5. Repeated Measures ANOVA

Question 1

What is repeated measures?

Answer 1

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.

Question 2

In a repeated measures data table, what do the rows represent?

Answer 2

The individuals

Question 3

In a repeated measures data table, what do the columns represent?

Answer 3

The levels of the repeated measures factor(s)

Question 4

What are the two ways to do the analysis of repeated measures?

Answer 4

Between-subjects (row by row) and within-subject (column by column)

Question 5

ANOVA model equation for repeated measures

Answer 5

Xijt = μ + ai + εij(1) + ct + (a c)it + εijt(2)

Question 6

In the equation Xijt = μ + ai + εij(1) + ct + (a c)it + εijt(2):

What does μ represent?

Answer 6

μ is the population mean

Question 7

In the equation Xijt = μ + ai + εij(1) + ct + (a c)it + εijt(2):

What does ai represent?

Answer 7

ai are the between-subjects effects

Question 8

In the equation Xijt = μ + ai + εij(1) + ct + (a c)it + εijt(2):

What does εij(1) represent?

Answer 8

εij(1) is the error term for the between-subjects effects (ai)

Question 9

In the equation Xijt = μ + ai + εij(1) + ct + (a c)it + εijt(2):

What does ct represent?

Answer 9

ct is the main effects

Question 10

In the equation Xijt = μ + ai + εij(1) + ct + (a c)it + εijt(2):

What does (a c)it represent?

Answer 10

(a c)it is the interaction

Question 11

In the equation Xijt = μ + ai + εij(1) + ct + (a c)it + εijt(2):

What does εijt(2) represent?

Answer 11

εijt(2) is the error term for within-subject effects (so for ct and (a c)it )

Question 12

What SAS procedure can be used for repeated measures?

Answer 12

PROC GLM

Question 13

What are the 5 different approaches to the analysis of repeated measures?

Answer 13

1. First Univariate approach (U1)
2. Multivariate approach (M)
3. Second Univariate approach (U2)
4. Third Univariate approach (U3)
5. Fourth Univariate approach (U4)

Question 14

What is the first univariate approach (U1)?

Answer 14

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)

Question 15

What is the multivariate approach (M)?

Answer 15

It uses a multivariate ANOVA (MANOVA) model built on the rows of the repeated measures data table, alias profile vectors

Question 16

What is the second univariate approach (U2)?

Answer 16

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

Question 17

What is the third univariate approach (U3)?

Answer 17

The within-subject effects are tested; the corresponding means are used to calculate the sum of squares

Question 18

What is the fourth univariate approach (U4)?

Answer 18

Contrasts are used to perform specific comparisons between two or more levels of the repeated measures factor(s)

Question 19

In the SAS code, what does PRINTE do?

Answer 19

It prints the Error SSCP Matrix
SSCP = sums of squares and cross products

Question 20

When doing a repeated measures in SAS, what type of model must you write under the MODEL statement?

Answer 20

a multivariate model (more than one variable on the left side of the equation)

Question 21

What do you have to write under the REPEATED statement?

Answer 21

The definition of repeated measures factors and their levels

Question 22

When writing the SAS code, how should you arrange the data under the INPUT statement?

Answer 22

Such that each profile vector can be read with one statement

Question 23

How do you find signs of heteroscedasticity in the Error SSCP Matrix?

Answer 23

In the ratio of the minimum and maximum diagonal elements

Question 24

How do you find signs of autocorrelations (i.e. correlation between errors)?

Answer 24

In the partial correlations; partial correlations close to 1.0 in absolute value indicate strong autocorrelation

Question 25

What is circularity?

Answer 25

The independence and homoscedasticity of orthonormalized contrasts

Question 26

Under the circularity condition, what happens to the classical ANOVA F-tests for within subjects?

Answer 26

They are valid, so they need not be modified. (their p values dont need to be adjusted)

Question 27

What is Box’s epsilon?

Answer 27

It is how we measure any discrepancy of the variance-covariance matrix of orthonormalized contrasts from the circularity condition

Question 28

What are the bounds of Box’s epsilon

Answer 28

1/(R-1) < Box’s epsilon < 1
where R is the number of repeated measures

Question 29

When is the circularity condition satisfied? (in terms of Box’s epsilon)

Answer 29

When Box’s epsilon = 1

Question 30

What is the value of Box’s epsilon when R = 2?

Answer 30

1

Question 31

What are the bounds of Box’s epsilon when R = 3?

Answer 31

0.5 < BOX < 1

Question 32

What does CONTRAST(1) mean on the line of the REPEATED statement in a SAS code?

Answer 32

CONTRAST(1) means that comparisons are made with respect to the first year, or the first level of the repeated measures factor in general