Chapter 12: ANCOVA Flashcards
What is ANCOVA?
When we measure covariates and include them in an analysis of variance we call it analysis of covariance: ANCOVA
Covariates
Continuous variables that are not part of the main experimental manipulation but have an influence on the dependent variable
Reasons for including covariates in ANOVA
1- To reduce within-group error variance
2- Eliminate confounds
ANCOVA: Equation
Outcome=bo+b1Dummy1+b2Dummy2+b3covariate+error
- Covariate: added as a predictor in ANCOVA
- this model tests the difference in group means adjusted for covariate
ANCOVA:
ANOVA table: Model 1
- how well the model fits when only the covariate is used in model
ANCOVA:
ANOVA table: Model 2
- the goodness of fit of model when covariate & dummy variables are used is used in model
- difference in R^2: the individual contribution of experimental groups
—> R^2 (M.2)- R^2 (M.1)
Constant
bo in ANCOVA
ANCOVA as ‘controlling’ for the covariate
- compares the predicted group means at the average value of the covariate, so the groups are being compared at a level of the covariate that is the same for each group
- ‘controlling for covariate’ analogy is not a good one
Assumptions of ANCOVA
- Linearity
- Normality
- Independence of error
- Homoscedasticity
- Independence of the covariate and experimental groups
- homogeneity of regression slopes
Independence of the covariates and predictor groups
- covariate must be independent of categorical predictor
- this situation arises mostly when participants aren’t randomly assigned to experimental conditions
- covariance must share no variance with experimental groups: the expected value of covariance will be the same for every group
—> group means for covariance will be equal
Solution for violation of:
The independence of covariate and experimental effect
- assign participants randomly to experimental groups
- or: match experimental groups on the covariate
Statistical Requirement:
Independence of Covariate and experimental effect
- no statistical requirement for experimental effect to be independent of covariate
- this assumption makes interpretation more straightforward
Temporal Additivity
- assumption that all experimental groups would experience the same change in covariate over time if the experimental groups had no effect
- according to Senn: the idea that ANCOVA is biased unless experimental groups are equal on the covariate applies only when there is temporal additivity
- when we have temporal additivity: make sure that the covariate is same in all experimental groups
Homogeneity of Regression Slopes
- relationship between outcome (dependent variable) & covariate is the same in each of our treatment groups
- visual representation: scatter plot of covariate vs outcome for each experimental group
Homogeneity of Regression Slopes
- how to check for it
- When an ANCOVA is conducted we look at overall relationship between outcome (dependent variable) & covariate
- fit a regression line to entire data set, ignoring to which group a person belongs
- imagine plotting a scatterplot for each group of participants with covariate on one axis and outcome on the other
Heterogeneity of regression slopes
- relationship between participant’s outcome and covariate is different in the different experimental groups
What are the consequences of violating the assumption of homogeneity of regression slopes?
I. Type I error rate is inflated and the power to detect effects is not maximized
—> This is especially true when group sizes are unequal and when the standardized regression slopes differ by more than .4
What to do when assumptions are violated?
- bootstrap (robust)
- post hoc (robust)
- R (main bits of ANCOVA can not be done using bootstrap or post-hoc test)
If assumption of Homogeneity of Regression Slopes is violated:
- use a multilevel model
ANCOVA: SPSS
- Testing the independence of the treatment variable and covariate
- Run ANOVA
- Outcome or Dependent Variable: Covariate
- Predictor or Independent Variable: Experimental groups
- if F of predictor is non-significant then assumption has not been violated
ANCOVA: Main Analysis: SPSS
I. Analyze
II. General Linear Model
III. Univariate
ANCOVA: Contrasts
- You can NOT enter your own codes
- Select one of the standard contrasts
ANCOVA: Other Options
- here you can get a limited range of Post-Hoc tests
ANCOA:
How to specify Post-Hoc test?
- select the independent variable and drag it to the box labeled: Display Means
- select compare main effects
- Select either Bonferroni or Sidak
- Sidak more power than Bonferroni
- Descriptive
- Parameter Estimates
- Homogeneity test