When is it important to use adjusted R-squared instead of R-squared?

R², the the Coefficient of Determination, is one of the most useful and intuitive statistics we have in linear regression.

It tells you how well the model predicts the outcome and has some nice properties. But it also has one big drawback.

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*First Published 4/29/09;*

*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.

- 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, Y_{1}, Y_{2}, …,Y_{m}, 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**.

*by Manolo Romero Escobar*

General Linear Model (GLM) is a tool used to understand and analyse linear relationships among variables. It is an umbrella term for many techniques that are taught in most statistics courses: ANOVA, multiple regression, etc.

In its simplest form it describes the relationship between two variables, “y” (dependent variable, outcome, and so on) and “x” (independent variable, predictor, etc). These variables could be both categorical (how many?), both continuous (how much?) or one of each.

Moreover, there can be more than one variable on each side of the relationship. One convention is to use capital letters to refer to multiple variables. Thus *Y* would mean multiple dependent variables and *X* would mean multiple independent variables. The most known equation that represents a GLM is: (more…)

One of the biggest challenges in learning statistics and data analysis is learning the lingo. It doesn’t help that half of the notation is in Greek (literally).

The terminology in statistics is particularly confusing because often the same word or symbol is used to mean completely different concepts.

I know it feels that way, but it really isn’t a master plot by statisticians to keep researchers feeling ignorant.

*Really.*

It’s just that a lot of the methods in statistics were created by statisticians working in different fields–economics, psychology, medicine, and yes, straight statistics. Certain fields often have specific types of data that come up a lot and that require specific statistical methodologies to analyze.

Economics needs time series, psychology needs factor analysis. Et cetera, et cetera.

But separate fields developing statistics in isolation has some ugly effects.

Sometimes different fields develop the same technique, but use *different names* or notation.

Other times different fields use the same name or notation on *different techniques* they developed.

And of course, there are those terms with slightly different names, often used in similar contexts, but with different meanings. These are never used interchangeably, but they’re easy to confuse if you don’t use this stuff every day.

And sometimes, there are different terms for subtly different concepts, but people use them interchangeably. (I am guilty of this myself). It’s not a big deal if you understand those subtle differences. But if you don’t, it’s a mess.

And it’s not just fields–it’s software, too.

SPSS uses different names for the exact same thing in different procedures. In GLM, a continuous independent variable is called a Covariate. In Regression, it’s called an Independent Variable.

Likewise, SAS has a Repeated statement in its GLM, Genmod, and Mixed procedures. They all get at the same concept there (repeated measures), but they deal with it in drastically different ways.

So once the fields come together and realize they’re all doing the same thing, people in different fields or using different software procedures, are already used to using their terminology. So we’re stuck with different versions of the same word or method.

So anyway, I am beginning a series of blog posts to help clear this up. Hopefully it will be a good reference you can come back to when you get stuck.

We’ve expanded on this list with a member training, if you’re interested.

If you have good examples, please post them in the comments. I’ll do my best to clear things up.