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When talking about longitudinal data, we may refer to data collected over time from the same subject / study unit repeatedly, thus there are correlations for the observations within the same subject, i.e., within-subject similarity.

When talking about time-series data, we also refer to the data collected over a series of time and it seems very similar to the longitudinal setting mentioned above.

I'm wondering if someone can provide a clear clarification between these two terms, what is the relationship and what are the differences?

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This may turn into a poll... I have worked on both types of data, and one key difference seems to be that longitudinal data is often used in causal analyses, to understand the impact of interventions or treatments, whereas time series are often used in forecasting. Of course, the difference is not clear-cut (you need to understand the underlying drivers to forecast, and IMO you haven't understood the drivers unless you can forecast well). But people who do signal detection in time series often don't care so much about forecasting, so they would probably reject my distinction. –  Stephan Kolassa Apr 11 at 15:24
    
Thanks for your comments. But I think the term "causal" may not be appropriate here rather the term "association" should be better? In terms of the purpose of the data analysis, I think your comments made some sense to me. But can't we use the longitudinal data to do forecast? Since it's also kind of time series data. –  askming Apr 11 at 15:32
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You do have a point re "causal" vs. "association", and of course longitudinal data can be used to forecast - it's just that I don't often see the two concepts together. Forecasters usually talk about time series. Apart from that, I couldn't put it any better than @gung. –  Stephan Kolassa Apr 11 at 15:57
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One of possible typical (not definitional) differences is that in time series you see and model time $t$ response as dependent on $t-1$ state; this is carryover effect. In longitudinal time analysis you usually consider time as permanent, evolutionary background factor. –  ttnphns Apr 11 at 16:00
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I doubt there are strict, formal definitions that a wide range of data analysts agree on.

In general however, time series connotes a single study unit observed at regular intervals over a very long period of time. A prototypical example would be the annual GDP growth of a country over decades or even more than a hundred years. For an analyst working for a private company, it might be monthly sales revenues over the life of the company. Because there are so many observations, the data are analyzed in great detail, looking for things like seasonality over different periods (e.g., monthly: more sales at the beginning of a month just after people have been paid; yearly: more sales in November and December, when people are shopping for the Christmas season), and possibly regime shifts. Forecasting is often very important, as @StephanKolassa notes.

Longitudinal typically refers to fewer measurements over a larger number of study units. A prototypical example might be a drug trial, where there are hundreds of patients measured at baseline (before treatment), and monthly for the next 3 months. With just 4 observations of each unit in this example, it is not possible to try to detect the kinds of features time series analysts are interested in. On the other hand, with patients presumably randomized into treatment and control arms, causality can be inferred once the non-independence has been addressed. As that suggests, often the non-independence is considered almost a nuisance, rather than the primary feature of interest.

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definitions of panel, x-section and TS are standard in stat textbooks. –  Aksakal Apr 11 at 17:45
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There are roughly three kinds of datasets:

  • cross section: different subjects at the same time; think of it as one row with many columns corresponding to different subjects;
  • time series: the same subject at different times; think of it as one column with rows corresponding to different time points;
  • panel (longitudinal): many subjects at different times, you have the same subject at different times, and you have many subjects at the same time; think of it as a table where rows are time points, and columns are subjects.
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Based on your comments, it seems the longitudinal data is a set of multiple time series data collected from different subjects? –  askming Apr 11 at 15:29
    
Generally, yes, you could see each subjects data as time series. In practice though, longitudinal data often has very few time points for each subject. They call the time points waves. For instance, it could be medical study where each patient has 4-5 observations at monthly intervals, and hundreds of patients over the course of years. That way panel data sets are often unbalanced (think of a very sparse table), so longitudinal studies have their own favorite methods to deal with this. –  Aksakal Apr 11 at 15:32
    
This is helpful given the question, but there are many other kinds of datasets that don't fall under any of these headings. However, they don't seem relevant to the question, and trying to classify every possible kind of dataset would be futile here. Examples: any dataset where the basic structure is subject x subject; any dataset that aren't two-dimensional. –  Nick Cox Apr 12 at 7:42
    
@NickCox, true, but I'm in econometrics, and these three are the ones with developed theories, and mostly used in our field –  Aksakal Apr 12 at 13:58
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No doubt you are, but nothing in the question obliges or even encourages a narrowly econometric viewpoint, nor was your specific perspective made explicit. –  Nick Cox Apr 12 at 14:26
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These two terms might not be related in the way the OP assumes--i.e., I don't think they are competing modes of analyses.

Instead time-series analysis describes a set of lower-level techniques which might be useful to analyze data in a longitudinal study.

The object of study in time series analysis is some time-dependent signal.

Most techniques to analyze and model / predict these time-dependent signals are built upon the premise that these signals are decomposable into various components. The two most important are:

  • cyclic components (eg, daily, weekly, monthly, seasonal); and

  • trend

In other words, time series analysis is based on exploiting the cyclic nature of a time-dependent signal to extract an underlying signal.

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