Multivariable vs multivariate regression [duplicate]

I am a little unsure about the semantics in this regard, and was hoping someone could cast some light on this.

As far as I understand, multivariable is basically one dependent variable and several independent variables, whereas multivariate is when several dependent variables are included. However, in the examples I've seen this is often related to repeated measures.

Instead, let's say I have some data (independent variables), and then three dependent variables. However, they are not repeated measures. Maybe I wanted to "measure" the health of a person, e.g. this could include: weight, lung capacity, and resting heart rate - just to pick some arbitrary ones.

So basically these dependent variables might give me some insight to a persons health. If one of the independent variables are smoking for example, then this might affect all three dependent variables, but maybe other independent variables will only influence one of each.

In this case, what am I performing ? Multivariable regression, multivariate regression, a mix, or...?

Multivariable regression is any regression model where there is more than one explanatory variable. For this reason it is often simply known as "multiple regression". In the simple case of just one explanatory variable, this is sometimes called univariable regression.

Unfortunately multivariable regression is often mistakenly called multivariate regression, or vice versa. Multivariate regression is any regression model in which there is more than one outcome variable. In the more usual case where there is just one outcome variable, this is also known as univariate regression.

Thus we can have:

• univariate multivariable regression. A model with one outcome and several explanatory variables. This is probably the most common regression model and will be familiar to most analysts, and is often just called multiple regression; sometimes (where the link function is the identity function) it is called the General Linear Model (not Generalized).

• univariate univariable regression. One outcome, one explanatory variable, often used as the introductory example in a first course on regression models.

• multivariate multivariable regression. Multiple outcomes, multiple explanatory variable. This is the scenario described in the question.

• multivariate univariable regression. Multiple outcomes, single explanatory variable. An example of this is Hotelling's T-Squared test, a multivariate counterpart of the T-test (thanks to @Dave for pointing this out).

The above is standard terminology in applied fields I have worked in: biostatistics, social sciences and psychology. I would not be surprises if other domains use the terms differemtly.

• Multiple outcomes, single explanatory variable...that’s a two-sample Hotelling $T^2$ test on a binary variable denoting group membership.
– Dave
Feb 2 '20 at 8:27
• @Dave Never heard of that. Thanks - something new to read about :) Feb 2 '20 at 9:06

Multivariate regression should refer to a situation like you’ve described where the response has multiple related dimensions such as lung capacity and heart rate. Determining how each predictor affects each dimension of the response falls to model building, but it doesn’t change the fact that the regression is multivariate.

I don’t know “multivariable regression” as a term, but I could see it referring to a multivariate regression of just a regular regression with multiple predictors of one response variable. As the latter is (much) more common, I would expect that if I saw “multivariable regression” in a job description. (Ditto if I read about “multivariate regression” in a job description, even if that’s not the typical use of the term.)

Yes, there’s plenty of abuse of terminology out there. The one that sends me through the roof is “multilinear regression”. “Multilinear” has a specific meaning in mathematics, and the “multiple linear regression” that someone means when she calls it “multilinear” is linear, not multilinear.