I got the answers from http://stats.stackexchange.com/questions/2358/explain-the-difference-between-multiple-regression-and-multivariate-regression

Example 1

Suppose that a university wishes to refine its admission criteria so that they admit 'better' students. Also, suppose that a student's grade Point Average (GPA) is what the university wishes to use as a performance metric for students. They have several criteria in mind such as high school GPA (HSGPA), SAT scores (SAT), Gender etc and would like to know which one of these criteria matter as far as GPA is concerned.

Solution: Multiple Regression

In the above context, there is one dependent variable (GPA) and you have multiple independent variables (HSGPA, SAT, Gender etc). You want to find out which one of the independent variables are good predictors for your dependent variable. You would use multiple regression to make this assessment.

Example 2

Instead of the above situation, suppose the admissions office wants to track student performance across time and wishes to determine which one of their criteria drives student performance across time. In other words, they have GPA scores for the four years that a student stays in school (say, GPA1, GPA2, GPA3, GPA4) and they want to know which one of the independent variables predict GPA scores better on a year-by-year basis. The admissions office hopes to find that the same independent variables predict performance across all four years so that their choice of admissions criteria ensures that student performance is consistently high across all four years.

Solution: Multivariate Regression

In example 2, we have multiple dependent variables (i.e., GPA1, GPA2, GPA3, GPA4) and multiple independent variables. In such a situation, you would use multivariate regression.

This means that:

Am I correct?

I got the answers from http://stats.stackexchange.com/questions/2358/explain-the-difference-between-multiple-regression-and-multivariate-regression

Example 1

Suppose that a university wishes to refine its admission criteria so that they admit 'better' students. Also, suppose that a student's grade Point Average (GPA) is what the university wishes to use as a performance metric for students. They have several criteria in mind such as high school GPA (HSGPA), SAT scores (SAT), Gender etc and would like to know which one of these criteria matter as far as GPA is concerned.

Solution: Multiple Regression

In the above context, there is one dependent variable (GPA) and you have multiple independent variables (HSGPA, SAT, Gender etc). You want to find out which one of the independent variables are good predictors for your dependent variable. You would use multiple regression to make this assessment.

Example 2

Instead of the above situation, suppose the admissions office wants to track student performance across time and wishes to determine which one of their criteria drives student performance across time. In other words, they have GPA scores for the four years that a student stays in school (say, GPA1, GPA2, GPA3, GPA4) and they want to know which one of the independent variables predict GPA scores better on a year-by-year basis. The admissions office hopes to find that the same independent variables predict performance across all four years so that their choice of admissions criteria ensures that student performance is consistently high across all four years.

Solution: Multivariate Regression

In example 2, we have multiple dependent variables (i.e., GPA1, GPA2, GPA3, GPA4) and multiple independent variables. In such a situation, you would use multivariate regression.

This means that:

Am I correct?

While this may be a useful distinction that should be used, I don't think there is a consensus on the terminology. I have heard statisticians refer to instances of one dependent variable and many predictors as "multivariate." In fact, I have heard this more than "multivariable."

Sometimes it's more useful to read the math than the description of the statistical techniques.

If a regression model contains of more than one explanatory (independent) variables, then it is known as multiple regression model. When, there is only one study variable, the regression is termed as Univariate regression. When there are more than one study variables, the regression is termed as multivariate regression.

Note that the multiple regressions is not same as multivariate regressions. The multiple regression is determined by the number of explanatory variables whereas multivariate regressions is determined by the number of study variables.

These all answers address the difference in how to run multiple or multivariate linear regression models (with some illustrative examples), which can be summarized as follows:

- simple liner regression: one y and one x variable (each is a series of n observations)

- multiple linear regression (MLR): one y and a number of x variables (each is a series of n observations)

- multivariate linear regression (MVLR) (or, more strictly: multivariate, multiple linear regression): a number of y and a number of x variables (each is a series of n observations).

But none of the answers tells clearly why one should run MVLR instead of a series of MLR. There is only a vague mentioning that they allow to test correlations between response variables and that the model "can learn more").

In fact, the coefficients, standard errors and p-values will be exactly the same (besides the fact that running one MLVR is faster than running a series of single MLR 'by hand').

What can be obtained from MLVR is covariance/variance matrix of all response/explanatory variables. A very nice explanation can be found here:

https://data.library.virginia.edu/getting-started-with-multivariate-multiple-regression/