Hypothetical violation of independence assumption for linear models

Question

Let's say I have measured the abscisic acid content (could be any metabolite really) of 10 individual plants which we can consider biologically independent. Then I repeat that measurement for all 10 plants on 15 different days over the course of a year. I then want to correlate, with a linear regression model, the values of abscisic acid with the amount of rainfall (from publicly available regional data, say) on each of those days. What is statistically appropriate?

Firstly, I believe that a mixed model should be used, with individual plant as a random effect, to account for the fact that I am repeatedly measuring the same plant over time.

On top of that, am I not also in danger of pseudo-replication from the fact that my rainfall has only been measured once for all 10 abscisic acid measurements (i.e. the abscisic acid vs rainfall measurements are not really independent on each day either)?

Answer 1

Observations are assumed to be independent, not predictors. For example, if you were doing some chemical fertilizer treatment, you'd probably not apply a random chemical or random dose of chemical to each plant, you'd most often choose a small number of doses, and have far fewer values of the predictor than you have observations.

But, I do think you've identified a potential issue here, it's just that it isn't about repetition in the rainfall measurements, it's about additional repetition besides rainfall in your observations: there are going to be a lot of other environmental variables, besides the one you measured, that are the same for all the plants because you're measuring them on the same day, and that's potentially going to introduce correlations in same-day measurements.

It probably makes sense to account for that using a random effect for observation day, in addition to a random effect for each plant.

That approach may not work well if there are other specific factors that have a substantial impact (you should try to measure and account for those directly, instead), or if it's not really right to make the choice of day as a "random" feature, for example if there are seasonal variations you may need to track time explicitly. It all depends on what you know about the system you're studying.