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I have some data taken at multiple time points from a set of individuals, but without tracking the individuals.

For example, at each of six time points, the heights of all individuals are recorded -- but there is no information on which height came from which individual. (This is similar to the R builtin data-set Loblolly if you ignore the Seed column which tracks individuals.)

I am trying to run a linear regression to analyze growth, and I know that the lack of independence is problematic for over-confidence in estimates of group differences and for adding excessive noise to the model. However, I am not sure how to go about correcting for those impacts.

In the simplest case, there would be just one group in which I am trying to model growth by predicting height by age. In more complicated cases, I will have two or more groups and will want to predict height by the interaction of age and group. The groups generally start as randomized and relatively homogenous, so the main expected difference is in growth rate. (Occasionally, I may need to handle differences in starting groups, but I think that is beyond the scope of this question.)

Yes: the correct answer is that I should track individuals and include them in the model. I am working on that. However, the production environment I am working in does not make that easy, and I don't have a Delorean to go back and get them from past experiments. So, I am forced to find at least a bit of guidance on correction methods, even if they are just back-of-the-envelope/rule-of-thumb guiding principles to let me know how much salt I should take with the results.

Here is the output from the R Loblolly dataset for those working in other systems that want a starting point (may just be too much time on StackOverflow, but I know it sometimes helps).

 height age Seed
   4.51   3  301
  10.89   5  301
  28.72  10  301
  41.74  15  301
  52.70  20  301
  60.92  25  301
   4.55   3  303
  10.92   5  303
  29.07  10  303
  42.83  15  303
  53.88  20  303
  63.39  25  303
   4.79   3  305
  11.37   5  305
  30.21  10  305
  44.40  15  305
  55.82  20  305
  64.10  25  305
   3.91   3  307
   9.48   5  307
  25.66  10  307
  39.07  15  307
  50.78  20  307
  59.07  25  307
   4.81   3  309
  11.20   5  309
  28.66  10  309
  41.66  15  309
  53.31  20  309
  63.05  25  309
   3.88   3  311
   9.40   5  311
  25.99  10  311
  39.55  15  311
  51.46  20  311
  59.64  25  311
   4.32   3  315
  10.43   5  315
  27.16  10  315
  40.85  15  315
  51.33  20  315
  60.07  25  315
   4.57   3  319
  10.57   5  319
  27.90  10  319
  41.13  15  319
  52.43  20  319
  60.69  25  319
   3.77   3  321
   9.03   5  321
  25.45  10  321
  38.98  15  321
  49.76  20  321
  60.28  25  321
   4.33   3  323
  10.79   5  323
  28.97  10  323
  42.44  15  323
  53.17  20  323
  61.62  25  323
   4.38   3  325
  10.48   5  325
  27.93  10  325
  40.20  15  325
  50.06  20  325
  58.49  25  325
   4.12   3  327
   9.92   5  327
  26.54  10  327
  37.82  15  327
  48.43  20  327
  56.81  25  327
   3.93   3  329
   9.34   5  329
  26.08  10  329
  37.79  15  329
  48.31  20  329
  56.43  25  329
   3.46   3  331
   9.05   5  331
  25.85  10  331
  39.15  15  331
  49.12  20  331
  59.49  25  331
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  • $\begingroup$ What variables are in the regression? $\endgroup$ – user0 Dec 28 '16 at 21:40
  • $\begingroup$ Edited to add model description. Looking to predict height by age (with interaction with group sometimes) $\endgroup$ – Mark Peterson Dec 28 '16 at 21:57
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This is not really an answer, but this is an interesting question.

I had a similar dataset where I did have individual tracking and I was not sure how to adjust the model (ultimately, I ended up sampling one from each individual to eliminate the dependencies, but I wondered whether there was a better way to do it).

With a ggplot2 plot of the Loblolly dataset such that each line corresponds to one seed:

Lollolly by seed

You can see that the coefficient on age for each individual should be similar. Therefore, at least with this dataset, even if you didn't have tracking, you would be able to make a statement about a confidence interval for the coefficient on age. Of course, you only know this for this dataset because you could plot according to individual. Without individual tracking you cannot know, but perhaps domain expertise could inform you here.

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