Confusing Statistical Term #9: Multiple Regression Model and Multivariate Regression Model

First Published 4/29/09;Stage 2
Updated 2/23/21 to give more detail.

Much like General Linear Model and Generalized Linear Model in #7, there are many examples in statistics of terms with (ridiculously) similar names, but nuanced meanings.

Today I talk about the difference between multivariate and multiple, as they relate to regression.

Multiple Regression

A regression analysis with one dependent variable and eight independent variables is NOT a multivariate regression model.  It’s a multiple regression model.

And believe it or not, it’s considered a univariate model.

This is uniquely important to remember if you’re an SPSS user. Choose Univariate GLM (General Linear Model) for this model, not multivariate.

I know this sounds crazy and misleading because why would a model that contains nine variables (eight Xs and one Y) be considered a univariate model?

It’s because of the fundamental idea in regression that Xs and Ys aren’t the same. We’re using the Xs to understand the mean and variance of Y. This is why the residuals in a linear regression are differences between predicted and actual values of Y. Not X.

(And of course, there is an exception, called Type II or Major Axis linear regression, where X and Y are not distinct. But in most regression models, Y has a different role than X).

It’s the number of Ys that tell you whether it’s a univariate or multivariate model. That said, other than SPSS, I haven’t seen anyone use the term univariate to refer to this model in practice. Instead, the assumed default is that indeed, regression models have one Y, so let’s focus on how many Xs the model has. This leads us to…

Simple Regression: A regression model with one Y (dependent variable) and one X (independent variable).

Multiple Regression: A regression model with one Y (dependent variable) and more than one X (independent variables).

References below.

Multivariate Regression

Multivariate analysis ALWAYS describes a situation with multiple dependent variables.

So a multivariate regression model is one with multiple Y variables. It may have one or more than one X variables. It is equivalent to a MANOVA: Multivariate Analysis of Variance.

Other examples of Multivariate Analysis include:

  • Principal Component Analysis
  • Factor Analysis
  • Canonical Correlation Analysis
  • Linear Discriminant Analysis
  • Cluster Analysis

But wait. Multivariate analyses like cluster analysis and factor analysis have no dependent variable, per se. Why is it about dependent variables?

Well,  it’s not really about dependency.  It’s about which variables’ mean and variance is being analyzed.  In a multivariate regression, we have multiple dependent variables, whose joint mean is being predicted by the one or more Xs. It’s the variance and covariance in the set of Ys that we’re modeling (and estimating in the Variance-Covariance matrix).

Note: this is actually a situation where the subtle differences in what we call that Y variable can help.  Calling it the outcome or response variable, rather than dependent, is more applicable to something like factor analysis.

So when to choose multivariate GLM?  When you’re jointly modeling the variation in multiple response variables.

References

In response to many requests in the comments, I suggest the following references.  I give the caveat, though, that neither reference compares the two terms directly. They simply define each one. So rather than just list references, I’m going to explain them a little.

  1. Neter, Kutner, Nachtsheim, Wasserman’s Applied Linear Regression Models, 3rd ed. There are, incidentally, never editions with slight changes in authorship. But I’m citing the one on my shelf.

Chapter 1, Linear Regression with One Independent Variable, includes:

“Regression model 1.1 … is “simple” in that there is only one predictor variable.”

Chapter 6 is titled Multiple Regression – I, and section 6.1 is “Multiple Regression Models: Need for Several Predictor Variables.” Interestingly enough, there is no direct quotable definition of the term “multiple regression.” Even so, it’s pretty clear. Go read the chapter to see.

There is no mention of the term “Multivariate Regression” in this book.

2. Johnson & Wichern’s Applied Multivariate Statistical Analysis, 3rd ed.

Chapter 7, Multivariate Linear Regression Models, section 7.1 Introduction. Here it says:

“In this chapter we first discuss the multiple regression model for the prediction of a single response. This model is then generalized to handle the prediction of several dependent variables.” (Emphasis theirs).

They finally get to Multivariate Multiple Regression in Section 7.7. Here they “consider the problem of modeling the relationship between m responses, Y1, Y2, …,Ym, and a single set of predictor variables.”

Misuses of the Terms

I’d be shocked, however, if there aren’t some books or articles out there where the terms are not used or defined  the way I’ve described them here, according to these references. It’s very easy to confuse these terms, even for those of us who should know better.

And honestly, it’s not that hard to just describe the model instead of naming it. “Regression model with four predictors and one outcome” doesn’t take a lot more words and is much less confusing.

If you’re ever confused about the type of model someone is describing to you, just ask.

Read More Explanations of Confusing Statistical Terms.

Four Critical Steps in Building Linear Regression Models
While you’re worrying about which predictors to enter, you might be missing issues that have a big impact your analysis. This training will help you achieve more accurate results and a less-frustrating model building experience.

Reader Interactions

Comments

  1. EDWARD IDIGO says

    Thank you for the clear explanation of the Multivariate Regression as against Multiple Regression.

    Please how do I cite this write up in the APA Style.

    THANK YOU

  2. Sean Daly says

    Hi there,

    I have a question…my dissertation committee is asking why I would choose MLR vs a multivariate analysis like MANCOVA or MANOVA. I have 8 IV’s and 5 DV’s in the model and thus ran five MLR’s, each with 8 IV’s and 1 DV. Can you help me explain to them why?

    Thank you.

    • Saikat Kar says

      There may be inter dependence between Y’s and when u are fitting MLR with individual Y’s ; you are ignoring that dependence between Y’s.

  3. Tyler says

    Hi Karen,

    I was wondering — what is the advantage of using multivariate regression instead of univariate regression for each dependent variable? Thanks

  4. neda says

    Hi
    I have a qusetion in this area. Shoud we care about the relstion ship between predictors which we are putting in multiple regression analysis or we can put all of them that has sinificant PValue in univariat univariable analysis in multiple regression ??

  5. sunny says

    Hello Karen,
    I would like to know whether it is possible to do difference in difference analysis by using multiple dependent and independent variables?

    Thank you.

    Sunny

  6. Hazel says

    Hi, I would like to know when will usually we need to us multivariate regression? It’s when there is two dependent variables?

  7. hayder says

    hi
    may I ask why the result of univariable regression differs from multivariable regression for the same tested values?

    thanks

  8. Ram says

    Hello Karen,
    “A regression analysis with one dependent variable and 8 independent variables is NOT a multivariate regression. It’s a multiple regression. Multivariate analysis ALWAYS refers to the dependent variable”…
    ………………..Can you please give some reference for this quote??

  9. Bonnie says

    Hi Karen,

    Just wondered what your take is on using the terms Univariate or Bivariate analysis when you are talking about testing an association between two variables (such as exposure and an outcome variable)? I have seen both terms used in the situation and I was wondering if they can be used interchangeably? Kind Regards Bonnie

    • Karen says

      Good question.

      When you’re talking about descriptive statistics, univariate means a single variable, so an association would be bivariate.

      But once you’re talking about modeling, the term univariate or multivariate refers to the number of dependent variables. You don’t ever tend to use bivariate in that context. But for example, a univariate anova has one dependent variable whereas a multivariate anova (MANOVA) has two or more.

      This is why a regression with one outcome and more than one predictor is called multiple regression, not multivariate regression.

  10. sylvia says

    Hi Karen,
    I have a question about multiple regression, when we choose predictors to include in the regression model based on univariate analysis, do we set the P-value at 0.1 or 0.2? Or it should be at the level of 0.05?

    Thanks

  11. Suresh Kumar says

    Hello there,
    My name is Suresh Kumar. Currently, I’m learning multivariate analysis, since i am only familiar with multiple regression. I want to ask you about my doubt in Factor Analysis (FA)in searching the dominant FACTOR not Factors. in Multiple Regression (MR)we can use t-test best on the residual of each independent variable.
    My doubt is whether FA is only to find factors not the dominant factor or we can also use it to find the dominant factor as what we can in MR. Instead of data reduction, what else can we do with FA?
    Once we have done getting the factors through FA, is it possible to use MR to find the influence or impact on something? or from FA we continue to Confirmatory FA and next using SEM?
    If FA to deal with dependent variables, then how to check the factors influencing the dependent variables?
    Are we dealing with multiple dependent variables and multiple independent variables if we want to find out the influencing factors?
    Thanking you in advance.
    Regards
    Suresh Kumar

    • Karen says

      Hi Suresh,

      Factor Analysis is doing something totally different than multiple regression. You’re right, it’s for data reduction, but specifically in a situation where theoretically there is a latent variable.

      You can then use the factor scores, in a MR, and that is equivalent to running an SEM.

      A really great book with all the details on this is Larry Hatcher’s book on Factor Analysis and SEM using SAS. I forget the exact title, but you can easily search for it. Even if you don’t use SAS, he explains the concepts and the steps so well, it’s worth getting.

      Best,
      Karen


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