A Hierarchical Bayesian Model with Crossed Random Effects for Predicting Daily Metabolizable Energy Intake in Wild Primates: Model Development and Selection
Akama Okioma C. *
Department of Mathematics, Multimedia University of Kenya, P.O. Box 15653–00503, Nairobi, Kenya and Kenya Institute of Primate Research, P.O. Box 24481-00502, Nairobi, Kenya.
Karanjah Anthony N.
Department of Mathematics, Multimedia University of Kenya, P.O. Box 15653–00503, Nairobi, Kenya.
Kihara Pius N.
Department of Statistics and Actuarial Sciences, Technical University of Kenya, P.O. Box 52428–00200, Nairobi, Kenya.
Kivai M. Stanislaus
Kenya Institute of Primate Research, P.O. Box 24481-00502, Nairobi, Kenya.
*Author to whom correspondence should be addressed.
Abstract
The current study involved the development and selection of a most appropriate hierarchical Bayesian model (HBM) for predicting daily metabolizable energy (ME) intake in the wild primate population and the estimation of the proportion of the variance of ME intake explained by individual identity and food species identity. Retrospective analysis of longitudinal feeding observations of focal animals in Tana River Mangabeys (Cercocebus galeritus) in Kenya, conducted in three study areas (Kitere, Mchelelo and Maramba) over a 15 month period (October 2014 - December 2015). A total of 20,486 feeding observations were made of 63 individually identified mangabeys eating 97 food species in the final dataset. The three models fitted to log-transformed daily ME intake with Hamiltonian Monte Carlo and the No-U-Turn Sampler (NUTS) in Stan were a fixed-effects-only model, a fixed-effects model with an individual-level random intercept, and a model with crossed random intercepts for individual and food species (the proposed model). The model performance was judged based on the Widely Applicable Information Criterion (WAIC) and Pareto-smoothed importance sampling leave-one-out cross validation (PSIS-LOO) and the intraclass correlation coefficients (ICCs) were calculated from posterior draws of best fitting model to partition the variance at the individual and food-species level. The crossed random-effects model substantially outperformed the alternatives with pairwise LOO z-scores of 59, well above the conventional threshold of 4 for significant differences, and with a WAIC improvement over the fixed-effects-only model of 15,080 points and over the individual-random-intercept model of 15,061 points. The variance partitioning revealed that the variance in log(ME) intake accounted for by food species identity was 64.9% (95% CI: 57.7–72.3%), while the variance accounted for by individual identity was 0.1% (95% CI: 0.0–0.2%). When considering behavioral predictors, the ingestion rate was the strongest positive predictor of ME intake, about +30% per SD, and the bout duration was the strongest negative predictor of ME intake, about −23% per SD, which was counterintuitive. Consequently, a crossed random-effects specification is necessary to provide an adequate model for the daily ME intake in this population, the usual individual-only mixed models would greatly underestimate a very important source of variation. The direct impact of food species identity on the ME intake is the main source of variability in the ME intake, not individual identity, and this can have direct implications for prioritization of dietary and habitat-based conservation strategies.
Keywords: Hierarchical Bayesian model, crossed random effects, metabolizable energy intake, intraclass correlation, WAIC, primate feeding ecology