Fitting a decaying exponential curve to a linear regression fixed factor in R

Question

I am looking at how landscape features might impact the presence of bat species using a binomial linear regression model in R. An example of one of my model is:

model <- lm(presence ~ distance_to_wood_rescaled
                      + distance_to_roost_rescaled
                      + Decid_%_cover
                      + Conif_%_cover
                      + Arable_%_cover
                      + Grass_%_cover
                      , data = data)

However, we know that as you move away from their roost you are less likely to record their presence. To account for this I wanted to use exp(distance_to_roost_rescaled) as its a decaying exponential curve. However, I am not sure if this is correct or whether this needs to be taking into account before being input into the model? How does one account for exponential decay of presence records as you move away from the roost?

EDIT

Answer 1

To fit a binomial, you need to do:

model <- glm(presence ~ distance_to_wood_rescaled
                      + distance_to_roost_rescaled
                      + Decid_%_cover
                      + Conif_%_cover
                      + Arable_%_cover
                      + Grass_%_cover
                      , data = data,family="binomial")

In the example code provided, you are fitted a regression model assuming gaussian distribution. To see whether you need to introduce the exponential, I guess you can visualize it like:

hist(data$distance_to_wood_rescaled[data$presence==1])

Assuming your presence is a binary 0,1.

Now regarding whether you need to model that exponential, it really depends on what your data looks like. My guess is that because we model the log-odds, you don't need to transform your data.

So let's assume a scenario where :

$$\frac{p}{1-p} = b_0 exp( b_1 * D)$$

where $D$ is the distance, $p$ is the probability of bat being present and $b_0$ and $b_1$ are constants.

We can simulate the data and you can see your distances will look like:

set.seed(222)
D = runif(1000,min=0,max=10)
OR = 5*exp(-1.25*D)
p = OR/(1+OR)
presence = rbinom(length(D),1,p)

par(mfrow=c(2,2))
plot(D,log(OR),ylab="Log odds (present)",xlab="distance",main="log odds vs Distance")
plot(D,p,ylab="Probability (present)",xlab="distance",main="prob vs dist")
hist(D[presence==1],cex.main=0.7,main="distribution of distances where present")
hist(D[presence==0],cex.main=0.7,main="distribution of distances where absent")

So you can see the distances of presence looks somewhat like an exponential decay. To summarize, in the linear model, we would model it as:

$$ log(\frac{p}{1-p}) = log(b_0) + b_1*D$$

So if your distances for present look like the above, I think the model you have now is ok, just remember to use glm() with binomial.

Answer 2

If you are adding a term exp(distance_to_roost_rescaled) to your linear model then you are fitting a model that is very restricted.

Namely, it has a fixed rate of decrease of a factor $1/e \approx 0.37$ for every unit change in the distance.

Instead, often one would like to use a model that is able to have variable rate of decrease. Or at least, often there is no a priori knowledge about the rate and the model estimates the rate for which the fit is best.

It is possible to fit exponential models with a GLM (generalized linear model) that allows to have a linear function wrapped inside a non-linear function. For more complicated cases you can use a method that uses some optimiser algorithm (e.g. gradient descent).

For the R statistical language, you can read into the functions glm and nls .