Understanding Multivariate Normality in Data with EDA

Understanding multivariate normality is crucial in multivariate statistical analysis, where many statistical techniques assume that the data follows a multivariate normal distribution. Exploratory Data Analysis (EDA) offers a practical framework for assessing whether this assumption holds, enabling data scientists and analysts to make more accurate inferences and decisions.

The Concept of Multivariate Normality

Multivariate normality extends the idea of the normal distribution to higher dimensions. A dataset with multiple variables is said to follow a multivariate normal distribution if every linear combination of its variables is normally distributed. In simpler terms, not only should each individual variable (univariate) be normally distributed, but every pair, triplet, or higher combination of variables should also follow a joint normal distribution.

Multivariate normal distributions are characterized by a mean vector and a covariance matrix. The mean vector describes the central tendency, while the covariance matrix describes the spread and the interrelationships between the variables.

This assumption underlies many statistical models, including:

• Multivariate Analysis of Variance (MANOVA)

• Linear Discriminant Analysis (LDA)

• Canonical Correlation Analysis

• Principal Component Analysis (PCA), to some extent

• Hotelling’s T² test

• Structural Equation Modeling

Violating this assumption can lead to biased estimates, misleading results, and incorrect conclusions, making it vital to assess multivariate normality before applying these techniques.

Importance of Exploratory Data Analysis in Assessing Multivariate Normality

EDA provides a combination of visual and quantitative tools to inspect data, detect anomalies, and uncover underlying patterns, all of which are essential in determining whether the data aligns with the assumptions of multivariate normality. The goal is not only to check for normality but to understand the nature of any deviations from it.

Steps to Assess Multivariate Normality Using EDA

1. Univariate Normality Checks

Before diving into multivariate methods, each variable should be checked for univariate normality. This step is necessary but not sufficient for multivariate normality.

Techniques include:

• Histograms and Density Plots: These plots help visualize the distribution of individual variables. A bell-shaped curve indicates normality.

• Box Plots: Useful for identifying outliers that may affect normality.

• Q-Q (Quantile-Quantile) Plots: These plots compare the quantiles of the data against a theoretical normal distribution.

• Shapiro-Wilk and Anderson-Darling Tests: Statistical tests that formally test the null hypothesis of normality for each variable.

While each variable might appear normally distributed on its own, multivariate normality requires checking the relationships between variables as well.

2. Scatterplot Matrix

Also known as a pair plot, this visual tool plots each variable against every other variable in the dataset.

• Elliptical Shapes: In a multivariate normal distribution, scatterplots between pairs of variables should exhibit an elliptical pattern.

• Non-linear Patterns or Clustering: Indicate possible deviations from multivariate normality, such as skewness or the presence of subgroups.

Scatterplot matrices provide an intuitive way to observe interactions and detect non-linear relationships or outliers that may distort the overall distribution.

3. Multivariate Outlier Detection

Multivariate outliers can significantly affect the multivariate normality assumption. Even if the variables individually appear normal, the presence of extreme values in the multivariate space can distort the analysis.

Common methods to detect multivariate outliers:

• Mahalanobis Distance: Measures the distance of a point from the mean of the distribution, taking into account the covariance structure. High Mahalanobis distances indicate potential outliers.

• Chi-Square Plot: When plotting Mahalanobis distances against the quantiles of the chi-square distribution, points lying far from the straight line may be outliers.

• Leverage and Influence Plots: These can detect points that disproportionately affect multivariate models.

4. Mardia’s Test for Multivariate Normality

Mardia’s test is one of the most widely used statistical methods to assess multivariate normality.

• Skewness Statistic: Measures asymmetry in the data. Significant skewness indicates deviation from normality.

• Kurtosis Statistic: Measures the “tailedness” of the distribution. Large deviations from expected kurtosis values under normality suggest non-normality.

Mardia’s test provides a formal statistical assessment, complementing the visual insights from EDA.

5. Royston’s Test

Royston’s test is an extension of the Shapiro-Wilk test for multivariate data. It’s particularly effective for small sample sizes and is more powerful than Mardia’s test in some scenarios.

6. Henze-Zirkler’s Test

This test is based on a consistent and affine-invariant statistic. It is often considered one of the most reliable methods for testing multivariate normality, especially for moderate to large sample sizes.

7. Q-Q Plots of Mahalanobis Distances

This specialized Q-Q plot compares the ordered Mahalanobis distances of observations to the expected chi-square distribution.

• If the data are multivariate normal, the points should lie roughly along a straight line.

• Deviations indicate non-normality, often pointing toward outliers or non-elliptical distribution shapes.

8. PCA Biplots

Principal Component Analysis can be a useful exploratory tool for assessing the shape of the data distribution in reduced dimensions.

• Symmetrical Biplots: Often suggest that the data is close to multivariate normality.

• Long Tails or Asymmetric Clustering: Indicate deviations.

Although PCA assumes linear relationships, it can still provide valuable insight into the structure and distribution of the data.

Dealing with Non-Normality

If multivariate normality is violated, analysts can take corrective actions:

• Data Transformation: Log, square root, or Box-Cox transformations can help normalize skewed variables.

• Outlier Treatment: Identifying and removing or down-weighting outliers may improve the multivariate distribution.

• Robust Statistical Methods: Some modern methods (e.g., robust PCA, bootstrapping) are less sensitive to non-normality.

• Non-Parametric Alternatives: When normality cannot be assumed, non-parametric tests that do not rely on distributional assumptions may be more appropriate.

Practical Applications

Understanding and testing for multivariate normality is essential in fields such as:

• Finance: For portfolio optimization, risk modeling, and asset pricing where returns are often assumed to follow a multivariate normal distribution.

• Marketing and Customer Analytics: Segmentation and predictive modeling often involve multivariate techniques.

• Medical and Biological Research: Multivariate measurements from clinical trials or genetic studies often require validation of normality.

• Engineering and Quality Control: Multivariate control charts assume normality for process monitoring.

Final Thoughts

Multivariate normality is a foundational assumption for many advanced statistical techniques. EDA provides a powerful toolkit to explore this assumption thoroughly through both visual and quantitative means. By leveraging techniques like scatterplot matrices, Mahalanobis distances, and Mardia’s test, analysts can validate or refute the normality assumption with confidence, ensuring the reliability and robustness of their multivariate analyses.