The Analysis Factor
Statistical Consulting, Resources, and Statistics Workshops for Researchers
The field of statistics has a terminology problem.
It affects students’ ability to learn statistics. It affects researchers’ ability to communicate with statisticians; with collaborators in different fields; and of course, with the general public.
It’s easy to think the real issue is that statistical concepts are difficult. That is true. It’s not the whole truth, though. [Read more…] about Why Statistics Terminology is Especially Confusing
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.
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 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.
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.
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.”
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.
Every time you analyze data, you start with a research question and end with communicating an answer. In fact, defining that research question is vital to getting to the right 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.
[Read more…] about Four Weeds of Data Analysis That are Easy to Get Lost In
Have you ever compared the list of 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.
[Read more…] about The Difference Between Model Assumptions, Inference Assumptions, and Data Issues
Updated Dec 18, 2020 to add more detail
In your statistics class, your professor made a big deal about unequal sample sizes in one-way Analysis of Variance (ANOVA) for two reasons.
1. Because she was making you calculate everything by hand. Sums of squares require a different formula* if sample sizes are unequal, but statistical software will automatically use the right formula. So we’re not too concerned. We’re definitely using software.
2. Nice properties in ANOVA such as the Grand Mean being the intercept in an effect-coded regression model don’t hold when data are unbalanced. Instead of the grand mean, you need to use a weighted mean. That’s not a big deal if you’re aware of it. [Read more…] about When Unequal Sample Sizes Are and Are NOT a Problem in ANOVA
Mean imputation: So simple. And yet, so dangerous.
Perhaps that’s a bit dramatic, but mean imputation (also called mean substitution) really ought to be a last resort.
It’s a popular solution to missing data, despite its drawbacks. Mainly because it’s easy. It can be really painful to lose a large part of the sample you so carefully collected, only to have little power.
But that doesn’t make it a good solution, and it may not help you find relationships with strong parameter estimates. Even if they exist in the population.
On the other hand, there are many alternatives to mean imputation that provide much more accurate estimates and standard errors, so there really is no excuse to use it.
This post is the first explaining the many reasons not to use mean imputation (and to be fair, its advantages).
First, a definition: mean imputation is the replacement of a missing observation with the mean of the non-missing observations for that variable.
True, imputing the mean preserves the mean of the observed data. So if the data are missing completely at random, the estimate of the mean remains unbiased. That’s a good thing.
Plus, by imputing the mean, you are able to keep your sample size up to the full sample size. That’s good too.
This is the original logic involved in mean imputation.
If all you are doing is estimating means (which is rarely the point of research studies), and if the data are missing completely at random, mean imputation will not bias your parameter estimate.
It will still bias your standard error, but I will get to that in another post.
Since most research studies are interested in the relationship among variables, mean imputation is not a good solution. The following graph illustrates this well:
This graph illustrates hypothetical data between X=years of education and Y=annual income in thousands with n=50. The blue circles are the original data, and the solid blue line indicates the best fit regression line for the full data set. The correlation between X and Y is r = .53.
I then randomly deleted 12 observations of income (Y) and substituted the mean. The red dots are the mean-imputed data.
Blue circles with red dots inside them represent non-missing data. Empty Blue circles represent the missing data. If you look across the graph at Y = 39, you will see a row of red dots without blue circles. These represent the imputed values.
The dotted red line is the new best fit regression line with the imputed data. As you can see, it is less steep than the original line. Adding in those red dots pulled it down.
The new correlation is r = .39. That’s a lot smaller that .53.
The real relationship is quite underestimated.
Of course, in a real data set, you wouldn’t notice so easily the bias you’re introducing. This is one of those situations where in trying to solve the lowered sample size, you create a bigger problem.
One note: if X were missing instead of Y, mean substitution would artificially inflate the correlation.
In other words, you’ll think there is a stronger relationship than there really is. That’s not good either. It’s not reproducible and you don’t want to be overstating real results.
This solution that is so good at preserving unbiased estimates for the mean isn’t so good for unbiased estimates of relationships.
A second reason is applies to any type of single imputation. Any statistic that uses the imputed data will have a standard error that’s too low.
In other words, yes, you get the same mean from mean-imputed data that you would have gotten without the imputations. And yes, there are circumstances where that mean is unbiased. Even so, the standard error of that mean will be too small.
Because the imputations are themselves estimates, there is some error associated with them. But your statistical software doesn’t know that. It treats it as real data.
Ultimately, because your standard errors are too low, so are your p-values. Now you’re making Type I errors without realizing it.
That’s not good.
A better approach? There are two: Multiple Imputation or Full Information Maximum Likelihood.