A likelihood test for inferring extinction from encounter data

Background/Question/Methods

For species which are rare or hard to locate, considerable periods of time may elapse between collections or reliable sight records of the species. In such cases, it may be difficult to know whether the time interval since the last recorded encounter represents merely the expected lapse of time between encounters, or whether it indicates that the species has gone extinct. To address this question, I propose a likelihood test that compares two models of the species encounter data, so that an inference may be made as to whether the species is extinct or not. In one model (the extinction model), encounters are supposed to follow a Poisson process with parameter λ_pre up until the time of the last encounter, t_k. After t_k, the species is supposed to become extinct, so λ_post≡0, and no encounters will have occurred from t_k until the present time, t_n. In the other model (the persistence or homogeneous model), encounters are supposed to follow a Poisson process with parameter λ_homo both before and after t_k.

Results/Conclusions

Because the extinction model contains an additional constraint, incorporating more information from the data in its formulation, it will necessarily have the higher likelihood. In comparing the models, we must therefore ask if the gain in likelihood of the extinction model is sufficient to offset the incorpration of the additional constraint. The gain in the natural logarithm of the likelihood of the extinction model is given by G_E=k[lnt_n-lnt_k], where k is the number of encounters during the entire period of observations, and t_n and t_k are as defined above. The significance of the gain may be evaluated either directly by a likelihood or support test (sensu Edwards), or by a likelihood ratio test (sensu Neyman & Pearson) comparing 2G_E to Χ² with one degree of freedom. The test is applied to encounter data for the West Indian monk seal (Monachus tropicalis). Since 1883, this species has been encountered about 14 times, the last time in 1952. This gives a value of G_E of 8.32, or Χ²_1df=16.64; viewed as either a support or likelihood ratio test, the data thus lend substantial support to the extinction model relative to the persistence model for this species. If attention is restricted to specimen records over the same period, the last record being from 1951, there is again substantial support for extinction over persistence (G_E=3.65; Χ²_1df=7.31).