Why not us conditional modes (BLUPs) for further analyses? De Waters et al. (2017 ;2019) used conditional modes (BLUPs) from a generalized linear mixed model to further examine individual differences. In their experiment, they performed a temporal delay task (a marshmallow test for adults with money), manipulating the amount of immediate reward and the duration of the temporal delay. After extracting the conditional modes for each participant, they correlated them with an fMRI signal. The beauty of this approach is that it allowed them to examine individual differences in the influence of amount and delay. To do this, the conditional modes of the random slopes for amount and delay were each correlated with the fMRI signal.
However, I am somewhat unsure whether such an approach is appropriate. For example, Kliegl et al. (2011) writes:
"[...] Conditional modes of different subjects are not independent observations, but values weighted by distance from the population mean. Therefore, statistical inference must refer to the estimates of the LMM parameters; it is not advisable to use the conditional modes for further inferential statistical purposes (e.g., to correlate them with each other or with other subject variables such as age or intelligence)."
I understand that conditional modes corresponding to shrinkage are not independent.
Question 1: However, I don't quite understand why further analysis of conditional modes is not appropriate. Could someone elaborate on this or provide a
Question 2: Does anyone have an idea how to analyze such data as from de Waters et al. (2017) without using the conditional modes?

The random effects are expressed as latent variables, and we don't actually know their value. We assume that they are drawn from a population of similar people and are drawn from a normal distribution with an estimated variance. We infer their value based on some observable data and a model. But we don't know their actual values, so when you assign a single value to a latent variable and use it in further analysis, you are treating it as if it is a known quantity with certainty.
Instead, you can keep it latent and still get the quantity of interest you described in the experimental study by adding the fMRI variable as a predictor in the appropriate multilevel model. Since (I assume) that fMRI is a person level variable, it's association with the outcome is the quantity that the researchers were interested in when looking at correlations. You can sort of get the same thing by reflating the random intercept estimate for the groups, as described in this excellent paper by George Leckie.
