# Analytical and simulation-based power analyses for mixed-design ANOVAs

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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.

# design ------- # mus CT <- c(34.12, 21, 17.44) BA <- c(36.88, 16.82, 8.75) ADM <- c(35.61, 14.39, 7.78) study <- data.frame("group" = gl(3,3, labels=c("CT", "BA", "ADM"))) study$time <- gl(3,1,9, labels=c("Intake", "8 weeks", "16 weeks")) study$DV <- c(CT, BA, ADM) study$SD <- 10 ggplot(study, aes(time, DV, group=group, linetype=group, shape=group)) + geom_line() + geom_point()

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`.

# analytical ---------- anova.pwr.mixed(data = study, Formula = "DV ~ time*group", n=10, m=3, rho=0.5) Terms power n.needed b group 0.197 NA w1 time 1.000 NA w2 time:group 0.617 NA # monte carlo ------------ anova.pwr.mixed.sim(data=study, Formula="DV ~ time*group + Error(subjects)", FactorA="group", n=10, rho=0.5, sims=100) terms power 1 group 0.19 2 time 1.00 3 time:group 0.64

## 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.

# compare samples <- seq(10,50,3) # n's to use analytical <- matrix(ncol=2, nrow=length(samples)) colnames(analytical) <- c("power", "n") for(i in samples) { j <- which(samples == i) analytical[j,1] <- anova.pwr.mixed(data = study, Formula = "DV ~ time*group", n=i, m=3, rho=0.5)$power[3] analytical[j,2] <- i } MC <- matrix(ncol=2, nrow=length(samples)) colnames(MC) <- c("power", "n") for(i in samples) { j <- which(samples == i) MC[j,1] <- anova.pwr.mixed.sim(data=study, Formula="DV ~ time*group + Error(subjects)", FactorA="group", n=i, rho=0.5, sims=500)$power[3] MC[j,2] <- i } # plot MC <- data.frame(MC) MC$method <- "MC" analytical <- data.frame(analytical) analytical$method <- "analytical" df <- rbind(analytical, MC) ggplot(df, aes(n, power, group=method, color=method)) + geom_smooth(se=F) + geom_point()

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

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