Estimating a Population from Sampling with Imperfect Detection

[MindsEyeSifter]

Hello All, thanks for looking at my post and question. I have jumped into deep statistical and probability waters which have outstripped my current state of knowledge so any help you can render would be greatly appreciated. Here is my question:

I am trying to get an estimate of the original population in this case of archaeological sites. To do so I have two methods by which I can sample an area. Using a mathematical model I built and based on known archaeological sites in a particular region I can characterize the sites based on two parameters which affect detection: site size and artifact density. From this I can derive a mean probability of detection as well as the standard deviation and the variance. The two methods have different detection probabilities and will strongly affect the interpretation if taken into account.

In order to estimate the population (true number of archaeological sites in a region) I could simply take the mean detection probability and proportionally adjust for it, but this would not produce very robust results. I have attempted to find other equations which would allow me to more accurately estimate the original population. However I do not understand the formulas enough to be able to determine if they would apply to my data and answer the questions I am asking.

The most promising comes from Thompson and Seber's book Adaptive Sampling (1996:223-226). In example 9.3 they present a modified version of the Horvitz-Thompson estimator. Thompson and Seber modified the original to take into account imperfect detectability of the sampling methods. Attached are the pertinent pages from their book. I am not proficient at typing out the formulas so I apologize for the low rez pdf.
Basically I am having a hard time understanding what all of the variables are and what I should do with them in order to evaluate the appropriateness of this formula for my work.
I am sure I am leaving out gobs of information so if you need any clarification please ask and thanks again for all the help!

 

[mmwave]

Thank you for reaching out for help with your question. It is clear that you have put a lot of effort into designing your study and trying to find an appropriate method for estimating the original population of archaeological sites in your region. I understand that the formulas and equations can be overwhelming and difficult to interpret, but I will do my best to explain them in simpler terms.

The formula you mentioned from Thompson and Seber's book is a modified version of the Horvitz-Thompson estimator. This estimator is commonly used in surveys to estimate the total number of individuals or objects in a population based on a sample. In your case, the population is the total number of archaeological sites in your region and the sample is the sites that have been detected.

The key variables in this formula are:

- n: The number of sites in your sample.
- y: The number of sites in your sample that have been detected.
- p: The probability of detecting a site in your sample. This is calculated based on the two parameters you mentioned - site size and artifact density.
- w: The weight assigned to each site in your sample. This is calculated by dividing 1 by the probability of detection for that site.
- N: The estimated total number of sites in your region.

The formula for the modified Horvitz-Thompson estimator is:

N = (n/y) * ∑(w * y)

This means that you can estimate the total number of sites in your region by multiplying the number of sites in your sample (n) by the ratio of detected sites (y) to total sites in the sample, and then multiplying this by the sum of the weights (w) assigned to each site that has been detected.

This modified version takes into account the imperfect detectability of your sampling methods, which is important for producing more accurate results. However, it is important to note that this estimator assumes that the probability of detection is the same for all sites in your region. If this is not the case, you may need to consider using a different method.

I hope this explanation has helped you better understand the formula and how it applies to your study. If you have any further questions or need clarification, please do not hesitate to ask. Best of luck with your research!