Correlation among repeated measures - I need an explanation

Question

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:

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

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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)?!

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Answer 1

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.

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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.

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

Answer 2

The help guide for G*Power 2 defines rho as "the population correlation between the individual levels of the repeated measures factor." Though there is no help currently available in G*Power 3 for the Repeated Measures ANOVA you are interested in, the options available under the "options button" suggests that the "Corr among rep measures" option is refering to rho.

Answer 3

I think @ocram is on the right track here. His first "possible structure" is also called compound symmetry (see here: What is compound symmetry in plain English? for more information). This is the correlational structure that goes with what's called a repeated measures ANOVA, that is, a mixed effects model with a random intercept only (no random effects / slopes for any other variable). However, it appears that the correlations between the sets differ from each other, so this covariance structure is not appropriate. It is possible to determine power for situations like this, but it isn't straightforward. It appears that GPower doesn't have routines specified that can handle your situation. The only way I know of to deal with these more complicated situations is to simulate. However, the answer you get from GPower may be good enough for your purposes.

Answer 4

As @ocram and @gung rightly point out, if your correlations vary greatly, a different statistical procedure may be more appropriate. That said, for your case I would suggest two possible ways to estimate a required sample size or achieved power using G*Power:

1. Use the most conservative estimate. A null correlation among repeated measures will yield a sample size that is equivalent to that of a between subject design divided by the number of groups: A between subjects comparison of your three groups ($f = .25$, $α = .05$, $1-β = .95$) would require a total sample size of $n = 252$. A within subjects comparison with an assumed correlation among repeated measures of $r = 0$ requires a total sample size of $n = 84$. Since each measurement is repeated in all three groups your effective sample size is $84\times3 = 252$. Thus, you could simply use 0 as an estimate for your correlation. However, that will most likely be an overly conservative estimate. Using the lowest correlation coefficient found in your data, would give you a more adequate but still conservative estimate of the required sample size.
2. Use the mean correlation. A less conservative estimate of the correlation among repeated measures is the mean correlation found in your data. Note, however, correlation coefficients are not normally distributed. The formulas you suggest are, therefore, not appropriate to determine the mean correlation. You need to perform a Fisher-transformation before averaging and afterwards retransform the mean to yield the correct mean correlation. But, I think, if the correlations you enter into the mean vary greatly, this approach may lead to sample size estimates that are too liberal. Comparing the results to those resulting from the most conservative estimate may be helpful.

Since it appears you are using R to conduct your analysis here is a function that will do the calculations for you:

rep.m.cor <- function(x, measure, formula, type = "min") {
  require("reshape2")

  fisher.z <- function(r) {
    return(0.5 * log((1+r)/(1-r)))
  }

  inv.fisher.z <- function(z) {
    return((exp(2*z) - 1)/(exp(2*z) + 1))
  }

  melt.data <- melt(x, measure.vars = measure, na.rm = FALSE)
  wide.data <- dcast(melt.data, formula = formula, mean, value.var = "value")

  correlations <- cor(wide.data[, -1])
  correlations <- correlations[upper.tri(correlations)]

  if(type == "mean") {
    m.correlations <- inv.fisher.z(mean(fisher.z(correlations)))
  } else if(type == "min") {
    m.correlations <- inv.fisher.z(min(fisher.z(correlations)))
  } else {
    stop("Type must be either 'min' or 'mean'.")
  }

  return(m.correlations)
}

x = A data.frame resulting from aggregation, for example aggregate(measure ~ subject * factor1 * factor2, data, mean).

measure = A string providing the name of the measure.

formula = A formula giving the factor for which the correlation should be calculated, for example subject ~ factor1. It is also possible to determine the correlation for the interaction of two or more factors (subject ~ factor1 + factor2; yes, it needs to be a "+"). Note that GPower can be used to perform power analyses for up to two repeated measures factors as long as one of them has only two levels. To do so, enter the larger number of factor levels into the field "Number of measurements" and multiply the effect size $f$ by $\sqrt{2}$ (2 corresponding to the number of levels of the other factor). If both factor have more than two levels GPower will underestimate the required sample size!

type = A string naming the estimation procedure for the correlation among repeated measures. "min" corresponds to the above option 1, "mean" corresponds to the above option 2.

I hope this helps.

P.S.: If you understand German, there are detailed step-by-step instructions (including screenshots) in this book supplement.