# Multivariate analysis

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Multivariate analysis (MVA) is based on the statistical principle of multivariate statistics, which involves observation and analysis of more than one statistical variable at a time. In design and analysis, the technique is used to perform trade studies across multiple dimensions while taking into account the effects of all variables on the responses of interest.

Uses for multivariate analysis includes:

• Design for capability (also known as capability-based design)
• Inverse design, where any variable can be treated as an independent variable
• Analysis of alternatives, the selection of concepts to fulfill a customer need
• Analysis of concepts with respect to changing scenarios
• Identification of critical design drivers and correlations across hierarchical levels

Multivariate analysis can be complicated by the desire to include physics-based analysis to calculate the effects of variables for a hierarchical "system-of-systems." Often, studies that wish to use multivariate analysis are stalled by the dimensionality of the problem. These concerns are often eased through the use of surrogate models, highly accurate approximations of the physics-based code. Since surrogate models take the form of an equation, they can be evaluated very quickly. This becomes an enabler for large-scale MVA studies: while a Monte Carlo simulation across the design space is difficult with physics-based codes, it becomes trivial when evaluating surrogate models, which often take the form of response surface equations.

## Cautions

While multivariate analysis has become a major statistical tool and is commonly used for adjustment — the process of correcting the main effect for multiple variables that confound the relation between exposure and outcome in an observational study. It is also true that the actual relations between variables is increasingly opaque to readers, researchers and editors.. The most common summary statistics of multivariate analyses — ratio measures such as odds ratio and relative risk — obscure the most fundamental measure of occurrence: the frequency of the outcome. Welsch, Schwartz and Woloshin (2005) explored this issue using 2 case studies of original investigations from 2 prominent medical journals. Both used population-based data of the highest quality and appropriately called on multivariate analysis to adjust for important confounders. In both cases, however, they demonstrated how statistical assumptions produced misleading results. They suggested guidelines for researchers who are communicating findings of multivariate analyses and for their readers.

## References

• KV Mardia, JT Kent, and JM Bibby (1979). Multivariate Analysis. Academic Press,.
• Welch, HG Schwartz, LM and Woloshin, S (2005).The exaggerated relations between diet, body weight and mortality: the case for a categorical data approach CMAJ,172 (7). doi:10.1503/cmaj.1041310.Full text