Multivariate Statistical Methods
What are multivariate statistical methods?
Many studies collect multiple measurements on the same individual, sample, product, or process. For example, a pharmaceutical tablet might be characterized by hardness, dissolution rate, weight, and moisture content. A customer survey might collect ratings for various features of a product. Because these variables are often related, analyzing them one at a time can miss important patterns.
Multivariate statistical methods analyze multiple variables simultaneously. By considering relationships among variables, these methods can reveal structure that is difficult or impossible to see in separate univariate analyses.
Multivariate methods are commonly used to:
- Explore relationships among variables.
- Identify patterns in complex data sets.
- Reduce many variables to a smaller set of underlying dimensions.
- Group similar observations.
- Classify observations into known categories.
- Model and predict multiple responses simultaneously.
A major benefit of multivariate analysis is dimension reduction, the ability to summarize information from many variables with only a few components, factors, or latent variables while preserving the most important features of the data.
Types of multivariate methods
Multivariate methods are often grouped into two broad categories: unsupervised learning and supervised learning.
Unsupervised learning
Unsupervised methods are used when the goal is to understand the structure of a data set without specifying a response variable. For the multivariate component, all variables are treated as response variables. The analysis focuses on discovering patterns, relationships, and groups within the data.
Common objectives include:
- Understanding how variables are related.
- Identifying underlying dimensions or latent factors.
- Finding natural groupings of observations.
- Visualizing high-dimensional data in just two or three dimensions.
Methods such as principal component analysis, factor analysis, and cluster analysis are common unsupervised learning techniques used for exploration and description.
Supervised learning
Supervised methods are used when one or more variables are designated as responses and the remaining variables are used as predictors. The goal is to build a model that explains, predicts, or classifies outcomes.
Common objectives include:
- Predicting future observations.
- Modeling relationships between predictors and responses.
- Classifying observations into predefined groups.
- Simultaneously analyzing multiple response variables.
Many supervised learning methods are specifically used with multiple response variables. Other supervised learning methods (for example, principal component regression) treat the predictor variables as a multivariate problem, then use the results in a univariate model.
Methods such as partial least squares regression, principal component regression, discriminant analysis, and multivariate multiple regression are common supervised learning techniques used for prediction and classification.
Summary of common multivariate methods
| Method | Category | Typical use |
| Principal component analysis (PCA) | Unsupervised | Reduce dimensionality and visualize multivariate relationships. |
| Exploratory (common) factor analysis | Unsupervised | Identify latent factors that explain observed variables. |
| Cluster analysis | Unsupervised | Group observations into clusters. |
| Principal component regression (PCR) | Supervised | Build regression models when predictors are highly correlated. |
| Partial least squares (PLS) regression | Supervised | Predict one or more responses using many correlated predictors. |
| Discriminant analysis | Supervised | Classify observations into known groups. |
| Confirmatory factor analysis | Supervised | Test a hypothesized factor analysis model. |
| Structural equation modeling (SEM) | Supervised | Use as a general framework for many modeling techniques. |
| MANOVA | Supervised | Model several response variables simultaneously using a single grouping variable. |
| Multivariate multiple regression | Supervised | Model several response variables simultaneously using multiple predictor variables. |
Choosing a method
The appropriate multivariate method depends on the objective of the analysis.
- Use PCA when you want to reduce dimensionality or visualize complex data. PCA is often a first step in the modeling process when dealing with multivariate data, followed by other methods.
- Use exploratory (common) factor analysis when you believe underlying latent constructs drive the observed variables.
- Use cluster analysis when you want to discover groups of similar observations.
- Use PLS or PCR when predictors are numerous and highly correlated.
- Use discriminant analysis when the goal is classification.
- Use confirmatory factor analysis to test a hypothesized factor analysis model.
- Use SEM for general framework models, such as confirmatory factor analysis, path models, measurement error models, mediation analysis, and latent growth curve models, among others.
- Use MANOVA or multivariate multiple regression when multiple response variables must be analyzed simultaneously.
By leveraging relationships among variables, multivariate methods provide a powerful framework for exploring, understanding, and modeling complex data. They are widely used in science, engineering, manufacturing, healthcare, marketing, and many other fields where important insights emerge only when variables are considered together rather than one at a time.