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I work for an environmental health nonprofit and I have moderate expertise in statistics. We want to estimate the total number of toxic industrial waste sites within a small African country. I would love to hear your thoughts on how we should start. At this stage, if you can recommend a book, an idea, or a general sampling method that's all I'm asking for. I basically need a place to start.

For instance, I thought that maybe we could we divide the country into three zones: low, medium, and high industrial zones (based on UN data). We could then create 100 equal-sized sectors within each zone, and randomly select 10 sectors from each of the three industrial zones. We would then survey these small sectors areas and find all toxic sites within those sectors. If we do this, could we estimate the total number of industrial sites along with a measure of uncertainty? Is there a name for this type of geographic sampling?

There is very little pre-existing data on the number of toxic waste sites. Also, to simplify things, assume it is very easy to identify a toxic waste site once you are on the ground with a team.

Slyron
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    Your idea is a form of hierarchical sampling. The problem has a spatial element to it, suggesting that results in one sector will be correlated with results in neighboring sectors. For a practical review of traditional and more modern sampling designs (including adaptive sampling, some kinds of spatial sampling, and other methods of "observational economy") see Steven Thompson's Sampling. – whuber Oct 30 '13 at 21:56

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Your approach seems reasonable, especially your choice to stratify your sampling. This will make it more efficient provided you can easily delineate the different industrial zones.

I don't have a book to recommend you, but you could model your uncertainty using the Poisson distribution, with the $\lambda =$ No. of Toxic Waste Sites per Square Kilometer. You could carry out your sampling program as you described and then find the maximum likelihood estimator for $\lambda_{Ai}$ where A is the area of a sampling sector in zone $i$. In particular, you would maximize the following formula wrt $\lambda_{Ai}$ where $N_i$= number of sectors sampled from zone $i$:

$\max\limits_{\lambda_{Ai}} \prod\limits_{j=1}^{N_i} \frac{e^{-\lambda_{Ai}}\lambda_{Ai}^{n_{ij}}}{n_{ij}!}$ where $n_{ij}$ is the number of toxic sites in sector $j$ of zone $i$. The value of $\lambda_{Ai}$ that maximizes the product is $\lambda_{Ai}^* = \frac{1}{N_i}\sum\limits_{j=1}^{N_i}{n_{ij}}$

You will get one estimate per zone, $\lambda_{Ai}^*$, which you can interpret as the frequency of toxic waste sites within a region of area $A_i$. Your uncertainty for the total number of sites in Zone $i$ with total area $A_{Ti}$can be modeled using your estimated $\lambda_{Ai}^*$ in the Poisson distribution: $Poisson(\lambda_{Ai}^*\frac{A_{Ti}}{A_i})$.

To get a country-wide estiamte, you would need to combine the $\lambda_{Ai}^*$ into another Poisson distribution: Total No. of Sites ~ $Poisson(\sum\limits_{i=1}^{N_{zones}}\lambda_{Ai}^*\frac{A_{Ti}}{A_i})$.

Refinements

The above should get you a decent estimate. However, if your country is small enough that your sample will cover an appreciable portion of the total land area or area within a zone, then you should reduce the total area for each zone by the sampled area in the above formula, so you are modleing the uncertainty on the remaining area (which is actually more accurate in both cases), then you add this uncertainty to your actual counts in the areas you've sampled.

Also, you will notice that you're using a point estimate of $\lambda$. There is some uncertianty in the actual value of this quantity, but including it requires using extended likelihood for predicting a Poisson variable. The formula is pretty simple, if Y is the total number of sites in zone $i$, then the likelihood function for Y is:

$L(Y_i) = e^{-(N+1)\hat\theta(Y_i)}\frac{\hat\theta(Y_i)^{Y_i+\sum\limits_{j=1}^{N_i}{n_{ij}}}}{Y_i!}$ Where $\hat\theta(Y_i) = \frac{A_{Ti}}{A_i}(Y_i + \frac{\sum\limits_{j=1}^{N_i}{n_{ij}}}{N_i+1})$ You need to normalize this formula to sum to 1 over the range of relevant Y. To get the country-wide estimate, you would need to use Monte-Carlo simulation for the sum of the $Y_i$ from each area based on the above formula. There are a couple inexpensive/free simulators out there.

  • Thank you so much for this suggestion. It sounds like you know quite a bit stats theory! Do you know of a practical example where what you suggested is worked through? – Slyron Oct 30 '13 at 20:31
  • See my link for extended likelihood, example 16.2. You will need to normalize the predictive likelihood L(y), as they did in the example, to sum to 1(i've clarified this in my answer) and then simulate from each distribution. I don't have a ready example of where this is used. Ill do a bit of searching and will get back to you. –  Oct 30 '13 at 21:03
  • You might find the following article helpful - its from epidemiology, but I think it is mathematically similar to estimating sites. http://www.biomedcentral.com/content/pdf/1471-2288-11-133.pdf –  Oct 31 '13 at 13:12
  • Hi, just wanted to let you know that I've not found any more relevant examples of this approach. I was trying to give you something that would be relatively simple to implement give your sampling scheme. Whuber is rather more advanced in stats than me, so I'd also take a look at the referenes given in the comment if you have the time. I tried to steer away from spatial statistics and go for something more basic. –  Nov 01 '13 at 17:58