The Analysis Factor
Statistical Consulting, Resources, and Statistics Workshops for Researchers
Just about everyone who does any data analysis has used a chi-square test. Probably because there are quite a few of them, and they’re all useful.
But it gets confusing because very often you’ll just hear them called “Chi-Square test” without their full, formal name. And without that context, it’s hard to tell exactly what hypothesis that test is testing. [Read more…] about What is a Chi-Square Test?
Missing data are a widespread problem, as most researchers can attest. Whether data are from surveys, experiments, or secondary sources, missing data abounds.
But what’s the impact on the results of statistical analysis? That depends on two things: the mechanism that led the data to be missing and the way in which the data analyst deals with it.
Here are a few common situations:
Subjects in longitudinal studies often start, but drop out before the study is completed. There are many reasons for this: they have moved out of the area (nothing related to the study), died (hopefully not related to the study), no longer see personal benefit to participating, or do not like the effects of the treatment.
Surveys suffer missing data in many ways. When participants refuse to answer the entire survey or parts of it; do not know the answer to, or accidentally skip an item. Some survey researchers even design the study so that some questions are asked of only a subset of participants.
Experimental studies have missing data when a researcher is simply unable to collect an observation. Bad weather conditions may render observation impossible in field experiments. A researcher becomes sick or equipment fails. Data may be missing in any type of study due to accidental or data entry error. A researcher drops a tray of test tubes. A data file becomes corrupt.
Most researchers are very familiar with one (or more) of these situations.
Missing data cause problems because most statistical procedures require a value for each variable. When a data set is incomplete, the data analyst has to decide how to deal with it.
The most common decision is to use complete case analysis (also called listwise deletion). This means analyzing only the cases with complete data. Individuals with data missing on any variables are dropped from the analysis.
It has advantages–it is easy to use, is very simple, and is the default in most statistical packages. But it has limitations.
It can substantially lower the sample size, leading to a severe lack of power. This is especially true if there are many variables involved in the analysis, each with data missing for a few cases.
Possibly worse, it can also lead to biased results, depending on why and in which patterns the data are missing.
The types of missing data fit into three classes, which are based on the relationship between the missing data mechanism and the missing and observed values. These badly-named classes are important to understand because the problems caused by missing data and the solutions to these problems are different for the three classes.
The first is Missing Completely at Random (MCAR). MCAR means that the missing data mechanism is unrelated to the values of any variables, whether missing or observed.
Data that are missing because a researcher dropped the test tubes or survey participants accidentally skipped questions are likely to be MCAR.
If the observed values are essentially a random sample of the full data set, complete case analysis gives the same results as the full data set would have. Unfortunately, most missing data are not MCAR.
At the opposite end of the spectrum is Missing Not at Random. Although you’ll most often see it called this, I prefer the term Non-Ignorable (NI). NI is a name that is not so easy to confuse with the other types, but it also tells you its primary feature. It means that the missing data mechanism is related to the missing values.
And this is something you, the data analyst, can’t ignore without biasing results.
It occurs sometimes when people do not want to reveal something very personal or unpopular about themselves. For example, if individuals with higher incomes are less likely to reveal them on a survey than are individuals with lower incomes, the missing data mechanism for income is non-ignorable. Whether income is missing or observed is related to its value.
But that’s not the only example. When the sickest patients drop out of a longitudinal study testing a drug that’s supposed to make them healthy, that’s non-ignorable.
Or an instrument can’t detect low readings, so gives you an error, also non-ignorable.
Complete case analysis can give highly biased results for NI missing data. If proportionally more low and moderate income individuals are left in the sample because high income people are missing, an estimate of the mean income will be lower than the actual population mean.
In between these two extremes is Missing at Random (MAR). MAR requires that the cause of the missing data is unrelated to the missing values but may be related to the observed values of other variables.
MAR means that the missing values are related to observed values on other variables. As an example of CD missing data, missing income data may be unrelated to the actual income values but are related to education. Perhaps people with more education are less likely to reveal their income than those with less education.
A key distinction is whether the mechanism is ignorable (i.e., MCAR or MAR) or non-ignorable. There are excellent techniques for handling ignorable missing data. Non-ignorable missing data are more challenging and require a different approach.
What’s the difference between Multilevel Models, Mixed Models, and Hierarchical Models?
I get this question a lot.
The answer: very little.
Centering a covariate –a continuous predictor variable–can make regression coefficients much more interpretable. That’s a big advantage, particularly when you have many coefficients to interpret. Or when you’ve included terms that are tricky to interpret, like interactions or quadratic terms.
For example, say you had one categorical predictor with 4 categories and one continuous covariate, plus an interaction between them.
First, you’ll notice that if you center your covariate at the mean, there is [Read more…] about Centering a Covariate to Improve Interpretability
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.
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