Repeated Measures Design? Or Not?

Repeated measures is one of those terms in statistics that sounds like it could apply to many design situations. In fact, it describes only one.

A repeated measures design is one where each subject is measured repeatedly over time, space, or condition on the dependent variable.

These repeated measurements on the same subject are not independent of each other. They’re clustered. They are more correlated to each other than they are to responses from other subjects. Even if both subjects are in the same condition.

That non-independent clustering needs to be accounted for in the analysis. It’s the only way to get accurate standard errors, p-values, and confidence intervals.

This gets confusing because this clustering can look very different in different studies.

Additionally, a few study designs look like they should be called repeated measures but, for the purpose of analysis, are not.

Three Design Elements that Define Repeated Measures

So, to reiterate, we have repeated measures if we have three important elements:

Multiple clustered measurements of the dependent variable
that are more correlated to each other than to others’ measurements because they’re measured on the same subject
repeatedly over time, space, or condition.

A great way to figure out if your study requires a repeated measures analysis (via mixed or marginal model) is to define these three elements. Here are a few examples:

Example 1

Measure the graduation rate in 50 high schools in the four years before and four years after an intervention.

Multiple clustered measurements of the DV: 8 measures of pass rate per school
Subject: school
Repeated over: time

Example 2

Measure the time it takes second language learners to read a sentence under 4 different grammar structures.

Multiple correlated measurements of the DV: 4 measures of reading time of each sentence per person
Subject: Person learning a language who reads the sentences
Repeated over: grammar structure condition

Studies that aren’t “repeated measures” even though something is measured repeatedly:

1. A single subject measured over time

A common design is to measure a single “subject” for many time points. An example is a data set with one company’s valuation measured each year for 10 years.

This design is missing element #2. The responses from our single subject cannot be more correlated to each other than to other subjects’ responses because there are no other subjects.

This is a time series. In a time series, there is serial autocorrelation — similarities in responses that are closer in time. Luckily, there are time series models that account for serial autocorrelation without accounting for clustered correlation.

In contrast, a design with 30 companies, each measured over 10 years, is repeated measures. You have 300 data points clustered into 30 sets of 10. (30 companies each have 10 non-independent measurements). Each company’s data may also have autocorrelation, and good repeated measures models can account for that too.

2. A new random sample of subjects at each time point

A second common design is a cross-sectional longitudinal design. For example, you measure the annual revenue of 30 companies for 10 years. But each year you randomly selected a new sample of 30 companies. That’s not repeated measures. With new subjects at each time point, we’re missing element #2.

When you cannot match a subject’s measurement in year 1 to the same subject’s measurement in year 2, that is not repeated measures. If each measurement in year 1 could be equally similar or different to any measurement in year 2, you have no clustered non-independence. A model that assumes independence would work here.

3. A predictor variable is measured repeatedly over time for each subject, but the dependent variable is measured once

I don’t know of a name for this design, but I see studies like this every so often. For example, a study where a researcher uses 30 companies’ revenues over 10 years to predict whether or not they had an IPO in year 11.

This is missing element #1. The repeated measurements have to be on the dependent variable. Since you have only a single measurement of the dependent variable, that’s not repeated measures, even though there are 10 measures of the predictor.

Conclusion

Once again, statistical vocabulary is inconsistent and less intuitive than we’d like. A good practice is to always be precise in describing your design, your variables, and your study. Define the terms you’re using rather than assuming your audience understands.

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