On blocking rules for the bootstrap with dependent data

We address the issue of optimal block choice in applicationsof the block bootstrap to dependent data. It is shown that optimalblock size depends significantly on context, being equal ton^1/3, n^1/4 and n^1/5 in the cases of variance or bias estimation,estimation of a onesided distribution function, and estimationof a two-sided distribution function, respectively. A clearintuitive explanation of this phenomenon is given, togetherwith outlines of theoretical arguments in specific cases. Itis shown that these orders of magnitude of block sizes can beused to produce a simple, practical rule for selecting blocksize empirically. That technique is explored numerically.     

Estimation of Individual Growth Curves by Empirically Weighted Least Squares

The simultaneous estimation of individual growth curves and a mean growth curve is accomplished by weighted least squares. A polynomial curve is fitted for each individual and the polynomial parameters are linear functions of parameters corresponding to covariates. A simple, computationally efficient variance-covariance estimator is derived. The resultant estimate is used in the weighted least squares estimation. The results are compared to empirical Bayes estimation.     

Predicted and realized grain yield responses to full-sib family selection in CIMMYT maize (Zea mays L.) populations

Summary The maize (Zea mays L.) improvement program of the International Maize and Wheat Improvement Center (CIMMYT) develops broad-based maize populations and, until recently, improved all of them through full-sib family selection with international testing. The purpose of this study was to estimate the genetic and genetic × environment variance components for ten of those populations and to measure expected yield improvement from full-sib selection. Mean yield ranged from 3.35–6.81 t ha^–1. For five populations the average yield in the last cycle was higher than in the initial cycles. Several populations showed no improvement or yielded less in the final cycle of selection, either because selection intensity was low or because strong selection pressure was applied simultaneously for several traits. Variation resulting from differences among family means within cycles and from interaction between families and locations within cycles were significant in all populations and cycles. Results indicate that variability among full-sib families was maintained throughout the cycles for all populations. The large _ge ² / _g ² ratio shown by most populations suggests that yield response per cycle could be maximized if the environments in which progenies are tested were subdivided and classified into similar subsets. The proportion of the predicted response realized in improved yield varied for each population.Journal Paper No. 8640, Nebraska Agric. Res. Div. Project No. 12-159. Research was supported in part by USAID/USDA/ CSRS Research Grant No. 86-CRSR-2-2789     

The biennial life strategy in a random environment

In a previous paper (J. Math. Biol. 26, 199–215 (1988)) we calculated the mean and variance of the long-run geometric growth rate of a discrete-time population model with two age classes in a random environment. The formula which was used in that paper as the starting point for the computation of the variance represents only the contribution of the one-period variances. Here we supplement these results by a calculation of the exact variance. All qualitative conclusions reached before are unaffected.     

Bias of heritability estimates from twin data in presence of errors of zygosity determination

Simple heritability estimators of continuous as well as discrete traits from twin data are known to overestimate the degree of genetic determination of the measured traits for several reasons. Errors of zygosity determination will, however, underestimate the true heritability. The bias due to wrong assignment of dizygous twin pairs into monozygous type is evaluated here, and the results indicate that this negative bias has a compensatory effect on the estimate of the degree of genetic determination when other factors of similarity between twin pairs are taken into account. It is shown that when an estimate of zygosity error is available, the bias due to this factor can be evaluated quantitatively, and hence the adjustment for zygosity error can be incorporated in the estimation of the degree of genetic determination of a trait. Although this theory is explicitly developed here for twin studies, the general principle also applies for other types of errors of determining the degree of biological relationships for estimation of heritability, in which case this type of error may be more important than the simple zygosity error.     

Confidence Intervals on Ratios of Variance Components for the Unbalanced Two Factor Nested Model

The model considered in this article is the two-factor nested unbalanced variance component model: for p = 1, 2, …, P; q = 1, 2, …, Q_p; and r = 1, 2, …, R_pq. The random variables Y_pqr are observable. The constant μ is an unknown parameter, and A_p, B_pq and C_pqr are (unobservable) normal and independently distributed random variables with zero means and finite variances σ²_A, σ²_B, and σ²_C, respectively. Approximate confidence intervals on ?_A and ?_B using unweighted means are derived, where The performance of these approximate confidence intervals are evaluated using computer simulation. The simulated results indicate that these proposed confidence intervals perform satisfactorily and can be used in applied problems.     

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.