SE for psi-covar relationship (categorical covar)

[Indermaur]

Hi,
Out of a single-season site-occupancy model containing 10 covariates, I want to show the effect of one categorical covariate on psi with histograms that include profile confidence intervals/SE. The covariate is called “sediment” with three factor levels (gravel, sand, mud), coded by two dummy variables with “gravel” as the reference sediment type (0 0).

Herefore, I calculated psi separately for the three sediment types as followed (it’s coded in R):
psi_gravel <- exp(b0+b1*0+b2*0)/(1+exp(b0+b1*0+b2*0)) #gravel (0 0)
psi_sand <- exp(b0+b1*1+b2*0)/(1+exp(b0+b1*1+b2*0)) #sand (1 0)
psi_mud <- exp(b0+b1*0+b2*1)/(1+exp(b0+b1*0+b2*1)) #mud/loam (0 1)

That part worked. However, it’s unclear to me, how I get the SE/CI for these habitat types and how the SE/CI are constrained to take values between 0 and 1. I was told that I could construct the SE based on the betas (b0=gravel, b1=sand, b2=mud) and the variance-covariance-matrix of the model. As SE for gravel is likely the intercept, I could grab this value from the presence-output.

When it comes to the other sediment types (sand, mud), several questions emerge:

a) In my case, the best-selected model I use for the predictions has 10 betas? Do I have to take the 2 columns/rows that correspond to b0 and b1 to calculate the SE for sediment type sand or mud (b0, b2), i.e. cut down the relevant columns of the 10x10 vcov-presence-matrix to a 2x2 vcov-matrix? Or do I have to construct the vcov-matrix by myself, based on b0 and b1 (or b0, b2)?

b) What does the formula for constructing the SE look like? And, has anybody done that so far already in R?

I appreciate any hint,
Cheers,
Lukas

[darryl]

Hi Lukas,
You can do that using something called the 'delta method'; we cover it in our book, it's in the MARK book and probably comes up on google pretty quickly.

If you want profile-likelihood confidence intervals you'll have to code things up yourself. Other option is to calculate intervals on the logit-scale and then transform them to the 0-1 scale. Check out some of the recent posts.
Darryl

[Indermaur]

Thanks Darryl,
I will go for it.

Another idea to get the SE/CI for the 3 sediment types could be to fit a the model without the intercept, and three dummy variables instead of 2. Reasonable?

Cheers,
Lukas

[darryl]

Yes, although depends what else is in the model if you're looking at the actual probabilities. It's at that point I suggest people start thinking about interpreting things in terms of odds ratios.