A Semiparametric Generalized Linear Mixed Model For Trends In Animal Counts
International Livestock Research Institute, Nairobi
Department of Bioinformatics,
University of Hohenheim, Nairobi
Monday, April 10, 2006
E205 Engineering Building
We model spatio-temporal trends in monthly ground counts of widlife conducted in three census blocks in Maasai Mara National Reserve in South-western Kenya. The data recorded during the counts included census block and number of individuals in each age group by sex. The vehicle was driven-off the transect path to within 200 m of each group and stopped to reliably assign individuals to age and sex classes and obtain accurate counts of animals. All sighted animals (n =555,799) were counted and aged while 275,555 were sexed . Monitoring spanned 174 months from July 1989 to December 2003 during which rainfall was also recorded using a network of 16 storage gauges. Survival rates were calculated as a ratio of young to adult and subadult females for each age class.
A suitable trend model for the counts of each age class and census block should accommodate possibly nonlinear and disparate long-term trends for different age classes and census blocks, seasonality in counts, effects of longer-term and monthly rainfall on trends, divergent responses of different age classes to rainfall and temporal autocorrelation. The model should also allow for non-normality of counts, many zeros and missing counts. To satisfy these requirements we use a semiparametric generalized linear mixed model with a negative binomial error variance and a log link function to model trends in the counts. The model incorporates age, census block and their interactions as fixed effects and a radial basis smoothing spline covariance structure. The model is thus equivalent to a semiparamteric regression model, allowing significance testing for the effects of age, block and their interactions as well as spline smoothing of the time series of counts. The radial smoother covariance structure is equivalent to an approximate low-rank thin-plate spline based on an automatic smoother due to Ruppert, Carroll and Wand. It uses the k-d tree method for knot construction and involves trying different choices to select a suitable bucket size.
The total variance is partitioned into four components corresponding to the correlations between temporal trends in counts of (1) the same age class across all the three blocks, (2) all age classes in the same block, (3) all age classes and blocks taken simultaneously and (4) serially correlated residual error. This is achieved by fitting the two-way model with factors age and block and assigning a separate spline component to each of the four effects (intercept, main effects for age and block, and the age-by-block interaction). Dummy coding of age class is used to simultaneously relate different components of rainfall and seasonality to different age classes.
The area sampled in each block and month is used as an offset variable to calculate numerical relative densities. The running months are divided by 100 to obtain variance components of similar magnitude and accelerate convergence. The denominator degrees of freedom are synthesized using the method of Kenward and Roger and the model fitted by restricted pseudo-likelihood in the SAS procedure GLIMMIX, which automatically computes the scale and overdispersion parameters of the negative binomial. The model is fitted by maximizing the residual log pseudo-likelihood with expansion about the current selections of the best linear unbiased predictors of the random effects based on the method of Wolfinger and O'Connell. Missing counts are assumed to be missing completely at random and hence ignorable. A similar model is used for trends in survival rates, with the number of breeders as the offset variable. The model produces intuitively pleasing and ecologically insightful results for all eight wildlife species included in the monitoring program.