Matthew Kay, Northwestern University mjskay@northwestern.edu
Lisa A. Elkin, University of Washington,
lelkin@cs.washington.edu
James J. Higgins, Kansas State
University, jhiggins@ksu.edu
Jacob O. Wobbrock, University of
Washington wobbrock@uw.edu
ARTool is an R package implementing the Aligned Rank Transform for conducting nonparametric analyses of variance on factorial models. This implementation is based on the ART procedure as used in the original implementation of ARTool by Wobbrock et al.
The package automates the Aligning-and-Ranking process using the art
function. It also automates the process of running a series of ANOVAs on
the transformed data and extracting the results of interest. It supports
traditional ANOVA models (fit using lm), repeated measures ANOVAs (fit
using aov), and mixed effects models (fit using lmer); the model
used is determined by the formula passed to art.
Note: The documentation of this package assumes some level of familiarity with when and why you may want to use the aligned rank transform; the ARTool page provides a more in-depth (and highly approachable) introduction to the aligned rank transform and the motivation for its use.
You can install the latest released version from CRAN with this R command:
install.packages("ARTool")Or, you can install the latest development version from GitHub with these R commands:
install.packages("devtools")
devtools::install_github("mjskay/ARTool")The general approach to using ART is to transform your data using art
, verify the ART procedure is appropriate to the dataset using summary
, and then run an ANOVA on the transformed data using anova .
First, let us load some example data:
library(ARTool)
data(Higgins1990Table5, package = "ARTool")Higgins1990Table5 is a data frame from an experiment in which the
effects of Moisture and Fertilizer on DryMatter in peat pots was
tested. Four pots were placed on each Tray , with Moisture varied
between Tray s and Fertilizer varied within Tray s. We can see the
basic structure of the data:
str(Higgins1990Table5)## 'data.frame': 48 obs. of 4 variables:
## $ Tray : Factor w/ 12 levels "t1","t2","t3",..: 1 1 1 1 2 2 2 2 3 3 ...
## $ Moisture : Factor w/ 4 levels "m1","m2","m3",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ Fertilizer: Factor w/ 4 levels "f1","f2","f3",..: 1 2 3 4 1 2 3 4 1 2 ...
## $ DryMatter : num 3.3 4.3 4.5 5.8 4 4.1 6.5 7.3 1.9 3.8 ...
head(Higgins1990Table5, n=8)## Tray Moisture Fertilizer DryMatter
## 1 t1 m1 f1 3.3
## 2 t1 m1 f2 4.3
## 3 t1 m1 f3 4.5
## 4 t1 m1 f4 5.8
## 5 t2 m1 f1 4.0
## 6 t2 m1 f2 4.1
## 7 t2 m1 f3 6.5
## 8 t2 m1 f4 7.3
To analyze this data using the aligned rank transform, we first
transform the data using art . We specify the response variable
(DryMatter ), the fixed effects and all of their interactions
(Moisture*Fertilizer, or equivalently
Moisture + Fertilizer + Moisture:Fertilizer), and any grouping terms
if present (here, (1|Tray) ).
While (1|Tray) has no effect on the results of the aligned rank
transformation, it will be used by anova to determine the type of
model to run: when grouping terms are present, mixed effects models are
run using lmer. If you wish to use a repeated measures ANOVA instead
of a mixed effects model, you can use an Error term instead (see below
for an example of this). If you do not having repeated measures, do not
include any grouping terms or error terms.
m <- art(DryMatter ~ Moisture*Fertilizer + (1|Tray), data=Higgins1990Table5)To verify that the ART procedure was correctly applied and is
appropriate for this dataset, we can look at the output of summary :
summary(m)## Aligned Rank Transform of Factorial Model
##
## Call:
## art(formula = DryMatter ~ Moisture * Fertilizer + (1 | Tray),
## data = Higgins1990Table5)
##
## Column sums of aligned responses (should all be ~0):
## Moisture Fertilizer Moisture:Fertilizer
## 0 0 0
##
## F values of ANOVAs on aligned responses not of interest (should all be ~0):
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0 0 0 0 0 0
We see that the columns sums of aligned responses and the F values of ANOVAs on aligned responses not of interest are all ~0, indicating that the alignment correctly “stripped out” effects not of interest. Thus, we can apply the ANOVA on the transformed data.
ARTool automatically selects the model to be used for the ANOVA. Because
we have included a grouping term, (1|Tray), ARTool will fit mixed
effects models using lmer and run the ANOVAs on them:
anova(m)## Analysis of Variance of Aligned Rank Transformed Data
##
## Table Type: Analysis of Deviance Table (Type III Wald F tests with Kenward-Roger df)
## Model: Mixed Effects (lmer)
## Response: art(DryMatter)
##
## F Df Df.res Pr(>F)
## 1 Moisture 23.833 3 8 0.00024199 ***
## 2 Fertilizer 122.402 3 24 1.1124e-14 ***
## 3 Moisture:Fertilizer 5.118 9 24 0.00064665 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
This particular study could also be analyzed using a repeated measures
ANOVA, yielding the same results (note that repeated measures ANOVAs and
mixed effects models will not always yield the same results). To instead
run a repeated measures ANOVA, add an Error term to the model as you
might for a call to aov:
m <- art(DryMatter ~ Moisture*Fertilizer + Error(Tray), data=Higgins1990Table5)
anova(m)## Analysis of Variance of Aligned Rank Transformed Data
##
## Table Type: Repeated Measures Analysis of Variance Table (Type I)
## Model: Repeated Measures (aov)
## Response: art(DryMatter)
##
## Error Df Df.res F value Pr(>F)
## 1 Moisture Tray 3 8 23.833 0.00024199 ***
## 2 Fertilizer Withn 3 24 122.402 1.1124e-14 ***
## 3 Moisture:Fertilizer Withn 9 24 5.118 0.00064665 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
For an example of how to run contrast tests on an art model, see this
vignette:
vignette("art-contrasts")This vignette is also available here.
Should you encounter any issues with this package, contact Matthew Kay (mjskay@northwestern.edu). If you have found a bug, please file it here with minimal code to reproduce the issue.
Kay M and Wobbrock J (2021). ARTool: Aligned Rank Transform for Nonparametric Factorial ANOVAs. R package version 0.11.0, https://github.com/mjskay/ARTool. DOI: 10.5281/zenodo.594511.
For the ART procedure used by art() and anova.art(), cite:
Wobbrock J, Findlater L, Gergle D and Higgins J (2011). “The Aligned Rank Transform for Nonparametric Factorial Analyses Using Only ANOVA Procedures.” In Proceedings of the ACM Conference on Human Factors in Computing Systems (CHI 2011), Vancouver, British Columbia (May 7-12, 2011). New York: ACM Press, pp. 143-146. https://depts.washington.edu/acelab/proj/art/. DOI: 10.1145/1978942.1978963.
For the ART-C contrast testing procedure used by art.con() and
artlm.con(), cite:
Elkin L, Kay M, Higgins J and Wobbrock J (2021). “An Aligned Rank Transform Procedure for Multifactor Contrast Tests.” arXiv eprint: 2102.11824.