Is it valid to add a regression predictor that shares a raw variable with a calculated response?

I want to know how forest growth rate has changed through time.

• $Growth Rate = \frac{(Biomass_t - Biomass_{t-1}) + Biomass.lost_t}{Year_t - Year_{t-1}}$ where $t$ is the time period of interest.

I'm also interested in how tree mortality and stem density affect the trend.

My data: 80 years of tree growth data for 37 forest plots. Plot-aggregated biomass of all trees and biomass loss due to all dead trees is known for each sampling period.

I'm using mixed effects models (using the lmer package in R) with random slopes and intercept for each plot to examine trends:

lmer(GrowthRate ~ Year + (1 + Year | Plot), data = dat, REML = F)


However, I'm wondering about the validity of adding additional predictors into my model (specifically, for examining effects of mortality and stem density).

My question:

To incorporate effects of mortality, I have to incorporate biomass loss into my model. Currently, I've done so using a "Loss Rate" (i.e., biomass lost due to mortality divided by period of time between samples).

• $Loss Rate = \frac{Biomass.lost_t}{Year_t - Year_{t-1}}$

But is this valid?

The updated model has a lower AIC and BIC, and the Loss Rate predictor is deemed to be significant (i.e., 95% bootstrapped CI do not cross 0). So, seemingly, this predictor is valid and informative.

But Growth Rate incorporates biomass loss in its calculation, so I have this hunch that I can't add Loss Rate (which also uses biomass loss in its calculation) as a predictor.

2 Questions:

1. Is it ok to add a predictor to a mixed model that incorporates a raw variable used to calculate the response?

• In this case, can I use Loss Rate as a predictor?
2. Since all other "signs" (i.e., AIC/BIC and CIs of estimates) indicate that this is ok, is there any way (e.g., some test) to determine the validity of adding this possibly-invalid predictor?

• Is this where expertise in the area of study comes into play, or is there a statistical approach for determining this?
• Is biomass lost in the GR response attributable to sources other than mortality? Or would the calculation of loss rate include the same value (per observation) used to calculate the response? i.e. is biomass lost to mortality a subset of biomass lost? – NatWH May 21 '18 at 13:35
• @NatWH sorry for the confusion. Anytime I mention biomass lost in my post I'm referring to all biomass lost. In this case, all biomass lost I have calculated is due to mortality. So "biomass lost to mortality" is equal to "biomass lost." (I just specified mortality to emphasize the connection of loss to mortality) – theforestecologist May 21 '18 at 13:39
• In that case I have a similar feeling to you that it is problematic, as reductions in growth rate will be tightly correlated with changes in loss rate because they share a term. But I am uncertain. – NatWH May 21 '18 at 14:56
• @NatWH yup, that's where I'm at. But knowing to what degree the loss impacts the growth doesn't seem so harmful if that's what I'm interested in, right? I'm just not sure either. Perhaps could you upvote the question to give it a bit more attention so we can both learn something today? :) – theforestecologist May 21 '18 at 14:57
• Indeed, and I have done so! – NatWH May 21 '18 at 15:01

I think that what you are doing simply has the effect of removing $Boimass.lost_t$ from the computation of $GrowthRate$. This is easier to see if we re-write your computation of $GrowthRate$. $$Growth Rate = \frac{Biomass_t - Biomass_{t-1}}{Year_t - Year_{t-1}} + \frac{Biomass.lost_t}{Year_t - Year_{t-1}}$$ Including the righmost fraction as a predictor of $GrowthRate$ in a regression model will simply remove the variance in $GrowthRate$ attributable to that part of the equation. If that is what you want, compute $GrowthRate$ without $Biomass.lost$.
A latent growth curve modeling approach may be useful in this situation. In that approach, you would model $Biomass$ directly as the dependent variable, creating separate growth trajectories for each plot. You could then include other predictors to see what effect they have on the trajectories.
• Thanks for the answer. Let's assume I define "ChangeRate" (CR) as GrowthRate (GR) without the Biomass.lost rate (LR) portion. Are you saying that the regression model GR ~ Age is essentially the same as CR ~ Age + LR ? – theforestecologist May 25 '18 at 16:43
• Wait, perhaps I have that backwards. I think what you're saying is that Gr ~ Age + LR is the same as CR ~ Age. Correct? – theforestecologist May 25 '18 at 16:45
• Also, FYI I've already modeled Biomass ~ Age + .... I chose to additionally examine growth rate to see if forests are not only more biomass rich but whether the rate in which they are accumulating biomass is also increasing. – theforestecologist May 25 '18 at 16:48