Aspects of the mineral nutrition of some native British plants — inter-site variation

Fifteen common native British plants were each sampled at a range of sites in Great Britain and green tissues analysed for several inorganic nutrients. Sampling criteria are discussed. The inter-site variation of each element within a species is assessed as a frequency distribution of raw data. Sample values of parameters including arithmetic mean, variance (coefficient of variation), skewness and kurtosis are presented. Their stability is assessed from nitrogen in sub-samples of Pteridium. This suggested sample sizes were adequate but some distributions had sufficient kurtosis to affect the variance. These parameters showed distinctions between macro- and micro-elements and between species. Some mean values sharply distinguished between species and may help to assess current theories of strategy and adaptation but a wider range of species is needed to clarify trends. Coefficients of variation are discussed and were relatively low for a nation-wide survey after allowing for sampling constraints. Coefficients of skewness and kurtosis showed two-thirds of the sample distributions were non-normal. Ecological aspects of the distributions are discussed and are relevant to studies along environmental gradients. Published hypotheses linking positive skewness to stress in the field are considered and two other postulates discussed. Distribution bounds such as those confining 95% of the values are discussed in relation to possible critical levels of nutrients.Nomenclature follows Clapham et al. (1981), Excursion flora of the British Isles. 3rd ed. University Press, Cambridge, except Chamaenerion.     

Nonlinear growth generates age changes in the moments of the frequency distribution: the example of height in puberty

Higher moments of the frequency distribution of child height and weight change with age, particularly during puberty, though why is not known. Our aims were to confirm that height skewness and kurtosis change with age during puberty, to devise a model to explain why, and to test the model by analyzing the data longitudinally. Heights of 3245 Christ's Hospital School boys born during 1927-1956 were measured twice termly from 9 to 20 years (n=129508). Treating the data as independent, the mean, standard deviation (SD), skewness, and kurtosis were calculated in 40 age groups and plotted as functions of age t. The data were also analyzed longitudinally using the nonlinear random-effects growth model H(t)=h(t-epsilon )+alpha, with H(t) the cross-sectional data, h(t) the individual mean curve, and epsilon and alpha subject-specific random effects reflecting variability in age and height at peak height velocity (PHV). Mean height increased monotonically with age, while the SD, skewness, and kurtosis changed cyclically with, respectively, 1, 2, and 3 turning points. Surprisingly, their age curves corresponded closely in shape to the first, second, and third derivatives of the mean height curve. The growth model expanded as a Taylor series in epsilon predicted such a pattern, and the longitudinal analysis showed that adjusting for age at PHV on a multiplicative scale largely removed the trends in the higher moments. A nonlinear growth process where subjects grow at different rates, such as in puberty, generates cyclical changes in the higher moments of the frequency distribution.     

Size hierarchy and mineral status of Rhinanthus angustifolius populations under different grassland management regimes

The size hierarchy and mineral concentrations of adult plants in flower and their seed crop have been investigated in three populations of Rhinanthus angustifolius under a grassland management regime consisting of haying without fertilization. The effect of this type of management on the nutrient budgets of soil or Rhinanthus plants was shown to be small initially, i.e. up to the twelfth year of changed management. Although the mean biomass of plants significantly diminished by 30% in this period, their nutrient concentration was not strongly affected. Small plants consistently had lower concentrations of N and Na than large plants. Prolongation of this management regime beyond twelve years let to a general reduction in the phosphorus concentration of the plants. An observed increase in the concentrations of K, Mn, and Cu, especially in small plants, is tentatively ascribed to an imbalance in mineral nutrition.Nomenclature of vascular plants follows Van der Meijden et al. (1983). Flora van Nederland.     

Latitudinal variation in the shape of the species body size distribution: an analysis using freshwater fishes

Many taxonomic and ecological assemblages of species exhibit a right-skewed body size-frequency distribution when characterized at a regional scale. Although this distribution has been frequently described, factors influencing geographic variation in the distribution are not well understood, nor are mechanisms responsible for distribution shape. In this study, variation in the species body size-frequency distributions of 344 regional communities of North American freshwater fishes is examined in relation to latitude, species richness, and taxonomic composition. Although the distribution of all species of North American fishes is right-skewed, a negative correlation exists between latitude and regional community size distribution skewness, with size distributions becoming left-skewed at high latitudes. This relationship is not an artifact of the confounding relationship between latitude and species richness in North American fishes. The negative correlation between latitude and regional community size distribution skewness is partially due to the geographic distribution of families of fishes and apparently enhanced by a nonrandom geographic distribution of species within families. These results are discussed in the context of previous explanations of factors responsible for the generation of species size-frequency distributions related to the fractal nature of the environment, energetics, and evolutionary patterns of body size in North American fishes.     

Linear mixed models with flexible distributions of random effects for longitudinal data

Normality of random effects is a routine assumption for the linear mixed model, but it may be unrealistic, obscuring important features of among-individual variation. We relax this assumption by approximating the random effects density by the seminonparameteric (SNP) representation of Gallant and Nychka (1987, Econometrics 55, 363-390), which includes normality as a special case and provides flexibility in capturing a broad range of nonnormal behavior, controlled by a user-chosen tuning parameter. An advantage is that the marginal likelihood may be expressed in closed form, so inference may be carried out using standard optimization techniques. We demonstrate that standard information criteria may be used to choose the tuning parameter and detect departures from normality, and we illustrate the approach via simulation and using longitudinal data from the Framingham study.