Oct 6
This week I looked at looking at the optimization problem piece by piece. I'm not sure I did it exactly how we were discussing, but I thought about each population and sampling characteristic involved in quadrat sampling and how they relate to each other. I also got limits for the upper and lower values of these characteristics. They are very rough and based on a paper that gives examples of quadrat sampling being used in the field.
I also looked at small examples that I could work out analytically to better understand how sample size and variance work. Doing this, I found that the expected sample size is the percent sampled times the total population size, which makes sense.
I also looked at variance, but haven't been able to calculate an explicit expression. I've found that it depends on number of quadrats in the population, % of quadrats sampled, between-quadrat variability (the variance of the populations in all of the quadrats), and possibly total sample size. I say possibly because when you divide the variance by between-quadrat variability the variance depends only on number of quadrats in the population and % sampled. When you don't do this, variance increases by n^2 as population size increases by n. I guess the way to think about this is that the effect of population size is wrapped up in between-quadrat variability, but I'm not positive.
I'm not sure precisely how variance changes with number of quadrats and percent sampled. Attached is a text file with (at the bottom) results for the variance divided by between-quadrat variability for 16, 9, and 4 quadrats. This data holds for any population.
![[paper clip]](/courses/source/wiki_images/paper_clip_tilt.png)
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