Most of recent books in longitudinal data analysis I have come through have mentioned the issue of unbalanced data but actually did not present a solution for it. Take for example:
- Hoffman, L. (2015). Longitudinal analysis: modeling within-person fluctuation and change (1 Edition). New York, NY: Routledge.
- Liu, X. (2015). Methods and applications of longitudinal data analysis. Elsevier.
Unbalanced measurements in longitudinal data occurs when participants of a study are not measured at the exact same points of time. We gathered big, complex and unbalanced data. Data comes from arousal level which is measured every minute (automatically) for a group of students while engaging in learning activities. Students were asked to report on what they felt while in the activities. Considering that not all students were participating in similar activities in the same time and not all of them were active in reporting their feelings, we end up with unstructured and uncontrolled data which does not reflect a systematic and regular longitudinal data. Add to this issue the complexity of the arousal level itself. Most of longitudinal data analysis assume the linearity (the outcome variable changes positively/negatively with the predictors). Clearly that does not apply to our case, since the arousal level fluctuates over time.
My questions:
Can you please specify a useful resource (e.g., book, article, forum of experts) to analysis unbalanced panel data?
Do you have yourself any idea on how one can handle unbalanced data analysis?
I do not have experience of this level of complexity in repeated measures , but it seems to me that you actually have multiple episodes - what you call events. So I am suggesting a three level structure with an identifier at level 3 for individuals; at level 2 for episodes within individuals and level 1 for repeated occasions within a specific episode; the minute by minute recording of repeated measurements. You could have variables measured at all three levels. The general approach is considered here
Fiona Steele (2008) Multilevel Models for Longitudinal Data, Journal of the Royal Statistical Society. Series A (Statistics in Society)Vol. 171, No. 1 (2008), pp. 5-19
https://www.jstor.org/stable/30130728?seq=1
Maybe go to the page of Tom Snijders (https://www.stats.ox.ac.uk/~snijders/), in chapter 15 these types of models are mentioned. Tom also mentions Robert Elashoff, Gang Li, Ning Li (2016). Joint Modeling of Longitudinal and Time-to-Event Data. Boca Raton: CRC Press. On his site Tom gives codes for MlWin and R to start from. Tom also refers to Pinheiro & Bates p. 251 for more information. I can not get to my books as my university is closed due to COVID.
Alaa Aldahdouh, I have not yet read Lesa Hoffman's book, but I do have it here. And I think you may be misunderstanding what she said in Section 3.C of the introduction. She says that if one uses conventional repeated measures ANOVA (with a wide file format), missing data for any time point causes the entire subject to be dropped, etc. But when you use multilevel modeling with a long (person-time) file format, that no longer happens. I fully expect that many of the examples throughout her book will in fact use unbalanced datasets. Ditto for another classic book in this area, Applied Longitudinal Data Analysis by Singer & Willett (2003). You can find code for examples from both of these books at the links given below.
- https://www.pilesofvariance.com/
- https://stats.idre.ucla.edu/other/examples/alda/
HTH.
Imbalance is a type of missingness and multilevel (or mixed) models handle these quite naturally under the forgiving MAR assumption, that it is the value of the response is not producing the missing data.
I have explained more with multiple answers here
https://www.researchgate.net/post/How_are_missing_data_handled_in_linear_mixed_effects_models
Thank you for all contributed in this thread.
It seems that either I really do not understand how unbalanced are handled (as Bruce Weaver ) or I failed to depict the issue correctly.
What I can say by now is that our data are based on event. The arousal level of each participant was measured several time within a period of two weeks. The number of measurements and the time of measurement (morning or evening) and the duration of each measurement (1h , 3h) are different within and between participants and depend on the engagement of each participant in a learning activity of his own. The arousal level is by nature fluctuating construct.
One solution is to align each event of a participant (each learning session) and then build an individual growth carve (which handled the missing and unbalanced data). I still have a concern to do that because alignment of each session could be unjustifiable and might produce erroneous results.
I think i am beginning to understand, but I need some more
You write "The number of measurements and the time of measurement (morning or evening) and the duration of each measurement (1h , 3h) are different within and between participants and depend on the engagement of each participant in a learning activity of his own."
So cannot you put variables - dummies or continuous or ordinal - to condition on these differences ?
You also write
"One solution is to align each event of a participant (each learning session) and then build an individual growth carve (which handled the missing and unbalanced data). I still have a concern to do that because alignment of each session could be unjustifiable and might produce erroneous results"
This is quite common problem and is concerned with the metric for time. For example, in a study of development of children you could put in day of birth but people usually use age, that is times since birth. You have to think about when the starting gun was fired to get the alignment. ; what is time zero?
Remember in a random effects mixed model there can be two aspects of this "time variable". Firstly occasions that follow each other so a measure at occasion 1 followed by occasion 2 etc, this defines the level 1 structure nested within individuals at level 2. And secondly you can additionally put in a continuous measure of time as a variable so that for individual 1 occasion 1 is 0, but occasion 2 is 3.12 hours and occasion 3 is 3.96 hours. If you have this decimal point time data, then use it as you will have have more power to detect effects. This decimal point is then included as polynomial in the fixed part to see overall trend and allowed to have a variable effect in the random part, so you can see whether the heterogeneity between people increases or decreases over time - the covariance is a key term here. Having got a good idea what needs to be explained , you can then include variables to try and account for this variation.
see my longish answer to this question for a sequence of models:
https://www.researchgate.net/post/What_is_the_best_alternative_to_ANCOVA_in_longitudinal_RCT
Singer and Willett have a section of their book, Chapter 5, called Treating TIME More Flexibly
Judith D. Singer John B. Willett
DOI:10.1093/acprof:oso/9780195152968.003.0005
https://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780195152968.001.0001/acprof-9780195152968-chapter-5
Thank you Kelvyn Jones for sharing your expertise. I read your comment carefully. The idea of adding two variables of the "time" sounds interesting.
I will try to explain our data by giving example.
Each one of our participants received a wearable device at the beginning of the experiment. The participants were instructed to activate the device before engaging in any learning activity (e.g., attending a course, joining an online discussion, meeting a colleague to learn together, etc). They were also instructed to deactivate the device and to fill in an online survey at the end of each learning activity/event/occasion. This device measures the participant's arousal level every minute. So, if the event lasts for 1 hour, the device will store 60 records. Each record contains a timestamp (as an integer) started from 1/1/1970. The students were free to use the device and to report on their events as much as and whenever they wanted.
As a result, student A, for example, reported 20 events. The first event lasted for 2 hours (120 arousal records) started at 12:00 PM in 1/1/2019. The second event lasted for 30 minutes (30 arousal records) started at 7:00 PM in 3/1/2019, .....
Student B, reported 10 events which are different in length and in duration and also different from those of student A or other students.
As far as I understand from your comment, your suggestion is to put two variables of time (timestamp and event count). Timestamp is nested within events. By doing so, we can analyze the development of emotion within timestamp and between events. Am I right?
I do not have experience of this level of complexity in repeated measures , but it seems to me that you actually have multiple episodes - what you call events. So I am suggesting a three level structure with an identifier at level 3 for individuals; at level 2 for episodes within individuals and level 1 for repeated occasions within a specific episode; the minute by minute recording of repeated measurements. You could have variables measured at all three levels. The general approach is considered here
Fiona Steele (2008) Multilevel Models for Longitudinal Data, Journal of the Royal Statistical Society. Series A (Statistics in Society)Vol. 171, No. 1 (2008), pp. 5-19
https://www.jstor.org/stable/30130728?seq=1
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