# Multidimensional linear regression (not multiple linear regression)

Let $$p$$ be a positive integer and suppose that each observation in my data set is a length-$$p$$ multivariate normal vector, and I have $$n$$ (an integer) observations of the length-$$p$$ multivariate normal vector. So $$\vec{Y} = \beta_0 + \beta_1 \vec{X}_{1} + \cdots + \beta_k \vec{X}_{k} + \vec{\epsilon},$$ with $$\vec{\epsilon} \sim N_p(\vec{0}, \Sigma)$$, $$\Sigma$$ is a covariance matrix of an observation-vector, $$\beta_i \in \mathbb{R}$$ (for $$i \in \{0,1,\cdots,k\}$$) and $$X_i \in \mathbb{R}^p$$. I am in a situation where this model looks relevant to my problem, but I have never been taught how to generalize the usual regression model into one where each observation is itself a vector of size $$p>1$$.

Is this called multivariate multiple regression? How can I find literature for it? If I look up multivariate-, or multidimensional linear regression I only get stuff on the multivariate linear regression model (the case where $$p=1$$).

• See if this page answers your question. The terminology is confusing. In current best practice, what you describe seems to be a true "multivariate" regression in the sense of having multiple, potentially correlated outcomes. The case with p = 1 but multiple predictors is best called "multiple" regression, not "multivariate" (usage isn't always consistent). What you want seems to be "multivariate multiple regression." Chapter 7 of Harrell's Regression Modeling Strategies outlines some approaches.
– EdM
Commented Apr 10, 2023 at 19:14
• Thanks EdM. If you put it into an answer, I can mark it as accepted Commented Apr 10, 2023 at 19:33
• stats.stackexchange.com/search?q=multivariate+regression -- but beware, because many questions about multiple regression are mis-tagged by their OPs as being multivariate. See stats.stackexchange.com/questions/18151 and stats.stackexchange.com/questions/80342 for basic accounts.
– whuber
Commented Apr 10, 2023 at 19:59

Much confusion can come from the too-frequent lack of distinction between "multivariate" and "multiple" regression. Although one might argue that "multivariate" can describe any situation with multiple variables, it's best current practice to restrict "multivariate" to situations with multiple outcome variables. See Hidalgo, B and Goodman, M (2013) American Journal of Public Health 103: 39-40, or this page or this page. Having more than one predictor variable is then "multiple" or "multivariable" regression. This ideal distinction, unfortunately, is too often neglected; at least once I have published "multivariate" when I should have said "multivariable."

For your application, a classic multivariate multiple regression model would seem to be OK. This page illustrates such a model. Fox and Weisberg have an online appendix to their text that explains in detail. The point estimates end up the same as with separate regressions for each outcome, but the (co)variances are adjusted to take the correlations into account.

More generally, there are several ways to deal with correlated outcomes. Chapter 7 of Frank Harrell's Regression Modeling Strategies provides a useful overview in a table. That chapter focuses on generalized least squares, which avoids the very strict no-missing-values requirement of classical multivariate multiple regression.