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 frequencies
Theory behind Confidence intervals for Frequencies
For frequencies, ANOFA does not lend itself to confidence intervals.
Hence, we decided to use Clopper & Pearson
(1934) pivot technique. This technique returns
stand-alone confidence intervals on a proportion, that is, an
interval which can be used to compare an observed proportion to a
theoretical value. In order to compare an observed proportion to another
proportion, it is necessary to adjust the wide (see Cousineau et al., 2021). Further, because
we want an interval for frequencies, we multiply the proportion by the
total number of data \(N\).
Herein, we use the Leemis & Trivedi
(1996) method which provides an analytical solution based on the
Fisher’s \(F\) distribution. It is
based on \(n\), the observed frequency
in a given cell, \(N\), the total
number of observations, and \(1-\alpha\), the confidence level (typically
95%). It is given by
\[\hat{\pi}_{\text{low}}=\left(
1+\frac{N-n+1}{n F_{1-\alpha/2}(2n,2(N-n+1)}
\right)^{-1}
< {\pi} <
\left(
1+\frac{N-n}{(n+1) F_{\alpha/2}(2(n+1),2(N-x)}\right)^{-1}
=\hat{\pi}_{\text{high}}\]
This interval wide is then multiply by \(N\) with \(\{n_{\text{low}}, n_{\text{high}} \} = N \,
\times\, \{ \hat{\pi}_{\text{low}}, \hat{\pi}_{\text{high}} \}\)
so that we obtain an interval on frequencies rather than on
probabilities.
Finally, the width of the intervals are adjusted for difference and
correlations using:
\[n_{\text{low}}^* = \sqrt{2}
\sqrt{2}(n-n_{\text{low}})+n\]
\[n_{\text{high}}^* = \sqrt{2}
\sqrt{2}(n_{\text{high}}-n)+n\]
The lower and upper length are adjusted separately because this
interval may not be symmetrical (equal length) on both side of the
observed frequency \(n\).
This is it.
Complicated?
Well, not really:
library(ANOFA)
w <- anofa( obsfreq ~ vocation * gender, LightMargolin1971)
anofaPlot(w)
Figure 1. The frequencies of
the Light & Margolin, 1971, data as a function of aspiration for
higher education and as a function of gender. Error bars show
difference-adjusted 95% confidence intervals.
You can select only some factors for plotting, with e.g.,
anofaPlot(w, obsfreq ~ gender)
Figure 2. The frequencies of
the Light & Margolin, 1971, data as a function of gender. Error bars
show difference-adjusted 95% confidence intervals.
Because of the interaction gender \(\times\) vocational aspiration,
the overall difference between boys and girls is small. Actually, they
differ only in their aspiration to go to the university (recall that
these are 1960s data…).
As is the case with any ggplot2 figure, you can
customize it at will. For example,
library(ggplot2)
anofaPlot(w, obsfreq ~ gender) +
theme_bw()
Figure 3. Same as Figure 2 with
some customization.
Here you go.