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.
Theory behind 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.
Complicated?
Well, not really:
library(ANOPA)
w <- anopa( {success;total} ~ Class * Difficulty, twoWayExample)
anopaPlot(w)
Figure 1. The proportions as a
function of class and Difficulty. Error bars show difference-adjusted
95% confidence intervals.
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.,
anopaPlot(w, ~ Difficulty )
Figure 2. The proportions as a
function of Difficulty only. Error bars show difference-adjusted 95%
confidence intervals.
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
Figure 3. Same as Figure 2 with
some visual improvements.
As you can see from this plot, Difficulty is very significant, and
the most different conditions are Easy vs. Difficult.
Here you go.