# Statistical differences between longitudinal and cross-sectional data

I would like to start a follow-up discussion to an existing question on cross-validated: Difference between longitudinal design and time series.

Most of the answers there tend to make distinctions between the data sets themselves and not between the analysis or modeling done with the data. Are there distinctions between the treatment of the data, and not the data itself?

For example, are there any differences in the inferential methods used on cross-sectional and longitudinal data, or is the only difference between the two the interpretation of the results and the verbiage you use to describe the study?

The names are applied largely based on the structural characteristics of the dataset. To quote from @JeromyAnglim's answer there:

If we think of designs made up of $n$ cases measured on $k$ occasions, then the following loose definition seems to me to be descriptive of the distinction:

• longitudinal designs: high $n$, low $k$
• time series: low $n$, high $k$

To those two possibilities, we can add cross-sectional datasets. They have only one measurement occasion per case (i.e., $k=1$, but $N>1$ and hopefully reasonably large).

However, we certainly need to take those features into account when analyzing the data, no matter what we call it. The key aspect from a statistical perspective is that longitudinal / time-series data are not independent, whereas cross-sectional data may be. If existing non-independence isn't taken into account, any inferences will be invalid (e.g., confidence intervals may be too narrow).

• If you have cross-sectional data, standard, statistics-101-level analyses (e.g., a $t$-test), can be applied.
• If you have time-series data ($n=1$, large $k$), analysts will often use ARIMA methods to identify auto-regressive, integrative, and moving average terms.
• With longitudinal data ($k>1$ , but small), there usually isn't enough information to estimate ARIMA terms. Instead, people will typically use mixed effects models.