delta method for model averaged estimates of survival

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delta method for model averaged estimates of survival

Postby t1820 » Sat Oct 17, 2020 7:00 pm


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 :oops:.

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: ... /app_3.pdf ... /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)

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.
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Re: delta method for model averaged estimates of survival

Postby jlaake » Sun Oct 18, 2020 2:46 pm

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.
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Re: delta method for model averaged estimates of survival

Postby t1820 » Mon Oct 19, 2020 4:12 pm

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.
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