# ANOPA: Analysis of Proportions using Anscombe transform

The library `ANOPA` provides easy-to-use tools to analyze proportions . With it, you can examine if proportions are significantly different (show an effect). In the case where there is more than one factor, you can also test if the interaction(s) are significant. You can also test simple effects (a.k.a. expected marginal analysis), as well as post-hoc tests (using Tukey’s Honestly Significant Difference test HSD). Finally, you can assess differences based on orthogonal contrasts. You can consult Laurencelle & Cousineau (2023) for details.

ANOPA also comes (a) with tools to make a plot of the proportions along with 95% confidence intervals [these intervals are adjusted for pair- wise comparisons; Cousineau, Goulet, & Harding (2021)]; (b) with tools to compute statistical power given some a priori expected proportions or sample size to reach a certain statistical power; (c) to generate random proportions if you wish to perform Monte Carlo simulations on proportions. In sum, eveything you need to analyse proportions!

The main function is `anopa()` which returns an omnibus analysis of the proportions for the factors given. For example, if you have a data frame `ArticleExample2` which contains a column called `s` where the number of successes per group are stored, and a column called `n` where the group sizes are stored, then the following performs an analysis of proportions as a function of the groups based on the columns `SES` and `MofDiagnostic`:

``````w <- anopa( {s; n} ~ SES * MofDiagnostic, ArticleExample2 )
summary(w)``````
``````##                         MS  df        F   pvalue correction    Fcorr pvalcorr
## SES               0.022242   2 6.394845 0.001670   1.004652 6.365237 0.001720
## MofDiagnostic     0.001742   1 0.500966 0.479076   1.002248 0.499842 0.479569
## SES:MofDiagnostic 0.007443   2 2.140035 0.117651   1.040875 2.055997 0.127965
## Error(between)    0.003478 Inf``````

As the results suggest (consult the first three columns), there is a main effect of the factor SES (F(2, inf) = 6.395, p = .002). A plot of the proportions can be obtained easily with

``anopaPlot(w) ``

or just the main effect figure with

``anopaPlot(w, ~ SES)``

If the interaction had been significant, simple effects can be analyzed from the expected marginal frequencies with `e <- emProportions(w, ~ SES | MofDiagnostic )`.

Follow-up analyses include contrasts examinations with `contrastProportions()`; finally, post-hoc pairwise comparisons can be obtained with `posthocProportions()`.

Prior to running an experiment, you might consider some statistical power planning on proportions using `anopaPower2N()` or `anopaN2Power()` as long as you can anticipate the expected proportions. A convenient effect size, the f-square and eta-square can be obtained with `anopaPropTofsq()`.

Finally, `toCompiled()`, `toLong()` and `toWide()` can be used to present the proportion in other formats.

# Installation

The official CRAN version can be installed with

``````install.packages("ANOPA")
library(ANOPA)``````

The development version 0.1.3 can be accessed through GitHub:

``````devtools::install_github("dcousin3/ANOPA")
library(ANOPA)``````

Note that the package `ANOPA` is named using UPPERCASE letters whereas the main function `anopa()` is written using lowercase letters.

``library(ANOPA)``

# In sum

As seen, the library `ANOPA` makes it easy to analyze proportions using the same general vocabulary found in ANOVAs.

The complete documentation is available on this site.

A general introduction to the `ANOPA` framework underlying this library can be found at Laurencelle & Cousineau (2023).

# References

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

Laurencelle, L., & Cousineau, D. (2023). Analysis of proportions using arcsine transform with any experimental design. Frontiers in Psychology, 13, 1045436. https://doi.org/10.3389/fpsyg.2022.1045436