Help with the Delta Method

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Help with the Delta Method

Postby heg90 » Fri Jun 14, 2019 6:21 pm

Hi all,

I've been reading and rereading the delta method sections in Ch 6 of the MARK book, as well as Appendix B specifically relating to the method, but I'm struggling to understand it and am having a hard time determining how I can apply it to my data.

My top model from a single season occupancy analysis was an interaction model between the variables elevation and % size class 5 trees (%SC5), resulting in the following equation:

- logit(psi) = Bo + B1(elev) + B2(SC5) + B3(elev*SC5)
- logit(psi) = -1.1506 - 3.4923(elev) + 1.0602(SC5) + 3.1208(elev*SC5)

The MARK book states that I can use the equation var(ln(OR) = var(B1) + var(B2) + 2cov(B3,B4), and although I've output that from MARK [var(B1) = 0.00781, var(B2) = 0.01599, cov(B1,B2) = -0.00060), I don't see how that equation can be applied for specific cases of different elevation/%SC5 values.

For example, one of the confidence intervals I am interested in generating is for the odds ratio I derived to understand how odds of occurrence changes as %SC5 changes when I hold elevation at a constant low value. I should note that I standardized both elevation and %SC5 around a mean of 0 and a 1 unit change represents one standard deviation away from the mean. I derived an OR = 0.127 by using the above logit(psi) equation and holding elev = -1 (i.e. 1 standard deviation below the mean) to represent low elevation, and let SC5 range from -1, 0, and 1. This allowed me to make the following statement:

"A 1 unit (1 sd = 13.12%) change in SC5 at low elevations is associated with a decrease in odds of occurrence by 87% (OR=0.127). Similarly, a 5% change in SC5 at low elevations is associated with a decrease in odds of occurrence if 54.4% (OR=0.456)."

[Side note: A 13.12% change doesn't make as much sense as a 5% change in SC5, so I used the beta from OR = 0.127 (-2.0636) and multiplied: e^-2.0636*0.3811 to isolate the effect at 5% (since 1sd/13.12 *n/5, n = 0.3811).]

So at this point I'm excited, I have my top model, I've generated my statements on effect size, but now I am very stuck on how to generate a confidence interval for the effect of a 5% change in SC5 at low elevation (OR=0.456), as well as the rest of the results I've generated for this interaction model (i.e. holding %SC5 constant and changing elevation).

Any insight would be greatly appreciated. I've tried rereading the information in the MARK book to see if it would click eventually, but I figured it was time to turn to the forum for help. I've tried reading through similar posts but can't seem to understand how to apply the delta method.
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Re: Help with the Delta Method

Postby cooch » Sun Aug 18, 2019 5:27 pm

Only just seeing this now -- not sure why it flew under my radar. Ah well...

The material reference in Appendix B, and Chapter 6, is largely there (in the majority) to explain *how* it all works. For your purposes, I suspect you can get there from here far easier, without doing much of anything by hand, by treating the envirionmental factors as individual covariates. Chapter 11 -- sidebar beginning on p. 30, then related bits later on about 'model averaging'.
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