Estimating the second-stage sample size and the most probable number of hot spots from a first-stage sample of heavy-metal contaminated soil

This article focuses on estimation of the second-stage sample size after collecting a first set of observations in contaminated soils. There are situations where investigators are not satisfied with the results from the first-stage sampling. For instance, it is possible that the number of hot spots of pollutants identified so far does not reflect properly what the investigators expected. In that case, further sampling may be required. We propose the use of Bayesʹ theorem and conditional probabilities to estimate the necessary sample size. One main advantage of this approach is the ability to use information from the previous investigation as well as expert opinions. The distribution of size of the second sample is formulated based on both the size of the first-stage sampling and confidence in the number of hot spots among the first-stage observations. The estimate of the required sample size is further derived from that distribution. A real data set of the heavy-metal contaminated soil located in Taoyuan County, Taiwan was used for illustration. When confidence in the first-stage sampling does not exceed 0.5, the chance of detecting more hot spots is 0.82 at least. In a simulation study, the probability of successfully identifying more hot spots in the second-stage sampling can be as high as 0.7 under reasonable prior assumptions. These indicate that our approach is of potential interest for estimating the second-stage sample size and obtaining more hot spots in contaminated soils.