Comparing two methods of calculating the probability of absence: the Negative Predictive Value, and a Credible Interval

If sampling fails to reveal the presence of an invasive species with potential to actually be present, how may we calculate the probability that it is truly absent, e.g. didymo (Didymosphenia geminate) in New Zealand’s North Island. In statistical terms this is a Bayesian question, concerning the probability of a hypothesis (presence/absence), given the obtained data (all results negative). “Classical” theory doesn’t answer this question, because it inverts the required considerations by calculating the probability of all samples being absent if the invasive was actually present. Accordingly, the Bayesian view of “probability” must be adopted in order to answer the question. That definition differs from classical probability in that it always includes an element of subjective belief, particularly in the choice of an appropriate “prior probability” (this is our belief as to the presence of the invasive organism before collecting new data). Bayesian methods can therefore be somewhat controversial – but we seem forced to use them. One Bayesian approach is to use the “Negative Predictive Value”, in which a point estimate of the probability of presence (or absence) prior to sample collection (the “prior probability”) is updated using data once collected using Bayes’ rule. This is in common use in medical studies, where the prior probability is the background disease prevalence, which is generally well understood. It is sometimes used in environmental ‘hot-spot’ investigations. An alternative approach is to recognise the uncertainty in the prior belief by using a distribution of prior probability and updating that using data once collected to give a Credible Interval in which the probability of presence (or absence) should lie – if all our assumptions have been satisfied. We will compare the merits of these approaches considering didymo, southern salt marsh mosquito (Aedes camptorhychus) and the sea squirt Styela clava.