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.
check_outliers(x, ...)
# S3 method for default
check_outliers(x, method = c("cook", "pareto"), threshold = 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, ...)
x  A model or a data.frame object. 

...  When 
method  The outlier detection method(s). Can be "all" or some of c("cook", "pareto", "zscore", "zscore_robust", "iqr", "eti", "hdi", "bci", "mahalanobis", "mahalanobis_robust", "mcd", "ics", "optics", "lof"). 
threshold  A list containing the threshold values for each method (e.g.

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.
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 (Zscores, Iglewicz, 1993; Interquartile range (IQR); Mahalanobis distance, Cabana, 2019; Robust Mahalanobis distance, Gnanadesikan & 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."
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 np degrees of freedom, the median point of
the quantile distribution can be used as a cutoff (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
).
Zscores ("zscore", "zscore_robust")
:
The Zscore, 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\
most extreme observations (assuming the data is normally distributed).
Importantly, the Zscore method is univariate: it is computed column by
column. If a dataframe is passed, the Zscore is calculated for each
variable separately, and the maximum (absolute) Zscore 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 boxandwhisker 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) the IQR below the
first or above the third quartile. Similar to the Zscore 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.
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 EqualTailed 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\
variable. See bayestestR::ci()
for more details
about the intervals.
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 & Van Zomeren, 1990),
we can use the alpha quantile of the chisquare 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
GnanadesikanKettenring pairwise estimator (Gnanadesikan \& 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 helpfile 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 others 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.
Isolation Forest:
The outliers are detected using the anomaly score of an isolation forest (a
class of random forest). The default threshold of 0.025 will classify as
outliers the observations located at qnorm(10.025) * MAD)
(a robust
equivalent of SD) of the median (roughly corresponding to the 2.5\
extreme observations). Requires the solitude package.
Local Outlier Factor:
Based on a K nearest neighbours algorithm, LOF compares the local density of
an point to the local densities of its neighbors instead of computing a
distance from the center (Breunig et al., 2000). Points that have a
substantially lower density than their neighbors are considered outliers. A
LOF score of approximately 1 indicates that density around the point is
comparable to its neighbors. Scores significantly larger than 1 indicate
outliers. The default threshold of 0.025 will classify as outliers the
observations located at qnorm(10.025) * SD)
of the logtransformed
LOF distance. Requires the dbscan package.
Default thresholds are currently specified as follows:
list(
zscore = stats::qnorm(p = 1  0.025),
iqr = 1.5,
ci = 0.95,
cook = stats::qf(0.5, ncol(x), nrow(x)  ncol(x)),
pareto = 0.7,
mahalanobis = stats::qchisq(p = 1  0.025, df = ncol(x)),
robust = stats::qchisq(p = 1  0.025, df = ncol(x)),
mcd = stats::qchisq(p = 1  0.025, df = ncol(x)),
ics = 0.025,
optics = 2 * ncol(x),
iforest = 0.025,
lof = 0.025
)
There is also a
plot()
method
implemented in the
seepackage. Please
note that the range of the distancevalues along the yaxis is rescaled
to range from 0 to 1.
Archimbaud, A., Nordhausen, K., \& RuizGazen, A. (2018). ICS for multivariate outlier detection with application to quality control. Computational Statistics & Data Analysis, 128, 184–199. doi: 10.1016/j.csda.2018.06.011
Gnanadesikan, R., \& Kettenring, J. R. (1972). Robust estimates, residuals, and outlier detection with multiresponse data. Biometrics, 81124.
Bollen, K. A., & Jackman, R. W. (1985). Regression diagnostics: An expository treatment of outliers and influential cases. Sociological Methods & Research, 13(4), 510542.
Cabana, E., Lillo, R. E., \& 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), 1518.
Iglewicz, B., & Hoaglin, D. C. (1993). How to detect and handle outliers (Vol. 16). Asq Press.
Leys, C., Klein, O., Dominicy, Y., \& Ley, C. (2018). Detecting multivariate outliers: Use a robust variant of Mahalanobis distance. Journal of Experimental Social Psychology, 74, 150156.
Liu, F. T., Ting, K. M., & Zhou, Z. H. (2008, December). Isolation forest. In 2008 Eighth IEEE International Conference on Data Mining (pp. 413422). IEEE.
Lüdecke, D., BenShachar, M. S., Patil, I., Waggoner, P., \& 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., \& Van Zomeren, B. C. (1990). Unmasking multivariate outliers and leverage points. Journal of the American Statistical association, 85(411), 633639.
data < mtcars # Size nrow(data) = 32
# For single variables 
outliers_list < check_outliers(data$mpg) # Find outliers
outliers_list # Show the row index of the outliers
#> Warning: 4 outliers detected (cases 18, 19, 20, 28).
#>
as.numeric(outliers_list) # The object is a binary vector...
#> [1] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0
filtered_data < data[!outliers_list, ] # And can be used to filter a dataframe
nrow(filtered_data) # New size, 28 (4 outliers removed)
#> [1] 28
# Find all observations beyond +/ 2 SD
check_outliers(data$mpg, method = "zscore", threshold = 2)
#> Warning: 2 outliers detected (cases 18, 20).
#>
# For dataframes 
check_outliers(data) # It works the same way on dataframes
#> Warning: 1 outliers detected (cases 9).
#>
# You can also use multiple methods at once
outliers_list < check_outliers(data, method = c(
"mahalanobis",
"iqr",
"zscore"
))
outliers_list
#> Warning: 6 outliers detected (cases 9, 15, 16, 17, 20, 31).
#>
# Using `as.data.frame()`, we can access more details!
outliers_info < as.data.frame(outliers_list)
head(outliers_info)
#> Distance_Zscore Outlier_Zscore Distance_IQR Outlier_IQR Distance_Mahalanobis
#> 1 1.189901 0 0 0 8.946673
#> 2 1.189901 0 0 0 8.287933
#> 3 1.224858 0 0 0 8.937150
#> 4 1.122152 0 0 0 6.096726
#> 5 1.043081 0 0 0 5.429061
#> 6 1.564608 0 0 0 8.877558
#> Outlier_Mahalanobis Outlier
#> 1 0 0
#> 2 0 0
#> 3 0 0
#> 4 0 0
#> 5 0 0
#> 6 0 0
outliers_info$Outlier # Including the probability of being an outlier
#> [1] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [8] 0.0000000 1.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [15] 0.6666667 0.6666667 0.6666667 0.3333333 0.3333333 0.6666667 0.0000000
#> [22] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [29] 0.0000000 0.3333333 0.6666667 0.0000000
# And we can be more stringent in our outliers removal process
filtered_data < data[outliers_info$Outlier < 0.1, ]
# We can run the function stratified by groups:
if (require("dplyr")) {
iris %>%
group_by(Species) %>%
check_outliers()
}
#> Loading required package: dplyr
#>
#> Attaching package: ‘dplyr’
#> The following objects are masked from ‘package:stats’:
#>
#> filter, lag
#> The following objects are masked from ‘package:base’:
#>
#> intersect, setdiff, setequal, union
#> Warning: 4 outliers detected (cases 42, 44, 69, 119).
#>
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
if (require("MASS")) {
check_outliers(model, method = c("mahalabonis", "mcd"))
}
if (require("ICS")) {
# This one takes some seconds to finish...
check_outliers(model, method = "ics")
}
}