
It states that the relationship between the outcome variable (DV)and the covariate has to be the same in each category of the IV
(See image in slide 3 for a representation of this)
- Parallel slopes model: each line representing the change in happiness for each group is parallel to each other.
- vertical height/difference of groups is determined by which treatmet is received
- If at least one isn’t parallel to the other two: assumption is violated
