# How to interpret regression coefficient if predictor itself is on a negative scale?

I'm looking at effects of tree mortality (using "Biomass loss") on forest growth patterns. I incorporate loss into a mixed effects model like so (using lmer in R):

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


However, BiomassLoss is negative (i.e., the range of the actual values is -50 to 0).

So when I examine my model estimates, how do I interpret the estimate for Biomassloss?

                               95% Confidence Intervals
Estimate        Lower        Upper
(Intercept)        7.97955622   6.44676081   9.51907782
Year               0.02984233   0.01092181   0.04854870
BiomassLoss        0.09893282   0.06394322   0.13372741


Normally (i.e., for a predictor on a positive scale), the estimate can be interpreted as:

For every unit increase of the predictor, the interecpt will increase by the value of the predictor's estimate.

But how does this work for a negatively-scaled predictor?

Does "every unit increase" of a negatively-scaled predictor mean:

• as it becomes less negative (i.e., as it increases in value) or

• as it becomes more negative (i.e., as it increases in magnitude) ?

In other words, do I interpret my results as:

• for every unit less of biomass lost my intercept increases by 0.0989

• for every unit more (i.e., every unit lost) my intercept increases by 0.0989

This is important so I can understand if mortality is having a positive or negative impact on the trend...

• As a follow-up, if I simply reverse the sign on my predictor values, would it impact the model? (in other words, is a mixed model "blind" to sign of individual predictors and more reactive to magnitude of predictor values?) – theforestecologist May 21 '18 at 14:20

It is your first option. That is, "for every unit less of biomass lost my intercept increases by 0.0989". To be more explicit, if the biomass loss went from, say, $-45$ to $-44$, and all else were held constant, you would expect the mean of the growth rate to increase by $0.0989$. The model doesn't 'know' what the numbers mean; as far as the fitting procedure goes, they're just numbers. So it reflects a positive increment no matter what the numbers actually correspond to.

If you reverse the sign on the data, the model will be the same, but the coefficient will be -0.0989 instead.

Update: It seems I skimmed over your quoted example of the "normal" interpretation. You state that it is for the intercept. That isn't correct (but may have been just a typo). The following formulation has been corrected:

Normally (i.e., for a predictor on a positive scale), the estimate can be interpreted as:

For every unit increase of the predictor, holding all else constant, the interecpt predicted mean value of the response will increase by the value of the predictor's estimate.

• This makes a lot of sense. Thank you for the simple explanation. +1 :) – theforestecologist May 21 '18 at 14:25
• @gung: Confused about "the intercept increases" interpretation provided as the first option and why this inadequate formulation was not corrected in your answer: the increase is in the mean growth rate not in the intercept. Also, in my view, the more meaningful interpretation in this context is: If biomass loss increases by 1-unit (e.g., from -40 to -41), the mean growth rate in a given year for a typical plot decreases by 0.09893282. – Isabella Ghement May 21 '18 at 15:11
• Thanks, @IsabellaGhement, it seems I missed that. I have addressed it now. Your interpretation is mathematically equivalent to mine--they will yield the same numbers--it's just framed differently. Your framing may well be preferable (ie, easier to follow) by people in the OP's field, though. – gung May 21 '18 at 16:19
• @theforestecologist, the intercept is conventionally understood to be the predicted value when all predictors are at 0 (& categorical variables are set at their reference levels). Thus the intercept only shows up when (among other things) biomass loss = 0. Ie, it isn't possible for biomass loss to go from -45 to -44 at the intercept. You seem to be using the term differently, perhaps because this is a mixed model & there are no random effects for biomass loss. – gung May 21 '18 at 16:43
• @gung has provided excellent explanations. The absence of an interaction between Year and BiomLoss implies that the effect of BiomLoss is the same each year. But, when interpreting the effect of BiomLoss on Growth Rate for a given year, you would say that a 1-unit increase in BiomLoss is associated with a 0.0989 increase in the mean value of Growth Rate. – Isabella Ghement May 21 '18 at 17:56