Estimating the spatial scales of regionalized variables by nested sampling, hierarchical analysis of variance and residual maximum likelihood

The variogram is essential for local estimation and mapping of any variable by kriging. The variogram itself must usually be estimated from sample data. The sampling density is a compromise between precision and cost, but it must be sufficiently dense to encompass the principal spatial sources of variance. A nested, multi-stage, sampling with separating distances increasing in geometric progression from stage to stage will do that. The data may then be analyzed by a hierarchical analysis of variance to estimate the components of variance for every stage, and hence lag. By accumulating the components starting from the shortest lag one obtains a rough variogram for modest effort. For balanced designs the analysis of variance is optimal; for unbalanced ones, however, these estimators are not necessarily the best, and the analysis by residual maximum likelihood (REML) will usually be preferable. The paper summarizes the underlying theory and illustrates its application with data from three surveys, one in which the design had four stages and was balanced and two implemented with unbalanced designs to economize when there were more stages. A Fortran program is available for the analysis of variance, and code for the REML analysis is listed in the paper. (c) 2005 Elsevier Ltd. All rights reserved.

Comparing sampling needs for variograms of soil properties computed by the method of moments and residual maximum likelihood

It has been generally accepted that the method of moments (MoM) variogram, which has been widely applied in soil science, requires about 100 sites at an appropriate interval apart to describe the variation adequately. This sample size is often larger than can be afforded for soil surveys of agricultural fields or contaminated sites. Furthermore, it might be a much larger sample size than is needed where the scale of variation is large. A possible alternative in such situations is the residual maximum likelihood (REML) variogram because fewer data appear to be required. The REML method is parametric and is considered reliable where there is trend in the data because it is based on generalized increments that filter trend out and only the covariance parameters are estimated. Previous research has suggested that fewer data are needed to compute a reliable variogram using a maximum likelihood approach such as REML, however, the results can vary according to the nature of the spatial variation. There remain issues to examine: how many fewer data can be used, how should the sampling sites be distributed over the site of interest, and how do different degrees of spatial variation affect the data requirements? The soil of four field sites of different size, physiography, parent material and soil type was sampled intensively, and MoM and REML variograms were calculated for clay content. The data were then sub-sampled to give different sample sizes and distributions of sites and the variograms were computed again. The model parameters for the sets of variograms for each site were used for cross-validation. Predictions based on REML variograms were generally more accurate than those from MoM variograms with fewer than 100 sampling sites. A sample size of around 50 sites at an appropriate distance apart, possibly determined from variograms of ancillary data, appears adequate to compute REML variograms for kriging soil properties for precision agriculture and contaminated sites. (C) 2007 Elsevier B.V. All rights reserved.

Estimating the spatial scale of herbicide and soil interactions by nested sampling, hierarchical analysis of variance and residual maximum likelihood

An unbalanced nested sampling design was used to investigate the spatial scale of soil and herbicide interactions at the field scale. A hierarchical analysis of variance based on residual maximum likelihood (REML) was used to analyse the data and provide a first estimate of the variogram. Soil samples were taken at 108 locations at a range of separating distances in a 9 ha field to explore small and medium scale spatial variation. Soil organic matter content, pH, particle size distribution, microbial biomass and the degradation and sorption of the herbicide, isoproturon, were determined for each soil sample. A large proportion of the spatial variation in isoproturon degradation and sorption occurred at sampling intervals less than 60 m, however, the sampling design did not resolve the variation present at scales greater than this. A sampling interval of 20-25 m should ensure that the main spatial structures are identified for isoproturon degradation rate and sorption without too great a loss of information in this field.

Estimating the spatial scales of regionalized variables by nested sampling, hierarchical analysis of variance and residual maximum likelihood

The variogram is essential for local estimation and mapping of any variable by kriging. The variogram itself must usually be estimated from sample data. The sampling density is a compromise between precision and cost, but it must be sufficiently dense to encompass the principal spatial sources of variance. A nested, multi-stage, sampling with separating distances increasing in geometric progression from stage to stage will do that. The data may then be analyzed by a hierarchical analysis of variance to estimate the components of variance for every stage, and hence lag. By accumulating the components starting from the shortest lag one obtains a rough variogram for modest effort. For balanced designs the analysis of variance is optimal; for unbalanced ones, however, these estimators are not necessarily the best, and the analysis by residual maximum likelihood (REML) will usually be preferable. The paper summarizes the underlying theory and illustrates its application with data from three surveys, one in which the design had four stages and was balanced and two implemented with unbalanced designs to economize when there were more stages. A Fortran program is available for the analysis of variance, and code for the REML analysis is listed in the paper. (c) 2005 Elsevier Ltd. All rights reserved.

Extending the Lincoln–Petersen estimator for multiple identifications in one source

The Lincoln–Petersen estimator is one of the most popular estimators used in capture–recapture studies. It was developed for a sampling situation in which two sources independently identify members of a target population. For each of the two sources, it is determined if a unit of the target population is identified or not. This leads to a 2 × 2 table with frequencies f11, f10, f01, f00 indicating the number of units identified by both sources, by the first but not the second source, by the second but not the first source and not identified by any of the two sources, respectively. However, f00 is unobserved so that the 2 × 2 table is incomplete and the Lincoln–Petersen estimator provides an estimate for f00. In this paper, we consider a generalization of this situation for which one source provides not only a binary identification outcome but also a count outcome of how many times a unit has been identified. Using a truncated Poisson count model, truncating multiple identifications larger than two, we propose a maximum likelihood estimator of the Poisson parameter and, ultimately, of the population size. This estimator shows benefits, in comparison with Lincoln–Petersen’s, in terms of bias and efficiency. It is possible to test the homogeneity assumption that is not testable in the Lincoln–Petersen framework. The approach is applied to surveillance data on syphilis from Izmir, Turkey.

Improved likelihood inference for the shape parameter in Weibull regression

We obtain adjustments to the profile likelihood function in Weibull regression models with and without censoring. Specifically, we consider two different modified profile likelihoods: (i) the one proposed by Cox and Reid [Cox, D.R. and Reid, N., 1987, Parameter orthogonality and approximate conditional inference. Journal of the Royal Statistical Society B, 49, 1-39.], and (ii) an approximation to the one proposed by Barndorff-Nielsen [Barndorff-Nielsen, O.E., 1983, On a formula for the distribution of the maximum likelihood estimator. Biometrika, 70, 343-365.], the approximation having been obtained using the results by Fraser and Reid [Fraser, D.A.S. and Reid, N., 1995, Ancillaries and third-order significance. Utilitas Mathematica, 47, 33-53.] and by Fraser et al. [Fraser, D.A.S., Reid, N. and Wu, J., 1999, A simple formula for tail probabilities for frequentist and Bayesian inference. Biometrika, 86, 655-661.]. We focus on point estimation and likelihood ratio tests on the shape parameter in the class of Weibull regression models. We derive some distributional properties of the different maximum likelihood estimators and likelihood ratio tests. The numerical evidence presented in the paper favors the approximation to Barndorff-Nielsen`s adjustment.

Improved maximum likelihood estimators in a heteroskedastic errors-in-variables model

This paper develops a bias correction scheme for a multivariate heteroskedastic errors-in-variables model. The applicability of this model is justified in areas such as astrophysics, epidemiology and analytical chemistry, where the variables are subject to measurement errors and the variances vary with the observations. We conduct Monte Carlo simulations to investigate the performance of the corrected estimators. The numerical results show that the bias correction scheme yields nearly unbiased estimates. We also give an application to a real data set.

Genetic parameters of milk, fat, and protein yields in the first three lactations, using an animal model and restricted maximum likelihood

Milk, fat, and protein yields of Holstein cows from the States of New York and California in the United States were used to estimate (co)variances among yields in the first three lactations, using an animal model and a derivative-free restricted maximum likelihood (REML) algorithm, and to verify if yields in different lactations are the same trait. The data were split in 20 samples, 10 from each state, with means of 5463 and 5543 cows per sample from California and New York. Mean heritability estimates for milk, fat, and protein yields for California data were, respectively, 0.34, 0.35, and 0.40 for first; 0.31, 0.33, and 0.39 for second; and 0.28, 0.31, and 0.37 for third lactations. For New York data, estimates were 0.35, 0.40, and 0.34 for first; 0.34, 0.44, and 0.38 for second; and 0.32, 0.43, and 0.38 for third lactations. Means of estimates of genetic correlations between first and second, first and third, and second and third lactations for California data were 0.86, 0.77, and 0.96 for milk; 0.89, 0.84, and 0.97 for fat; and 0.90, 0.84, and 0.97 for protein yields. Mean estimates for New York data were 0.87, 0.81, and 0.97 for milk; 0.91, 0.86, and 0.98 for fat; and 0.88, 0.82, and 0.98 for protein yields. Environmental correlations varied from 0.30 to 0.50 and were larger between second and third lactations. Phenotypic correlations were similar for both states and varied from 0.52 to 0.66 for milk, fat and protein yields. These estimates are consistent with previous estimates obtained with animal models. Yields in different lactations are not statistically the same trait but for selection programs such yields can be modelled as the same trait because of the high genetic correlations.

String-averaging expectation-maximization for maximum likelihood estimation in emission tomography

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)