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I am currently studying the effect that a pollutant has on plant growth. The plants come from a few different regions, and it is assumed that plants from the same region share more in common than plants from different regions. Furthermore, from each region we sample a few plants and then take measurements relating to time on them. This means that the repeated measurements from the plants also introduce dependence in the data.

We thus have a multilevel model problem, where plants are nested in regions and also repeated measurements are taken for each plant. (Think of it as one measurement per year per plant.) Furthermore, we have region specific variables (levels of the pollutant, rainfall, temperature, etc) that, within a given year, are shared between all plants in a region. It is the effect of such a region level pollutant on the (individual level) plant growth that is the main interest.

Now, I have read that a good practice is to keep the model maximal when it comes to specifying random effects in linear mixed models (Barr et al. 2013), but I am unsure how to apply this to multilevel models. Should I introduce the random slopes on the individual plant level? Or should they be on the region level since that's the level the variables are measured on? Or perhaps on both?

Specifying a full model on all levels leads to convergence issues (as does specifying a full model on any of the levels), so a truly full model is not an option.

Thankful for advice and/or references to articles where the topic is discussed.

References

Barr, D. J., Levy, R., Scheepers, C., & Tily, H. J. (2013). Random effects structure for confirmatory hypothesis testing: Keep it maximal. Journal of memory and language, 68(3): 255-278.

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Indeed, it is many times the case that even if your design is complex, such as in your case, a multilevel longitudinal design, the actual correlations in your data do not support the use of all the random effects that you could possibly use according to the design.

This is why often the modeling building approach that is used to select the random-effects structure of the model starts from simple random intercepts term and then builds it up. Moreover, the choice of the random-effects structure is affected by the chosen fixed effects. Therefore, the steps that are often used to build the model are:

  1. Start with by specifying an elaborate structure for your fixed effects, containing all the covariates you want to consider. In the case of longitudinal data, you may want to consider a nonlinear time effect and possible interactions of the (nonlinear) time effect with other covariates.

  2. Given this elaborate model for the fixed effects, you start building your random effects, starting with intercepts in the lower level, adding slopes in the lower level (possibly also nonlinear slopes), then adding intercepts in the upper levels, and then slopes (also possible nonlinear) in the upper level. At each step, you can see if you need the extra random effects using likelihood ratio tests.

  3. Then having selected the random-effects structure you can return in the fixed effects and see if you want to simplify the model by excluding the complex terms (i.e., interactions and nonlinear terms).

    You can find more information also in my course notes (Chapter 3 in general, and Section 3.9 more specifically).

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  • $\begingroup$ Thank you for this answer and for your very helpful lecture notes. Wouldn't a random intecept at higher levels be more more helpful in combating the dependencies in the data than adding a random slope at a lower level for some covariate that is not the main interest of the study? $\endgroup$ – Phil Oct 8 '18 at 8:56
  • $\begingroup$ Typically you would expect that repeated measurements in the lower level are more correlated than measurements in the higher levels. $\endgroup$ – Dimitris Rizopoulos Oct 8 '18 at 11:39

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