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Outlier Analysis
An outlier is a data point that differs significantly from other data points within a give data set. Sometimes referred to as abnormalities, anomalies, or deviants, outliers can occur by chance in any given distribution. In large samples, there is an expectation of a small number of outliers and their presence alone does not suggest any anomaly and should not generate concern over the entire data set. However, outliers can also be indicative of measurement error, a skewed distribution, or data points from a different underlying distribution. Many statistical tests are sensitive to the presence of outliers and therefore the ability to detect an outlier is an important part of data analysis. Typically outliers are recording and measurement errors or incorrect distribution assumptions but can also reveal unknown data structures or suggest evidence of some novel phenomenon.
Outliers can have negative effects on data analyses, such as analyses of variance (ANOVAs) or regressions. They increase error variance and reduce the power of statistical tests and when they are not distributed across the data set, but generally fall on one extreme, they function to decrease normality. Therefore, they can influence tests that rely on distribution assumptions or introduce bias into parameter estimates. In such cases, it is important to identify outliers so that they can be dealt with appropriately, resulting in improved statistical analysis. However, outliers can also be valuable data points that reveal important information about the data set, its creation, or the data points themselves. For example, if the outlier is due to a mistake in data entry or instrument error, then researchers can correct those issues by appropriately entering the data or expunging the poor measurements. Other outliers point to normal and expected deviations in the population, such as extremes in human height or weight. Outliers could also suggest faults in a system, changes to how a system behaves, or abnormal behavior of the data in the system. Since the information contained in outliers is potentially so valuable, it is important that researchers, including communication researchers, know how to detect outliers, analyze them to determine why the outlier exists, and understand their impact. This entry examines the detection and analysis of outliers and outlier labeling methods.
Outlier Detection and Analysis
Outlier detection methods create probabilistic, statistical, or algorithmic models that characterize the normal behavior of the data and then based on that analysis identify what values should be considered outliers. Researchers must determine which model type to use for outlier detection and are influenced by several factors, including data type, data size, and the need for interpretability. Interpretability is important because it can explain why a data point is an outlier, providing the researcher valuable information about how to handle the outlier. The choice of the underlying data model is extremely important because outliers can only be determined based on the underlying distribution of the data. If the data are not modeled correctly, then data points will be erroneously characterized as outliers or as normal parts of the data sets.
In probabilistic and statistical models, the data are modeled as a probability distribution whereby the parameters of the model are learned through the data set itself. Such models generate probabilities that data points are from different clusters of the data set, providing evidence of outliers by determining which data points have very low fit within those clusters. Probabilistic models can be applied to almost any data type; since the models are based on probability, the issues of normality are already accounted for within the tests. However, probabilistic models generally try to fit data to a particular type of distribution, depending on the model choice, which may not be appropriate for the data set. Such models are also harder to interpret, which can lead to poor understandings of why a data point is being considered an outlier. Statistical models typically rely on assumptions of normality; therefore, an analyst must make sure that the data meet the assumptions of the test. The difficulty of outlier detection increases when there exists a significant relationship among the data points. In such instances, time-series and network data analyses are used because the relationship patterns among the data points helps determine the outliers. Algorithms (and meta-algorithms) have become valuable in detecting outliers in large sets of multivariate data, typically in fields outside of communication. However, in communication research, outlier detection typically occurs with univariate data, and the most common outlier detection tests are designed to accommodate such data.
Most outlier analyses begin with a determination of the normality of the data set. Checking the data for normality is important because it allows the analyst to choose the proper underlying model for the distribution and therefore increase the accuracy of detecting outliers. However, formal tests of normality can be influenced by the presence of outliers. Therefore, in addition to running formal tests of normality, it is generally considered good practice to plot the data using a normal probability plot and visually inspect it for outliers. Scatter plots, box plots, and histograms can also be used as graphical tools to check normality and inspect the data for outliers.
When the distribution is normal, the most basic form of outlier detection is extreme value analysis. Extreme value analysis determines a specific type of outlier, one that is too small or too large to realistically belong to a data set. However, if the underlying distribution is not normal, extreme values in a data set may not be outliers. For example, a bimodal data set in which the data bunch around the extremes would make scores falling in the center of the distribution outliers, rather than those at the ends. So while extreme value analysis is a common form of outlier detection, it is important to note that it relies on a normal distribution for its accuracy.
Once the underlying distribution of the data set has been determined, there are formal tests that can be used to detect outliers. Formal tests, sometimes referred to as tests of discordancy, attempt to detect outliers by generating values that are then tested for significance. A common test for identifying a single outlier is the Grubbs test. The Grubbs test determines if the maximum or minimum value of a data set is an outlier. Another common test, the Tietjen-Moore test, is similar to the Grubbs test but is used to test cases of multiple outliers. Both the Grubbs and Tietjen-Moore tests require the number of outliers to be specified by the data analyst before the test is run. If a data analyst does not know the number of outliers present in the data, then the generalized extreme studentized deviate (ESD) can be used. The ESD can detect single or multiple outliers by setting an upper bound on the number of outliers and allowing the test to determine how many outliers are present.
Such formal tests are quite powerful when the data set meets distribution assumptions, but they can become problematic when those assumptions break down, when the distribution is unknown, or when it is not a specific distribution type, such as normal, gamma, or exponential distributions. Tests of discordancy can also be vulnerable to masking or swamping problems. Masking is when one outlier masks a second outlier because the two outliers’ proximity to one another influences the detection test enough to suggest those points as part of the data set. Conversely, swamping occurs when the test identifies too many outliers by considering a second data point an outlier only under the presence of the first one. So, if the first outlier is deleted from the data set, then the second observation is no longer considered an outlier.
Outlier Labeling Methods
Outlier labeling methods are informal detection tests that can also be used to detect outliers. Such tests generate an interval or criterion for outlier detection, rather than relying on a statistical hypothesis. Once the interval or criterion is set, any value outside of those limits is considered an outlier. However, what constitutes an outlier is often subjective due to the inexact application of what should be considered sufficient deviation from the data set. One common method of outlier labeling is to use the known properties of the normal curve and define outliers by number of standard deviations from the mean. Typically any value that falls more than three standard deviations from the mean is considered an outlier. In a normal distribution, a value will only fall beyond three standard deviations less than 0.3% of the time.
Using z scores is another common outlier labeling method. Like the standard deviation method, the z score approach assumes a normal distribution and provides a reasonable criterion for identifying outliers. However, z scores do not work well with small distributions, especially any data set with fewer than 10 observations, and it is easily influenced by extreme scores. In such cases, the z score approach can easily fall victim to issues of masking. Therefore, some analysts use a modified z score that uses the median and median of the absolute deviation of the median instead of the mean and standard deviation. This method is less susceptible to the influence of extreme scores and can avoid the problem of masking.
Tukey’s method is another common outlier labeling approach. It utilizes a box-plot and quartiles to determine outliers. The approach defines an interquartile range (IQR) as the distance between the lower and upper quartiles and then sets “fences” at 3 IQRs below the first quartile and above the third. Any data point falling outside the “fences” is considered an outlier. Quartiles are useful because they are resistance to extreme values and therefore Tukey’s method does not have the inherent issues of z scores or the standard deviation method. It can also be used to evaluate distributions that are skewed because it makes no assumptions about the distribution and does not depend on the mean or standard deviation. There is also an adjusted version of Tukey’s test that can handle extremely skewed distributions.
The median absolute deviation (MAD) method is another common outlier labeling approach. It is a robust method unaffected by extreme values. It is similar to the standard deviation method except it uses the median and the MAD rather than mean and standard deviation to generate a standard score for a value (a MADe). MADe values greater than three are typically considered outliers.
While both the formal and informal tests can help determine if a value is an outlier, it is still up to the data analyst to determine what the outlier means in comparison with the overall data set. Outlier detection provides direction for which data point(s) a data analyst must inspect with more scrutiny. It will be up to the analyst to determine if the outlier is simply poorly entered data, a systemic problem with the data generation, or a novel phenomenon worth further investigation.
Matthew J. Gill
See also Data; Data Cleaning; Data Reduction; Data Trimming; Errors of Measurement; Normal Curve Distribution; Skewness
Further Readings
Aggarwal, C. C. (2013). Outlier analysis. New York, NY: Springer.
Barnett, V., & Lewis, T. (1994). Outliers in statistical data. Chichester, UK: Wiley.
Hawkins, D. M. (1980). Identification of outliers. London, UK: Chapman and Hall.
Rousseeuw, P. J., & Leroy, A. M. (2003). Robust regression and outlier detection. New York, NY: Wiley-Interscience.
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