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ARTool

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R Package for Aligned Rank Transform for Nonparametric Factorial ANOVAs

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# ARTool: R Package for the Aligned Rank Transform for Nonparametric Factorial ANOVAs

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*Matthew Kay, Northwestern University <mjskay@northwestern.edu>*<br>
*Lisa A. Elkin, University of Washington,
<lelkin@cs.washington.edu>*<br> *James J. Higgins, Kansas State
University, <jhiggins@ksu.edu>*<br> *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](https://depts.washington.edu/acelab/proj/art/) 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](https://depts.washington.edu/acelab/proj/art/) provides a more
in-depth (and highly approachable) introduction to the aligned rank
transform and the motivation for its use.

## Installation

You can install the latest released version from CRAN with this R
command:

``` r
install.packages("ARTool")
```

**Or**, you can install the latest development version from GitHub with
these R commands:

``` r
install.packages("devtools")
devtools::install_github("mjskay/ARTool")
```

## Example

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:

``` r
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:

``` r
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 ...

``` r
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

### Step 1: Transform the data

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.

``` r
m <- art(DryMatter ~ Moisture*Fertilizer + (1|Tray), data=Higgins1990Table5)
```

### Step 2: Verify appropriateness of ART

To verify that the ART procedure was correctly applied and is
appropriate for this dataset, we can look at the output of `summary` :

``` r
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.

### Step 3: Run the ANOVA

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:

``` r
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

### Alternative model: Repeated Measures ANOVA

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`:

``` r
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

## Contrast tests

For an example of how to run contrast tests on an `art` model, see
`vignette("art-contrasts")`.

## Problems

Should you encounter any bugs in this package, please file it
[here](https://github.com/mjskay/ARTool/issues/new) with minimal code to
reproduce the issue.

## Citations

Kay, M., Elkin, L. A., Higgins, J. J., and Wobbrock, J. O. (2025).
*ARTool: Aligned Rank Transform for Nonparametric Factorial ANOVAs*. R
package version 0.11.2, <https://github.com/mjskay/ARTool>. DOI:
[10.5281/zenodo.594511](https://dx.doi.org/10.5281/zenodo.594511).

For the *ART* procedure used by `art()` and `anova.art()`, cite:

Wobbrock, J. O., Findlater, L., Gergle, D., and Higgins, J. J. (2011).
The Aligned Rank Transform for Nonparametric Factorial Analyses Using
Only ANOVA Procedures. *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](https://dx.doi.org/10.1145/1978942.1978963).

For the *ART-C* contrast testing procedure used by `art.con()` and
`artlm.con()`, cite:

Elkin, L. A., Kay, M, Higgins, J. J., and Wobbrock, J. O. (2021). An
Aligned Rank Transform Procedure for Multifactor Contrast Tests.
*Proceedings of the ACM Symposium on User Interface Software and
Technology (UIST 2021)*. Virtual Event (October 10-14, 2021). New York:
ACM Press, pp. 754-768. DOI:
[10.1145/3472749.3474784](https://dx.doi.org/10.1145/3472749.3474784)

ARTool

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