Probably the most useful tools for data analysis is a plot with suitable error bars (Cousineau, Goulet, & Harding, 2021). In this vignette, we show how to make confidence intervals for proportions.

For proportions, ANOPA is based on the Anscombe transform . This measure has a known theoretical standard error which depends only on sampe size \(n\):

\[SE_{A}(n) = 1/\sqrt{4(n+1/2)}.\]

Consequently, when the groups’ sizes are similar, homogeneity of variances holds.

From this, we can decomposed the total test statistic \(F\) into a component for each cell of the design. We thus get

\[\left[ A + z_{0.5-\gamma/2} \times SE_{A}(n), \; A + z_{0.5+\gamma/2} \times SE_{A}(n) \right]\]

in which \(SE_{A}(n)\) is the theoretical standard error based only on \(n\), and \(\gamma\) is the desired confidence level (often .95).

This technique returns *stand-alone* confidence intervals,
that is, intervals which can be used to compare the proportion to a
fixed point. However, such *stand-alone* intervals cannot be used
to compare one proportion to another proportion (Cousineau et al., 2021). To compare an observed
proportion to another observed proportion, it is necessary to adjust
them for pair-wise differences (Baguley,
2012). This is achieved by increasing the wide of the intervals
by \(\sqrt{2}\).

Also, in repeated measure designs, the correlation is beneficial to improve estimates. As such, the interval wide can be reduced when correlation is positive by multiplying its length by \(\sqrt{1-\alpha_1}\), where \(\alpha_1\) is a measure of correlation in a matrix containing repeated measures (based on the unitary alpha measure).

Finally, the above returns confidence intervals for the
*transformed* scores. However, when used in a plot, it is
typically more convenient to plot proportions (from 0 to 1) rather than
Anscombe-scores (from 0 to \(\pi/2
\approx\) 1.57). Thus, it is possible to rescale the vertical
axis using the inverse Anscombe transform and be shown proportions.

This is it.

Well, not really:

Because the analyses `summary(w)`

suggests that only the
factor `Difficulty`

has a significant effect, you may select
only that factors for plotting, with e.g.,

As is the case with any `ggplot2`

figure, you can
customize it at will. For example,

```
library(ggplot2)
anopaPlot(w, ~ Difficulty) +
theme_bw() + # change theme
scale_x_discrete(limits = c("Easy", "Moderate", "Difficult")) #change order
```

As you can see from this plot, Difficulty is very significant, and the most different conditions are Easy vs. Difficult.

Here you go.

Baguley, T. (2012). Calculating and graphing within-subject confidence
intervals for ANOVA. *Behavior Research Methods*, *44*,
158–175. https://doi.org/10.3758/s13428-011-0123-7

Cousineau, D., Goulet, M.-A., & Harding, B. (2021). Summary plots
with adjusted error bars: The superb framework with an implementation in
R. *Advances in Methods and Practices in Psychological
Science*, *4*, 1–18. https://doi.org/10.1177/25152459211035109