In this post we will discuss univariate and multivariate outliers. A univariate outlier is a data point that consists of an extreme value on one variable. A multivariate outlier is a combination of unusual scores on at least two variables. Both types of outliers can influence the outcome of statistical analyses. Outliers exist for four reasons. Incorrect data entry can cause data to contain extreme cases. A second reason for outliers can be failure to indicate codes for missing values in a dataset. Another possibility is that the case did not come from the intended sample. And finally, the distribution of the sample for specific variables may have a more extreme distribution than normal.
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In many parametric statistics, univariate and multivariate outliers must be removed from the dataset. When looking for univariate outliers for continuous variables, standardized values (z scores) can be used. If the statistical analysis to be performed does not contain a grouping variable, such as linear regression, canonical correlation, or SEM among others, then the data set should be assessed for outliers as a whole. If the analysis to be conducted does contain a grouping variable, such as MANOVA, ANOVA, ANCOVA, or logistic regression, among others, then data should be assessed for outliers separately within each group. For continuous variables, univariate outliers can be considered standardized cases that are outside the absolute value of 3.29. However, caution must be taken with extremely large sample sizes, as outliers are expected in these datasets. Once univariate outliers have been removed from a dataset, multivariate outliers can be assessed for and removed.
Multivariate outliers can be identified with the use of Mahalanobis distance, which is the distance of a data point from the calculated centroid of the other cases where the centroid is calculated as the intersection of the mean of the variables being assessed. Each point is recognized as an X, Y combination and multivariate outliers lie a given distance from the other cases. The distances are interpreted using a p < .001 and the corresponding χ2 value with the degrees of freedom equal to the number of variables. Multivariate outliers can also be recognized using leverage, discrepancy, and influence. Leverage is related to Mahalanobis distance but is measured on a different scale so that the χ2 distribution does not apply. Large scores indicate the case if further out however may still lie on the same line. Discrepancy assesses the extent that the case is in line with the other cases. Influence is determined by leverage and discrepancy and assesses changes in coefficients when cases are removed. Cases > 1.00 are likely to be considered outliers.