The Difference Between Crossed and Nested Factors

by Karen

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One of those tricky, but necessary, concepts in statistics is the difference between crossed and nested factors.

As a reminder, a factor is just any categorical independent variable. In experiments, or any randomized designs, these factors are often manipulated. Experimental manipulations (like Treatment vs. Control) are factors.

Observational categorical predictors, such as gender, time point, poverty status, etc., are also factors. Whether the factor is observational or manipulated won’t affect the analysis, but it will affect the conclusions you draw from the results.

When there is only one factor in a design, you don’t have to worry about crossing and nesting. But when there are at least two factors, you need to understand whether they are fixed or crossed, because it will affect the analyses you can and should conduct.

Two factors are crossed when every category of one factor co-occurs in the design with every category of the other factor. In other words, there is at least one observation in every combination of categories for the two factors.

A factor is nested within another factor when each category of the first factor co-occurs with only one category of the other. In other words, an observation has to be within one category of Factor 2 in order to have a specific category of Factor 1. All combinations of categories are not represented.

If two factors are crossed, you can calculate an interaction. If they are nested, you cannot because you do not have every combination of one factor along with every combination of the other.

If you’re not sure whether two factors in your design are crossed or nested, the easiest way to tell is to run a cross tabulation of those factors.

Here is an example. In this study, 27 men in their early 20s were randomized into one of three physical training groups. The subjects in every group–endurance, strength, and concurrent training regimens–were measured on a number of physical health measures at two time points: pre and post.

Group and Time are Crossed

crossed-factors

The two factors of interest–Training group and Time–are crossed, as there are 9 observations from each training group in each time. In other words, each Training group is represented at every Time point. This cross tabulation table shows this.

crossed-factors-2

Subjects and Time are Crossed

However, there is a third factor that needs to be taken into account because it’s a repeated measures study: Subject.

If the same nine people in each group were not measured at both times, subject would not be an important factor and we could stop there. Groups and time would still be crossed. We would miss out on some of the efficiency advantages that we get from repeated measures, though, so let’s keep going.

In repeated measures, subject itself becomes a factor. Subject is crossed with time because each subject appears in every time point. Again, this is easy to see in the cross tabulation. Every subject has at least one value at every time point.

Subjects are Nested within Group

But you can see in the Subject*Groups cross tabulation that each subject has observations in only one group. Subjects 1-9 are in the Endurance training group only. Subjects 10-18 are in the Strength training group, etc. Because each subject was assigned to only one training group, subject and group are not crossed. Rather, subject is nested within training group.

In traditional multivariate approaches of analyzing repeated measures data, we ignore issues of nesting and crossing and use different names for these same concepts. Factors that Subject is nested within, like Training group, are called between-subjects factors. Factors that Subject is crossed with, like Time, are called within-subjects factors.

nested-factorsThose concepts are helpful and valid. But being able to translate them into which factors are crossed and which are nested will allow you to see the bigger design and analysis issues. This becomes very, very important when you expand your analysis of repeated measures beyond traditional approaches to mixed models approaches.

It also becomes extremely important in clustered designs, which don’t necessarily have repeated measures, but do have some sort of nesting of individuals within some larger group.

The combinations of nesting and crossing in designs with many factors can get quite complex. It gets even more confusing when some of these factors should (or could) be treated as fixed or random. Remember to use the cross tabulations to help you sort it out.

rm-500Learn more about repeated measures analysis using mixed models in our most popular workshop, Analyzing Repeated Measures Data: GLM and Mixed Models Approaches.

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{ 6 comments… read them below or add one }

Cleber November 27, 2016 at 4:42 pm

Thank you very much, you removed my doubts

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Mark March 8, 2016 at 3:18 pm

How does a nested ANOVA or nested ANCOVA handle the fact the subject observations are violating the assumption of independent observation. Meaning, since the students are grouped within a classroom (nested) how does ANCOVA/ANOVA get around this lack of independent observations.

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Karen March 11, 2016 at 7:22 pm

Hi Mark,

The only way to do that is with a mixed model. ANOVA can’t do it.

Reply

Mark April 7, 2016 at 7:06 pm

But wouldn’t the example of the participants in the nested groups be violating the assumption of independence? Could it be possible that the participants be competing? It is not as if they are separated into different rooms. For example, participants one through ten go to rooms one through ten. Wouldn’t the observation be influenced by the other participants in the group? Is it assumed in the example that the observations are independent and the fitness participants ignoring each other?

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Marcia February 9, 2016 at 11:33 am

Very Nice

Thank you

Reply

Josh November 14, 2014 at 5:51 pm

Nice explanation!

Reply

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Please note that Karen receives hundreds of comments at The Analysis Factor website each week. Since Karen is also busy teaching workshops, consulting with clients, and running a membership program, she seldom has time to respond to these comments anymore. If you have a question to which you need a timely response, please check out our low-cost monthly membership program, or sign-up for a quick question consultation.

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