# Outliers detection (check for influential observations)

Source:`R/check_outliers.R`

`check_outliers.Rd`

Checks for and locates influential observations (i.e.,
"outliers") via several distance and/or clustering methods. If several
methods are selected, the returned "Outlier" vector will be a composite
outlier score, made of the average of the binary (0 or 1) results of each
method. It represents the probability of each observation of being
classified as an outlier by at least one method. The decision rule used by
default is to classify as outliers observations which composite outlier
score is superior or equal to 0.5 (i.e., that were classified as outliers
by at least half of the methods). See the **Details** section below
for a description of the methods.

## Usage

```
check_outliers(x, ...)
# S3 method for default
check_outliers(
x,
method = c("cook", "pareto"),
threshold = NULL,
ID = NULL,
...
)
# S3 method for numeric
check_outliers(x, method = "zscore_robust", threshold = NULL, ...)
# S3 method for data.frame
check_outliers(x, method = "mahalanobis", threshold = NULL, ID = NULL, ...)
```

## Arguments

- x
A model or a data.frame object.

- ...
When

`method = "ics"`

, further arguments in`...`

are passed down to`ICSOutlier::ics.outlier()`

. When`method = "mahalanobis"`

, they are passed down to`stats::mahalanobis()`

.- method
The outlier detection method(s). Can be

`"all"`

or some of`"cook"`

,`"pareto"`

,`"zscore"`

,`"zscore_robust"`

,`"iqr"`

,`"ci"`

,`"eti"`

,`"hdi"`

,`"bci"`

,`"mahalanobis"`

,`"mahalanobis_robust"`

,`"mcd"`

,`"ics"`

,`"optics"`

or`"lof"`

.- threshold
A list containing the threshold values for each method (e.g.

`list('mahalanobis' = 7, 'cook' = 1)`

), above which an observation is considered as outlier. If`NULL`

, default values will be used (see 'Details'). If a numeric value is given, it will be used as the threshold for any of the method run.- ID
Optional, to report an ID column along with the row number.

## Value

A logical vector of the detected outliers with a nice printing
method: a check (message) on whether outliers were detected or not. The
information on the distance measure and whether or not an observation is
considered as outlier can be recovered with the `as.data.frame`

function.

## Details

Outliers can be defined as particularly influential observations. Most methods rely on the computation of some distance metric, and the observations greater than a certain threshold are considered outliers. Importantly, outliers detection methods are meant to provide information to consider for the researcher, rather than to be an automatized procedure which mindless application is a substitute for thinking.

An **example sentence** for reporting the usage of the composite method
could be:

*"Based on a composite outlier score (see the 'check_outliers' function
in the 'performance' R package; Lüdecke et al., 2021) obtained via the joint
application of multiple outliers detection algorithms (Z-scores, Iglewicz,
1993; Interquartile range (IQR); Mahalanobis distance, Cabana, 2019; Robust
Mahalanobis distance, Gnanadesikan and Kettenring, 1972; Minimum Covariance
Determinant, Leys et al., 2018; Invariant Coordinate Selection, Archimbaud et
al., 2018; OPTICS, Ankerst et al., 1999; Isolation Forest, Liu et al. 2008;
and Local Outlier Factor, Breunig et al., 2000), we excluded n participants
that were classified as outliers by at least half of the methods used."*

## Note

There is also a
`plot()`

-method
implemented in the
see-package. **Please
note** that the range of the distance-values along the y-axis is re-scaled
to range from 0 to 1.

## Model-specific methods

**Cook's Distance**: Among outlier detection methods, Cook's distance and leverage are less common than the basic Mahalanobis distance, but still used. Cook's distance estimates the variations in regression coefficients after removing each observation, one by one (Cook, 1977). Since Cook's distance is in the metric of an F distribution with p and n-p degrees of freedom, the median point of the quantile distribution can be used as a cut-off (Bollen, 1985). A common approximation or heuristic is to use 4 divided by the numbers of observations, which usually corresponds to a lower threshold (i.e., more outliers are detected). This only works for Frequentist models. For Bayesian models, see`pareto`

.**Pareto**: The reliability and approximate convergence of Bayesian models can be assessed using the estimates for the shape parameter k of the generalized Pareto distribution. If the estimated tail shape parameter k exceeds 0.5, the user should be warned, although in practice the authors of the`loo`

package observed good performance for values of k up to 0.7 (the default threshold used by`performance`

).

## Univariate methods

**Z-scores**`("zscore", "zscore_robust")`

: The Z-score, or standard score, is a way of describing a data point as deviance from a central value, in terms of standard deviations from the mean (`"zscore"`

) or, as it is here the case (`"zscore_robust"`

) by default (Iglewicz, 1993), in terms of Median Absolute Deviation (MAD) from the median (which are robust measures of dispersion and centrality). The default threshold to classify outliers is 1.959 (`threshold = list("zscore" = 1.959)`

), corresponding to the 2.5\ (assuming the data is normally distributed). Importantly, the Z-score method is univariate: it is computed column by column. If a dataframe is passed, the Z-score is calculated for each variable separately, and the maximum (absolute) Z-score is kept for each observations. Thus, all observations that are extreme on at least one variable might be detected as outliers. Thus, this method is not suited for high dimensional data (with many columns), returning too liberal results (detecting many outliers).**IQR**`("iqr")`

: Using the IQR (interquartile range) is a robust method developed by John Tukey, which often appears in box-and-whisker plots (e.g., in`geom_boxplot`

). The interquartile range is the range between the first and the third quartiles. Tukey considered as outliers any data point that fell outside of either 1.5 times (the default threshold is 1.7) the IQR below the first or above the third quartile. Similar to the Z-score method, this is a univariate method for outliers detection, returning outliers detected for at least one column, and might thus not be suited to high dimensional data. The distance score for the IQR is the absolute deviation from the median of the upper and lower IQR thresholds. Then, this value is divided by the IQR threshold, to “standardize” it and facilitate interpretation.**CI**`("ci", "eti", "hdi", "bci")`

: Another univariate method is to compute, for each variable, some sort of "confidence" interval and consider as outliers values lying beyond the edges of that interval. By default,`"ci"`

computes the Equal-Tailed Interval (`"eti"`

), but other types of intervals are available, such as Highest Density Interval (`"hdi"`

) or the Bias Corrected and Accelerated Interval (`"bci"`

). The default threshold is`0.95`

, considering as outliers all observations that are outside the 95% CI on any of the variable. See`bayestestR::ci()`

for more details about the intervals. The distance score for the CI methods is the absolute deviation from the median of the upper and lower CI thresholds. Then, this value is divided by the difference between the upper and lower CI bounds divided by two, to “standardize” it and facilitate interpretation.

## Multivariate methods

**Mahalanobis Distance**: Mahalanobis distance (Mahalanobis, 1930) is often used for multivariate outliers detection as this distance takes into account the shape of the observations. The default`threshold`

is often arbitrarily set to some deviation (in terms of SD or MAD) from the mean (or median) of the Mahalanobis distance. However, as the Mahalanobis distance can be approximated by a Chi squared distribution (Rousseeuw and Van Zomeren, 1990), we can use the alpha quantile of the chi-square distribution with k degrees of freedom (k being the number of columns). By default, the alpha threshold is set to 0.025 (corresponding to the 2.5\ Cabana, 2019). This criterion is a natural extension of the median plus or minus a coefficient times the MAD method (Leys et al., 2013).**Robust Mahalanobis Distance**: A robust version of Mahalanobis distance using an Orthogonalized Gnanadesikan-Kettenring pairwise estimator (Gnanadesikan and Kettenring, 1972). Requires the bigutilsr package. See the`bigutilsr::dist_ogk()`

function.**Minimum Covariance Determinant (MCD)**: Another robust version of Mahalanobis. Leys et al. (2018) argue that Mahalanobis Distance is not a robust way to determine outliers, as it uses the means and covariances of all the data - including the outliers - to determine individual difference scores. Minimum Covariance Determinant calculates the mean and covariance matrix based on the most central subset of the data (by default, 66\ is deemed to be a more robust method of identifying and removing outliers than regular Mahalanobis distance.**Invariant Coordinate Selection (ICS)**: The outlier are detected using ICS, which by default uses an alpha threshold of 0.025 (corresponding to the 2.5\ value for outliers classification. Refer to the help-file of`ICSOutlier::ics.outlier()`

to get more details about this procedure. Note that`method = "ics"`

requires both ICS and ICSOutlier to be installed, and that it takes some time to compute the results.**OPTICS**: The Ordering Points To Identify the Clustering Structure (OPTICS) algorithm (Ankerst et al., 1999) is using similar concepts to DBSCAN (an unsupervised clustering technique that can be used for outliers detection). The threshold argument is passed as`minPts`

, which corresponds to the minimum size of a cluster. By default, this size is set at 2 times the number of columns (Sander et al., 1998). Compared to the other techniques, that will always detect several outliers (as these are usually defined as a percentage of extreme values), this algorithm functions in a different manner and won't always detect outliers. Note that`method = "optics"`

requires the dbscan package to be installed, and that it takes some time to compute the results.

## Threshold specification

Default thresholds are currently specified as follows:

```
list(
zscore = stats::qnorm(p = 1 - 0.001),
zscore_robust = stats::qnorm(p = 1 - 0.001),
iqr = 1.7,
ci = 0.999,
eti = 0.999,
hdi = 0.999,
bci = 0.999,
cook = stats::qf(0.5, ncol(x), nrow(x) - ncol(x)),
pareto = 0.7,
mahalanobis = stats::qchisq(p = 1 - 0.001, df = ncol(x)),
mahalanobis_robust = stats::qchisq(p = 1 - 0.001, df = ncol(x)),
mcd = stats::qchisq(p = 1 - 0.001, df = ncol(x)),
ics = 0.001,
optics = 2 * ncol(x),
lof = 0.001
)
```

## References

Archimbaud, A., Nordhausen, K., and Ruiz-Gazen, A. (2018). ICS for multivariate outlier detection with application to quality control. Computational Statistics and Data Analysis, 128, 184-199. doi:10.1016/j.csda.2018.06.011

Gnanadesikan, R., and Kettenring, J. R. (1972). Robust estimates, residuals, and outlier detection with multiresponse data. Biometrics, 81-124.

Bollen, K. A., and Jackman, R. W. (1985). Regression diagnostics: An expository treatment of outliers and influential cases. Sociological Methods and Research, 13(4), 510-542.

Cabana, E., Lillo, R. E., and Laniado, H. (2019). Multivariate outlier detection based on a robust Mahalanobis distance with shrinkage estimators. arXiv preprint arXiv:1904.02596.

Cook, R. D. (1977). Detection of influential observation in linear regression. Technometrics, 19(1), 15-18.

Iglewicz, B., and Hoaglin, D. C. (1993). How to detect and handle outliers (Vol. 16). Asq Press.

Leys, C., Klein, O., Dominicy, Y., and Ley, C. (2018). Detecting multivariate outliers: Use a robust variant of Mahalanobis distance. Journal of Experimental Social Psychology, 74, 150-156.

Liu, F. T., Ting, K. M., and Zhou, Z. H. (2008, December). Isolation forest. In 2008 Eighth IEEE International Conference on Data Mining (pp. 413-422). IEEE.

Lüdecke, D., Ben-Shachar, M. S., Patil, I., Waggoner, P., and Makowski, D. (2021). performance: An R package for assessment, comparison and testing of statistical models. Journal of Open Source Software, 6(60), 3139. doi:10.21105/joss.03139

Rousseeuw, P. J., and Van Zomeren, B. C. (1990). Unmasking multivariate outliers and leverage points. Journal of the American Statistical association, 85(411), 633-639.

## Examples

```
data <- mtcars # Size nrow(data) = 32
# For single variables ------------------------------------------------------
outliers_list <- check_outliers(data$mpg) # Find outliers
outliers_list # Show the row index of the outliers
#> OK: No outliers detected.
#> - Based on the following method and threshold: zscore_robust (3.09).
#> - For variable: data$mpg
#>
#>
as.numeric(outliers_list) # The object is a binary vector...
#> [1] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
filtered_data <- data[!outliers_list, ] # And can be used to filter a dataframe
nrow(filtered_data) # New size, 28 (4 outliers removed)
#> [1] 32
# Find all observations beyond +/- 2 SD
check_outliers(data$mpg, method = "zscore", threshold = 2)
#> 2 outliers detected: cases 18, 20.
#> - Based on the following method and threshold: zscore (2).
#> - For variable: data$mpg.
#>
#> -----------------------------------------------------------------------------
#> Outliers per variable (zscore):
#>
#> $`data$mpg`
#> Row Distance_Zscore
#> 18 18 2.042389
#> 20 20 2.291272
#>
# For dataframes ------------------------------------------------------
check_outliers(data) # It works the same way on dataframes
#> OK: No outliers detected.
#> - Based on the following method and threshold: mahalanobis (31.26).
#> - For variables: mpg, cyl, disp, hp, drat, wt, qsec, vs, am, gear, carb
#>
#>
# You can also use multiple methods at once
outliers_list <- check_outliers(data, method = c(
"mahalanobis",
"iqr",
"zscore"
))
outliers_list
#> 1 outlier detected: case 31.
#> - Based on the following methods and thresholds: mahalanobis (3.09), iqr
#> (1.7), zscore (31.26).
#> - For variables: mpg, cyl, disp, hp, drat, wt, qsec, vs, am, gear, carb.
#> Note: Outliers were classified as such by at least half of the selected methods.
#>
#> -----------------------------------------------------------------------------
#> The following observations were considered outliers for two or more variables
#> by at least one of the selected methods:
#>
#> Row n_Zscore n_IQR
#> 1 31 1 2
# Using `as.data.frame()`, we can access more details!
outliers_info <- as.data.frame(outliers_list)
head(outliers_info)
#> Row Distance_Zscore Outlier_Zscore Distance_IQR Outlier_IQR
#> 1 1 1.189901 0 0.4208483 0
#> 2 2 1.189901 0 0.2941176 0
#> 3 3 1.224858 0 0.5882353 0
#> 4 4 1.122152 0 0.5882353 0
#> 5 5 1.043081 0 0.3915954 0
#> 6 6 1.564608 0 0.6809025 0
#> Distance_Mahalanobis Outlier_Mahalanobis Outlier
#> 1 8.946673 0 0
#> 2 8.287933 0 0
#> 3 8.937150 0 0
#> 4 6.096726 0 0
#> 5 5.429061 0 0
#> 6 8.877558 0 0
outliers_info$Outlier # Including the probability of being an outlier
#> [1] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [8] 0.0000000 0.3333333 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [15] 0.0000000 0.3333333 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [22] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [29] 0.0000000 0.0000000 0.6666667 0.0000000
# And we can be more stringent in our outliers removal process
filtered_data <- data[outliers_info$Outlier < 0.1, ]
# We can run the function stratified by groups using `{dplyr}` package:
if (require("poorman")) {
iris %>%
group_by(Species) %>%
check_outliers()
}
#> Loading required package: poorman
#>
#> I'd seen my father. He was a poor man, and I watched him do astonishing things.
#> - Sidney Poitier
#>
#> Attaching package: ‘poorman’
#> The following objects are masked from ‘package:stats’:
#>
#> filter, lag
#> OK: No outliers detected.
#> - Based on the following method and threshold: mahalanobis (18.47).
#> - For variables: Sepal.Length, Sepal.Width, Petal.Length, Petal.Width
#>
#>
if (FALSE) {
# You can also run all the methods
check_outliers(data, method = "all")
# For statistical models ---------------------------------------------
# select only mpg and disp (continuous)
mt1 <- mtcars[, c(1, 3, 4)]
# create some fake outliers and attach outliers to main df
mt2 <- rbind(mt1, data.frame(
mpg = c(37, 40), disp = c(300, 400),
hp = c(110, 120)
))
# fit model with outliers
model <- lm(disp ~ mpg + hp, data = mt2)
outliers_list <- check_outliers(model)
if (require("see")) {
plot(outliers_list)
}
insight::get_data(model)[outliers_list, ] # Show outliers data
}
```