Correlation among repeated measures – I need an explanation

I am asked in G*Power to enter the Correlation among repeated measures. I have repeated an experiment with the same subjects under 3 conditions (Set1, Set2 and Set3).

I calculate the correlation this way:

lala=cbind(  
  Set1.Weber,
  Set2.Weber,
  Set3.Weber
)
cor(lala)

And get

           Set1.Weber Set2.Weber Set3.Weber
Set1.Weber  1.0000000  0.3683676  0.1283023
Set2.Weber  0.3683676  1.0000000 -0.0959547
Set3.Weber  0.1283023 -0.0959547  1.0000000

as a result.

Is that correct? In G*Power there is only one parameter for the correlation. Should I use the average correlation. In the example:

(0.37+0.13+0.10)/3

?

Here is a screenshot:

enter image description here

Edit: As a response to the very good answers so far I want to specify the question:


Do I understand that correct that the ANOVA assumes that the correlation should be equal in the matrix and not like in my case (3 different values)? So it maybe not super correct but I can use an average as a poormans solution?! I could use (|a|+|b|+|c|)/3 or sqrt(a²+b³+c²)?!


Answer

Correlation measures association between two random variables and a correlation matrix collects pairwise correlations.

For example, in dimension 3 we have
$$
\textrm{Cor} \left(
\begin{array}{c}
X_1 \\ X_2 \\ X_3
\end{array}
\right) = \left(
\begin{array}{cccc}
\textrm{Cor}(X_1, X_1) & \textrm{Cor}(X_1, X_2) & \textrm{Cor}(X_1, X_3) \\
\textrm{Cor}(X_2, X_1) & \textrm{Cor}(X_2, X_2) & \textrm{Cor}(X_2, X_3) \\
\textrm{Cor}(X_3, X_1) & \textrm{Cor}(X_3, X_2) & \textrm{Cor}(X_3, X_3)
\end{array}
\right).
$$
It is symmetric and there are $1$’s along the diagonal.


When measurements are taken several times on the same individual, we usually expect some positive association that can be quantified by correlation. In the mixed model methodology, in particular, it is typical to put some structure on a correlation matrix.

One possible structure would be
$$
\textrm{Cor} \left(
\begin{array}{c}
X_1 \\ X_2 \\ X_3
\end{array}
\right) = \left(
\begin{array}{cccc}
1 & \rho & \rho \\
& 1 & \rho \\
& & 1
\end{array}
\right),
$$
with $0 \leq \rho \leq 1$, that is, correlation is the same regardless of the lag between pairs of repeated
measures. That’s the compound symmetry (CS) structure.

Another popular example when repeated measures are taken at equally-spaced time points is the AR(1) structure:
$$
\textrm{Cor} \left(
\begin{array}{c}
X_1 \\ X_2 \\ X_3
\end{array}
\right) = \left(
\begin{array}{cccc}
1 & \rho & \rho^2 \\
& 1 & \rho \\
& & 1
\end{array}
\right),
$$
that is, correlation decreases with distance in time.

In these two example, only one parameter has to be estimated to get the three entries of the matrix. This generalises to higher dimension.


More details, for example, in the doc for proc mixed, p3955.

Attribution
Source : Link , Question Author : Dr. Manuel Kuehner , Answer Author : ocram

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