Multilevel, Hierarchical, and Mixed Models–Questions about Terminology

by Karen Grace-Martin

Multilevel models and Mixed Models are generally the same thing. In our recent webinar on the basics of mixed models, Random Intercept and Random Slope Models, we had a number of questions about terminology that I’m going to answer here.

If you want to see the full recording of the webinar, get it here. It’s free.

Q: Is this different from multi-level modeling?

A: No. I don’t really know the history of why we have the different names, but the difference in multilevel modeling and mixed modeling is similar to the difference between linear regression and ANOVA.

They’re the same thing at the mathematical root of it, but they tend to focus in different areas of the output, interpret the results a little bit differently, and use different notation and vocabulary to describe the model and its results.

The biggest difference is in the logic of how the model is set up and described. The vocabulary, notation, and even the structure of the model are described differently in mixed and multilevel models. But mathematically, they’re running the same thing.

The only real difference is that the multilevel model descriptions and logic don’t work for every possible design that work for mixed. The multilevel modeling approach tends to focus on designs where all the random factors are nested — children nested within classes, which are nested within schools, which are nested within districts, for example. These are described as ‘levels.’ Mixed models would describe them as ‘random factors.’

Multilevel models have a harder time (though it’s not impossible) making sense in designs with multiple random factors that are semi-nested or crossed with each other. But if you work in a field that only ever uses the fully nested design, you may find the multilevel way of thinking about it easier to wrap your head around. It’s more targeted

That said, for the most part, mainstream statistical software, like Stata, R, SAS, and SPSS, use the mixed framework. All the vocabulary, the notation, and what you need to specify. For example, you’re specifying which predictors you need fixed or random effects for, not which level each variable is measured at. So if you’re used to the language and logic of multilevel models, it can be a little hard to translate into mixed.

Some stand-alone software, like HLM and MLWin, have you specify the model according to “levels.”

Q: You mentioned MultiLevel Models as a specific type of mixed model, how would the use of MLM differ to whats been performed here?

It would be performed exactly the same way. We would describe it differently though.

For example, we’d talk about the County level (level 2) and the Observation level (level 1). The outcome, number of jobs (n thousands), would be measured at the Observation level. In other words, we have one measurement per county in each decade.

Rural seems like it’s measured at the County level, but it’s not. That’s because some counties switch from being rural to urban over the course of the study.

Decade is measured at the Measurement level.

Q: So for panel data with a few time points a mixed model can be used?

Yes. Panel data is an econometric term. It describes a study design where individuals (whether people, companies, or counties) are measured repeatedly over time. Some panel data sets have cohorts (aka panels) of these individuals who all start at the same time. But it is simply a specific description of a type of design with non-independent observations that fits into the mixed modeling framework.

Q: Don’t all of these models fall under the “hierarchical linear model” label?  With nested observations?

This example does. Each observation is nested within a county. Not all mixed models do, though. Linguistics, psychology, ecology, and agriculture, in particular, seem to have many situations where you have partial nesting, crossed random factors, etc. Nesting of random factors doesn’t always work.

Q: There are no nested data here right?

There is some nesting in this design. We have these factors in the model, and I’ve specified each as Fixed(F) or Random (R):

Decade (F)
Rural (F)
County (R)

Decade is crossed with both Rural and County. Each county is measured at every Decade (except for some sparse missing data, which isn’t really enough to call them not crossed). Likewise, every value of Rural (yes and no) is measured at every Decade.

Counties are mostly but not completely nested within Rural. That means that most counties are classified as either Yes, Rural for all 5 Decades. And other counties are classified as No, Urban for all 5 Decades. But some switch from Yes, Rural to No, Urban at some point during the 50 years of the study. The mixed model can handle this semi-nesting just fine.

And of course, the observations themselves, which are not specified in the model, are nested within the combination of County and Decade.

Q: What is the difference between fixed and random effects?

There is a lot to this question, so I’m actually going to send you to some other resources I’ve put together that answer it in more detail:

Fixed and Random Factors in Mixed Models: What is the Difference?

Specifying Fixed and Random Factors in Mixed Models

The Difference Between Random Factors and Random Effects

 

Random Intercept and Random Slope Models
Get started with the two building blocks of mixed models and see how understanding them makes these tough models much clearer.

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