Difference between Univariate Linear Regression and Simple Linear Regression? Is there any difference between Univariate Linear Regression and Simple Linear Regression? If so, what is the difference exactly? It seems both of them are exactly same. I would appreciate if anyone could cite a scientific paper that defines Univariate Linear Regression.
 A: Simple Linear Regression is defined in as model with a single explanatory variable (i.e., the independent variable).
According to this answer,, Univariate Linear Regression refers to a model with a single response variable (i.e., the dependent variable). This answer corroborates the theory.
Now, here is a claim that says Simple regression necessarily has a single dependent variable too, but I cannot verify the claim. A model with one explanatory variable and more than one response variable will still be called simple (and multivariate), I think.
I have seen the terms 'Simple' and 'Univariate' used interchangeably, and was under the impression that there is no difference. But I reckon it is best to keep that distinction.
A: Purist view: In general, there is clear difference between those two terms.
Univariate might have e.g. two explanatory variables (or five etc.) but always just one response variable. But it can also have just one expl. variable. In that scenario it is a simple linear regression as well.
Now, the question remains whether simple linear regression can have multiple response variables. According to Wikipedia (citing multiple text books), no, although not stated explicitly. (Subquestion: Can regression have more response than explanatory variables?)
Practical view: If you have one response variable (univariate) and multiple explanatory ones, you should call it (univariate) multiple regression. Wherever you have more than one variable, you need to be explicit about it. Otherwise one variable is expected, thus the term univariate is redundant.
In conclusion, univariate r. and simple linear regression describe (most likely) the same kind of regression, but in most cases it will not make sense to use the former term. (The opposite is only true when you contrast it with multivariate regression.)
A: Univariate Model : 1) Simple :  1 DV & 1 IV
               2) Multiple: 1 DV & many IV 

Multivariate Model: 1) Simple Multivariate:   many DV & 1 IV
                2) Multiple Multivariate: many DV & many IV 

