5. Repeated Measures ANOVA Flashcards
What is repeated measures?
It is an extension of paired observations that are made on the same “individual” or “experimental unit” for a given random variable of interest.
i.e. the same X is observed repeatedly on the same individual, at different times or on different parts of it, or both.
In a repeated measures data table, what do the rows represent?
The individuals
In a repeated measures data table, what do the columns represent?
The levels of the repeated measures factor(s)
What are the two ways to do the analysis of repeated measures?
Between-subjects (row by row) and within-subject (column by column)
ANOVA model equation for repeated measures
Xijt = μ + ai + εij(1) + ct + (a c)it + εijt(2)
In the equation Xijt = μ + ai + εij(1) + ct + (a c)it + εijt(2):
What does μ represent?
μ is the population mean
In the equation Xijt = μ + ai + εij(1) + ct + (a c)it + εijt(2):
What does ai represent?
ai are the between-subjects effects
In the equation Xijt = μ + ai + εij(1) + ct + (a c)it + εijt(2):
What does εij(1) represent?
εij(1) is the error term for the between-subjects effects (ai)
In the equation Xijt = μ + ai + εij(1) + ct + (a c)it + εijt(2):
What does ct represent?
ct is the main effects
In the equation Xijt = μ + ai + εij(1) + ct + (a c)it + εijt(2):
What does (a c)it represent?
(a c)it is the interaction
In the equation Xijt = μ + ai + εij(1) + ct + (a c)it + εijt(2):
What does εijt(2) represent?
εijt(2) is the error term for within-subject effects (so for ct and (a c)it )
What SAS procedure can be used for repeated measures?
PROC GLM
What are the 5 different approaches to the analysis of repeated measures?
- First Univariate approach (U1)
- Multivariate approach (M)
- Second Univariate approach (U2)
- Third Univariate approach (U3)
- Fourth Univariate approach (U4)
What is the first univariate approach (U1)?
It is based on separate univariate ANOVAs perfomed for a given level of the repeated measures factor(s) (i.e. using one column Xijt of the repeated measures data table for a given t)
What is the multivariate approach (M)?
It uses a multivariate ANOVA (MANOVA) model built on the rows of the repeated measures data table, alias profile vectors
What is the second univariate approach (U2)?
The between-subjects effects (ai) are assessed using the means calculated across all levels of the repeated measures factor(s) for each of the individuals
What is the third univariate approach (U3)?
The within-subject effects are tested; the corresponding means are used to calculate the sum of squares
What is the fourth univariate approach (U4)?
Contrasts are used to perform specific comparisons between two or more levels of the repeated measures factor(s)
In the SAS code, what does PRINTE do?
It prints the Error SSCP Matrix
SSCP = sums of squares and cross products
When doing a repeated measures in SAS, what type of model must you write under the MODEL statement?
a multivariate model (more than one variable on the left side of the equation)
What do you have to write under the REPEATED statement?
The definition of repeated measures factors and their levels
When writing the SAS code, how should you arrange the data under the INPUT statement?
Such that each profile vector can be read with one statement
How do you find signs of heteroscedasticity in the Error SSCP Matrix?
In the ratio of the minimum and maximum diagonal elements
How do you find signs of autocorrelations (i.e. correlation between errors)?
In the partial correlations; partial correlations close to 1.0 in absolute value indicate strong autocorrelation
What is circularity?
The independence and homoscedasticity of orthonormalized contrasts
Under the circularity condition, what happens to the classical ANOVA F-tests for within subjects?
They are valid, so they need not be modified. (their p values dont need to be adjusted)
What is Box’s epsilon?
It is how we measure any discrepancy of the variance-covariance matrix of orthonormalized contrasts from the circularity condition
What are the bounds of Box’s epsilon
1/(R-1) < Box’s epsilon < 1
where R is the number of repeated measures
When is the circularity condition satisfied? (in terms of Box’s epsilon)
When Box’s epsilon = 1
What is the value of Box’s epsilon when R = 2?
1
What are the bounds of Box’s epsilon when R = 3?
0.5 < BOX < 1
What does CONTRAST(1) mean on the line of the REPEATED statement in a SAS code?
CONTRAST(1) means that comparisons are made with respect to the first year, or the first level of the repeated measures factor in general
