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Multivariate statistics is a form of statistics encompassing the simultaneous observation and analysis of more than one outcome variable. The application of multivariate statistics is multivariate analysis.
Multivariate statistics concerns understanding the different aims and background of each of the different forms of multivariate analysis, and how they relate to each other. The practical implementation of multivariate statistics to a particular problem may involve several types of univariate and multivariate analyses in order to understand the relationships between variables and their relevance to the actual problem being studied.
In addition, multivariate statistics is concerned with multivariate probability distributions, in terms of both
Certain types of problem involving multivariate data, for example simple linear regression and multiple regression, are NOT usually considered as special cases of multivariate statistics because the analysis is dealt with by considering the (univariate) conditional distribution of a single outcome variable given the other variables.
There are many different models, each with its own type of analysis:
There is a set of probability distributions used in multivariate analyses that play a similar role to the corresponding set of distributions that are used in univariate analysis when the normal distribution is appropriate to a dataset. These multivariate distributions are:
The Inverse-Wishart distribution is important in Bayesian inference, for example in Bayesian multivariate linear regression. Additionally, Hotelling's T-squared distribution is a univariate distribution, generalising Student's t-distribution, that is used in multivariate hypothesis testing.
Anderson's 1958 textbook, An Introduction to Multivariate Analysis,^{[2]} educated a generation of theorists and applied statisticians; Anderson's book emphasizes hypothesis testing via likelihood ratio tests and the properties of power functions: Admissibility, unbiasedness and monotonicity.^{[3]}^{[4]}
There are an enormous number of software packages and other tools for multivariate analysis, including:
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