Linear Regression

Statistical contrasts are a tool for testing specific hypotheses and model effects, particularly comparing specific group means.

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What are goodness of fit statistics? Is the definition the same for all types of statistical model? Do we run the same tests for all types of statistic model?

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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, newer 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, …,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.

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

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Every time you analyze data, you start with a research question and end with communicating an answer. But in between those start and end points are twelve other steps. I call this the Data Analysis Pathway. It’s a framework I put together years ago, inspired by a client who kept getting stuck in Weed #1. But I’ve honed it over the years of assisting thousands of researchers with their analysis.

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Have you ever compared the list of model assumptions for linear regression across two sources? Whether they’re textbooks, lecture notes, or web pages, chances are the assumptions don’t quite line up.

Why? Sometimes the authors use different terminology. So it just looks different.

And sometimes they’re including not only model assumptions, but inference assumptions and data issues. All are important, but understanding the role of each can help you understand what applies in your situation.

Model Assumptions

The actual model assumptions are about the specification and performance of the model for estimating the parameters well.

1. The errors are independent of each other
2. The errors are normally distributed
3. The errors have a mean of 0 at all values of X
4. The errors have constant variance
5. All X are fixed and are measured without error
6. The model is linear in the parameters
7. The predictors and response are specified correctly
8. There is a single source of unmeasured random variance

Not all of these are always explicitly stated. And you can’t check them all. How do you know you’ve included all the “correct” predictors?

But don’t skip the step of checking what you can. And for those you can’t, take the time to think about how likely they are in your study. Report that you’re making those assumptions.

Assumptions about Inference

Sometimes the assumption is not really about the model, but about the types of conclusions or interpretations you can make about the results.

These assumptions allow the model to be useful in answering specific research questions based on the research design. They’re not about how well the model estimates parameters.

Is this important? Heck, yes. Studies are designed to answer specific research questions. They can only do that if these inferential assumptions hold.

But if they don’t, it doesn’t mean the model estimates are wrong, biased, or inefficient. It simply means you have to be careful about the conclusions you draw from your results. Sometimes this is a huge problem.

But these assumptions don’t apply if they’re for designs you’re not using or inferences you’re not trying to make. This is a situation when reading a statistics book that is written for a different field of application can really be confusing. They focus on the types of designs and inferences that are common in that field.

It’s hard to list out these assumptions because they depend on the types of designs that are possible given ethics and logistics and the types of research questions. But here are a few examples:

1. ANCOVA assumes the covariate and the IV are uncorrelated and do not interact. (Important only in experiments trying to make causal inferences).
2. The predictors in a regression model are endogenous. (Important for conclusions about the relationship between Xs and Y where Xs are observed variables).
3. The sample is representative of the population of interest. (This one is always important!)

Data Issues that are Often Mistaken for Assumptions

And sometimes the list of assumptions includes data issues. Data issues are a little different.

They’re important. They affect how you interpret the results. And they impact how well the model performs.

But they’re still different. When a model assumption fails, you can sometimes solve it by using a different type of model. Data issues generally stay around.

That’s a big difference in practice.

Here are a few examples of common data issues:

1. Small Samples
2. Outliers
3. Multicollinearity
4. Missing Data
5. Truncation and Censoring
6. Excess Zeros

So check for these data issues, deal with them if the solution doesn’t create more problems than you solved, and be careful with the inferences you draw when you can’t.

Go to the next article or see the full series on Easy-to-Confuse Statistical Concepts

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You think a linear regression might be an appropriate statistical analysis for your data, but you’re not entirely sure. What should you check before running your model to find out?

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