Sampling bias in the determination of first and last occurrences

Background: Past studies of first and last occurrence dates of phenological events have revealed close associations with climatic parameters. Consequently, it is widely acknowledged that recent shifts in the beginning, duration or ending of such events are a response to present climate change. In addition, in recent times, there have tended to be many more observers than in earlier times, especially in urban areas. Furthermore, the number of individuals (plants or animals) observed has often changed markedly. In many situations it is not possible to obtain the average first or last occurrence date of a group of individuals, and only the most extreme occurrence is recorded. This common observational difficulty leads to sampling bias that needs to be taken into account. Aim: Our aim is to use statistical models to quantify the sampling bias and its dependence on sample size and the variability and correlation amongst the individuals under consideration. Methods: n-dimensional multivariate normal distribution and two-way fixed-effects analysis of variance models were developed to examine the dependence of the sampling bias on the above factors. Our results are compared with real data. Results: For first and last occurrence observations, which are the most common index in many phenological studies, we found that changes in observational practice and sample size can, in certain circumstances, easily produce changes in bias that can swamp (or indeed reverse) any climatic change effects. Conclusions: Our new, realistic statistical models allow the sampling bias to be quantified and calculated in terms of the number of individuals under observation, their variability and the degree of correlation between individuals.