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The biogeography of plant-animal interactions is a novel topic on which many disciplines converge (e.g., reproductive biology, biogeography, and evolutionary biology). Narrative reviews have indicated that tropical columnar cacti and agaves have highly specialized pollination systems, while extratropical species have generalized systems. However, this dichotomy has never been quantitatively tested. We tested this hypothesis using traditional and phylogenetically informed meta-analysis. Three effect sizes were estimated from the literature: diurnal, nocturnal, and hand cross-pollination (an indicator of pollen limitation). Columnar cactus pollination systems ranged from purely bat-pollinated in the tropics to generalized pollination, with diurnal visitors as effective as nocturnal visitors in extratropical regions; even when phylogenetic relatedness among species is taken into account. Metaregressions identified a latitudinal increase in pollen limitation in columnar cacti, but this increase was not significant after correcting for phylogeny. The currently available data for agaves do not support any latitudinal trend. Nectar production of columnar cacti varied with latitude. Although this variation is positively correlated with pollination by diurnal visitors, it is influenced by phylogeny. The degree of specificity in the pollination systems of columnar cacti is heavily influenced by ecological factors and has a predictable geographic pattern.

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Influence analysis is important in modelling and identification of special patterns in the data. It is well established in ordinary regression. However, analogous diagnostics are generally not available for the multilevel regression model, in which estimation involves a complex iterative algorithm. This paper studies the local influence of small perturbations on the parameter estimates in the multilevel regression model with application to growth curves. The estimation is based on the iterative generalized least-squares (IGLS) method suggested by Goldstein (Biometrika 73 (1986) 43). The generalized influence function and generalized Cook statistic (Biometrika 84(1) (1997) 175) of IGLS of unknown parameters under some specific simultaneous perturbations are derived to study the joint influence of subject units on parameter estimators. The perturbation scheme is introduced through a variance-covariance matrix of error variables. A one-step approximation formula is suggested for simplifying the computations. The method is examined on growth-curve data. r

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