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Hello,

I am having some trouble understanding an application of the delta method. Sorry in advance, as I'm a newbie to this method and could be way off here .

I’ve fit a set of CJS models in Rmark and have used model averaging (given low dAIC scores between the top-ranked models), which has generated estimates of survival for a treatment and control group. The models also include a few other covariates but the main question is whether the survival estimates are different between the treatment and control groups. Model averaging results in a total of 4 unique estimates of Phi (for treatment and control groups, while also controlling for another categorical covariate with 2 levels).

A reviewer has asked that I use the delta method to generate 95% confidence intervals for the survival estimates in order “to clearly show that the estimates actually differ” for treatment/control groups. So, I’m using the deltamethod function in R, implementing the following lines of code which is based on code from these documents: http://www.phidot.org/software/mark/doc ... /app_3.pdf http://www.phidot.org/software/mark/doc ... /app_3.pdf

Code: Select all

#(note that taking indices 1, 37, 64, and 100 just results in extracting the 4 unique estimates of Phi).
cjs.mods <- collect.models(type="CJS")
Mod.Avg.Phi <- model.average(cjs.mods, "Phi",vcv=T, drop=F)
betas=(Mod.Avg.Phi)$estimates
deltamethod(~exp(x1)/(1+exp(x1)),mean=betas$estimate[1],cov=betas$se[1]^2)
deltamethod(~exp(x1)/(1+exp(x1)),mean=betas$estimate[37],cov=betas$se[37]^2)
deltamethod(~exp(x1)/(1+exp(x1)),mean=betas$estimate[64],cov=betas$se[64]^2)
deltamethod(~exp(x1)/(1+exp(x1)),mean=betas$estimate[100],cov=betas$se[100]^2)

My understanding is that this will output estimates of SE for the real parameters for survival. I am then multiplying those by 1.96 to find a upper/lower confidence limits for a 95% CI around each estimate of Phi (so I take the relevant estimate of Phi from the model averaging results, e.g. 0.96, and add +/- 1.96 times the result from the relevant deltamethod output above). In doing so, I find that for treatment groups 95% confidence intervals around survival are ~ 0.89-1, whereas for control group they are ~0.5-.88. I think this gives me the information the reviewer is requesting and confirms that the two estimates “actually differ.”

My question is whether I am doing anything appropriately here or if I've gone off on a tangent based on a bad interpretation of the instruction documents. I admit that my understanding of how the delta method actually works is limited and I’m having a pretty hard time interpreting the appendix document. I’m also unsure whether I’ve integrated the model averaged estimates in appropriately.

Thank you for any assistance.

model.average is for real parameters so it gives you what you want. Not sure what reviewer is asking for but what you are doing is wrong for 2 reasons. First, you are treating reals as betas and second, what you were doing ignored covariances. You should just be able to compare model averaged real parameters.

Hi Jeff,

Thanks for your help. I see--so the UCL and LCL reported in the results of Mod.Avg.Phi$estimates are already exactly what I am after. And as described in Appendix C, those were internally computed as follows:

Confidence intervals for the model-averaged estimates were somewhat more challenging. To provide valid intervals for bounded parameters (e.g., 0 < ϕ < 1), the model-average variance-covariance matrix of the real parameters are transformed to a variancecovariance matrix for the estimates transformed into the appropriate link space using the Delta-method (see Appendix B). Then asymptotic 95% normal confidence intervals are constructed for the transformed link values and the interval end points are then back-transformed into real parameters.

Thanks again for your help with this.