Help with the Delta Method
Posted: Fri Jun 14, 2019 6:21 pmHi 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.
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.