Post provided by Izzy McCabe

Predictions of insect populations from phenology models and sampling can help growers manage pests and beneficial insects on their farms. Phenology models relate accumulated heat units after winter to pest development and thereby inform timing of control actions, while sampling relies on regular on-site data collection (daily, weekly, etc.) to measure pest population densities and estimate potential damage. A drawback of these methods is that phenology and sampling are often considered independently, and growers must rely on their intuition and experience to connect the dots. One way to address this is to establish a maximum tolerable number of individual pests captured during sampling for each heat accumulation value over time, based on a phenology model and an economic threshold. However, data collected by growers come from a variable number of sampling units, and counts are hard to compare with a dynamic threshold. A challenge is thus to find a statistical procedure that allows growers to see the probability of exceeding their maximum tolerable pest density from their sampling data as early in the season as possible. This started an offshoot of our work, which culminated in publishing “Sequential testing of complementary hypotheses about population density” in MEE about our new statistical method, the Sequential Test of Bayesian Posterior Probabilities, or the STBP.

Data for pest sampling comes regularly in batches, so rather than traditional hypothesis testing tools which consider all samples at the same time, we searched the literature for sequential analysis, which iteratively updates predictions as new data arrives. The existing methods, however, had limitations. The most popular sequential data analysis procedure is the Sequential Probability Ratio Test (SPRT), introduced in 1945 by Abraham Wald, but it requires fixed sample sizes and only tests non-complementary hypotheses. For example, a typical SPRT might compare whether the average pests per sample is more likely to be 8 vs 10. Instead, what we need is a method to test complementary hypotheses, for example whether the average pests per sample is more likely to be greater than or less than 9. Further, the original methodology only accounted for testing samples one-at-a-time, and requires complicated modifications to support designs where data should be evaluated in batches, which still limits you to a fixed and predetermined sample size for each batch. We believed that the problem of iteratively updating our knowledge and predictions based on new data aligns with Bayesian statistics, and we found methods using Bayes’ Rule to the domain of sequential hypothesis testing in Morgan & Cressie’s Variable Probability Ratio Test (VPRT). Their method, while providing compelling properties, still requires non-complementary hypothesis ratios and comes with the typical computational complexities of Bayesian methods. The STBP can be seen as building on the VPRT and aims to provide an intuitive procedure that explicitly supports variable sample sizes over time, explicit threshold-style hypotheses, and even dynamic trajectories as hypotheses. The STBP works by making a crucial simplification of typical Bayesian methods: instead of the specification of proper prior probability distributions, it assigns a single probability to the parameter of inference being greater (or less) than a threshold and the complementary probability for it being less (or greater) than that threshold. This is in comparison to the typical procedure of giving each possible value of the parameter its own probability density. This reduces the resulting posterior probability from an entire distribution to a single number, which we can then plug back into the equation as the prior probability for the next iteration. It also eliminates the computational exercise of integrating over constantly changing joint probability density functions which forms the major limitation for typical Bayesian methods. Integrals only need be calculated over your desired likelihood function, for which all commonly used distributions already have closed-form solutions in the form of their cumulative density functions. While our method had a greater rate of false positives compared to the SPRT in one-at-a-time sequential designs, it outperformed it when sequential sampling bouts are made of multiple samples. In general, the STBP requires less sampling and errors scaled more favorably with increasing sample sizes at each sampling time than the SPRT. We also tested our method as a procedure for invasive species monitoring, where it demonstrated higher power for equal sample sizes than conventional approaches, with further reductions in required sampling when prior belief in the absence of the invasive species is already high. We also found that our simplified model quickly converges on the results produced by more complex Bayesian techniques requiring proper conjugate priors just by increasing sample size.

Because our method allows for both variable sample sizes and sequential comparisons for changing-but-related hypotheses, it has potential applications in continuous monitoring for signal processing, fraud detection, forensics, and clinical tests (like that of changes in glucose levels, heart rate, or mortality). For example, if the expected patient mortality rates for certain clinical procedures were continuously compared to those of individual practitioners through STBP, we found that Harold Shipman’s serial killings could have been detected with great confidence in the early 80s for female victims and early 90s for male victims. Unfortunately, the infamous “Dr. Death” was convicted in January 2000 when he had killed nearly 300 people.

Our method represents innovation in a section of the ecological and agricultural literature which has remained largely stagnant for decades. The SPRT, introduced in 1945, is still the most widely used method for sequential analysis, and the fixed-sample size method used in invasive species monitoring has been used unchanged since its introduction in 1993. We plan on working to integrate the STBP into integrated pest management strategies in the Pacific Northwest, and hope to see others find places in their corners of science and industry where it can improve their processes and research. To facilitate this, we are working on an R package that will be easy to use, extensible, and available from the CRAN repository in the coming weeks. The source code can be found in this GitHub repository.

Read the full article here.

Thanks to Emily Rampone for her valuable revision to this post!