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Member Training: Goodness of Fit Statistics

March 4th, 2021 by


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

February 20th, 2021 by

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.
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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:

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, …,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.

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


Member Training: Missing Data

December 1st, 2020 by

Missing data causes a lot of problems in data analysis. Unfortunately, some of the “solutions” for missing data cause more problems than they solve.

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Member Training: Preparing to Use (and Interpret) a Linear Regression Model

November 1st, 2020 by

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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Member Training: Using Open Data in Research: Opportunities and Challenges

October 1st, 2020 by

Open data, particularly government open data is a rich source of information that can be helpful to researchers in almost every field, but what is open data? How do we find what we’re looking for? What are some of the challenges with using data directly from city, county, state, and federal government agencies?

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Same Statistical Models, Different (and Confusing) Output Terms

January 7th, 2020 by

Learning how to analyze data can be frustrating at times. Why do statistical software companies have to add to our confusion?Stage 2

I do not have a good answer to that question. What I will do is show examples. In upcoming blog posts, I will explain what each output means and how they are used in a model.

We will focus on ANOVA and linear regression models using SPSS and Stata software. As you will see, the biggest differences are not across software, but across procedures in the same software.

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