The Weibull threshold model
The relationships between annual estimates of burrow occupancy and the probability that plague was detected in gerbils (1 = plague detected that year, 0 = plague not detected) is assessed with generalized nonlinear regression models (Lindsey, 2001) with binomial error (Davis et al., 2004). The nonlinear regression function is the cumulative Weibull distribution function given by
where represents a threshold value and ηand β are shape parameters. Since the binomial probability of this function is zero below the threshold, the log likelihood is infinite. To be able to fit the model, the probability below the threshold was set to a very small value (0.001). To account for serial dependence among years, year was added as a first-order autoregressive variable (exponential serial dependence) in the generalized nonlinear autoregression model (Lindsey, 1999). In the model exp(η) and exp(β) were estimated to make the parameters positive.
To estimate the models, we applied the method proposed by Lindsey (2001). One starts with the simplest model without serial dependence. In a subsequent step, serial dependence is added. Estimates from the previous model serves as initial estimates for the new model.
Model selection was based on Akaike’s information criterion (AIC, Burnham and Anderson, 1998). AIC can be used to compare different types of candidate models, given the response data ( y ) are equal. Models that differed in AIC by 2 or less were considered to be equally good approximations of the data.
Initial least square estimates for the parameters (model without serial dependence) were obtained with the NLIN procedure in SAS 9.1 (SAS Institute Inc., 2005). The model with serial dependence was performed using the GAR (Generalized Autoregression Models) functions in the R-program (R Development Core Team, 2006). R is a fast S-Plus clone freely available to the public ( GAR is part of the repeated package (Non-normal Repeated Measurements Models) written by J. Lindsey (1997). In addition the package rmutil (Utilities for Nonlinear Regression and Repeated Measurements Mode) is required (runs in background). Both packages can be downloaded from
References
Burnham KP, Anderson DR (1998) Model selection and inference. A practical information-theoretic approach. Springer-Verlag, New York
Davis S, Begon M, De Bruyn L, Ageyev VS, Klassovskiy NL, Pole SB, Viljugrein H, Stenseth NC, Leirs H (2004) Predictive thresholds for plague in Kazakhstan. Science 304:736-738
Lindsey JK (1997) Applying Generalized Linear Models. Springer Verlag, Heidelberg
Lindsey JK (2001) Nonlinear models in medical statistics. Oxford University Press, Oxford, UK
Lindsey JK (1999) Models for repeated measurements. Oxford University Press, Oxford, UK
R Development Core Team (2006) R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria
SAS Institute Inc. (2005) The NLIN procedure. SAS/STAT Users Guide, Version 9.1, Cary, NC, USA