Analytical and simulation-based power analyses for mixed-design ANOVAs
In this post I show some R-examples on how to perform power analyses for mixed-design ANOVAs. The first example is analytical — adapted from formulas used in G*Power (Faul et al., 2007), and the second example is a Monte Carlo simulation. The source code is embedded at the end of this post.
Both functions require a dataframe, containing the parameters that will be used in the power calculations. Here is an example using three groups and three time-points.
Here is a plot of our hypothetical study design.
Now, we will use this design to perform a power analysis using
anova.pwr.mixed and anova.pwr.mixed.sim.
Comparison of analytical and monte carlo power analysis
Now let’s compare the two functions’ results on the time x group-interaction. Hopefully, the two methods will yield the same result.
Luckily, the analytical results are consistent with the Monte Carlo results.
References
Faul, F., Erdfelder, E., Lang, A. G., & Buchner, A. (2007). G* Power 3: A flexible statistical power analysis program for the social, behavioral, and biomedical sciences.Behavior research methods, 39(2), 175-191.
Source code
Written by Kristoffer Magnusson, a researcher in clinical psychology. You should follow him on Twitter and come hang out on the open science discord Git Gud Science.
Published May 22, 2013 (View on GitHub)
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Archived Comments (2)
Hi, Nice post. Could you please provide a bit more detail (and/or a reference) for the computation on the second line below? Thank you.
B <- matrix(stdevs, ncol=4, nrow=4)
sigma <- t(B) * B * rho
That computation is used to construct the covariance matrix based on the given standard deviations and correlation coefficient between each time point (rho). Here's an expanded calculation, hope it helps
First, we have that
$latex \rho_{X,Y}=\mathrm{corr}(X,Y)={\mathrm{cov}(X,Y) \over \sigma_X \sigma_Y} $
and thus we can easily solve for Cov(X,Y).
$latex \mathrm{cov}(X,Y) = \sigma_X \sigma_Y \mathrm{corr}(X,Y)$
And we want our covariance matrix have compound symmetry but allow heterogeneous variances, so that we can explicitly violate the sphericity assumption if we want to. This is usefull if want want to simulate the effects of violating this assumption. However, sphericity can still hold even without compound symmetry. So our covariance matrix will look like this
$latex \Sigma = \begin{pmatrix}
\sigma_1^2 & \sigma_2\sigma_1\rho & \sigma_3\sigma_1\rho \\
\sigma_2\sigma_1\rho & \sigma_2^2 & \sigma_3\sigma_2\rho \\
\sigma_3\sigma_1\rho & \sigma_3\sigma_2\rho & \sigma_3^2
\end{pmatrix}
$
So t(B)*B*rho is just an vectorized implementation of this, i.e:
$latex B = \begin{pmatrix}
\sigma_1 & \sigma_1 & \sigma_1 \\
\sigma_2 & \sigma_2 & \sigma_2 \\
\sigma_3 & \sigma_3 & \sigma_3 \\
\end{pmatrix}
$
and t(B) is
$latex \mathbf{B}^\top = \begin{pmatrix}
\sigma_1 & \sigma_2 & \sigma_3 \\
\sigma_1 & \sigma_2 & \sigma_3 \\
\sigma_1 & \sigma_2 & \sigma_3 \\
\end{pmatrix}
$
and thus t(B) * B is
$latex \mathbf{B}^\top*\mathbf{B} = \begin{pmatrix}
\sigma_1^2 & \sigma_2\sigma_1 & \sigma_3\sigma_1 \\
\sigma_2\sigma_1 & \sigma_2^2 & \sigma_3\sigma_2 \\
\sigma_3\sigma_1 & \sigma_3\sigma_2 & \sigma_3^2
\end{pmatrix}
$
And if we multiply this by the correlation coefficient of the repeated measures, and replace the diagonal since it has also been multiplied with rho, then we have the covariance matrix as described above.