Chris M. Theobald

A bayesian approach to regional and local-area prediction from crop variety trials

The inclusion of covariates in models for analyzing variety × environmental data sets allows the estimation of variety yields for specific locations within a region as well as for the region as a whole. Here we explore a Bayesian approach to the estimation of such effects and to the choice of variety using a possibly incomplete variety × location × year data set that includes location × year covariates. This approach allows expert knowledge of the crop and uncertainty about local circumstances to be incorporated in the analysis. It is implemented using Markov chain Monte Carlo simulation. An example is used to illustrate the approach and investigate its robustness.

Univariate analysis of test day milk yields of British Holstein‐Friesian heifers using Gibbs sampling

Estimates of posterior distributions of genetic and phenotypic parameters and functions of them for individual test day milk yield are obtained for 28 873 British Holstein‐Friesian heifers, the progeny of 40 proven and 649 unproven sires, using restricted maximum likelihood (REML) and Gibbs sampling methods with a univariate sire model. Results from the two methods are then compared. It is found that the posterior expectations and REML estimates of the parameters are fairly similar. The marginal posterior expectation and REML estimate of heritability for lactation milk yield are 0.49 and 0.50, respectively. Heritability estimates for individual test day records range from 0.27 to 0.40, whilst the posterior expectations are between 0.28 and 0.42. Generally, heritabilities for test day records were lowest at the start and highest in the second half of the lactation. Gibbs sampling requires substantially more computing than REML estimation, but provides a more informative analysis.

The Bayesian choice of crop variety and fertilizer dose

Recent contributions to the theory of optimizing fertilizer doses in agricultural crop production have introduced Bayesian ideas to incorporate information on crop yield from several environments and on soil nutrients from a soil test, but they have not used a fully Bayesian formulation. We present such a formulation and demonstrate how the resulting Bayes decision procedure can be evaluated in practice by using Markov chain Monte Carlo methods. The approach incorporates expert knowledge of the crop and of regional and local soil conditions and allows a choice of crop variety as well as of fertilizer level. Alternative dose–response functions are expressed in terms of a common interpretable set of parameters to facilitate model comparisons and the specification of prior distributions. The approach is illustrated with a set of yield data from spring barley nitrogen–response trials and is found to be robust to changes in the dose–response function and the prior distribution for indigenous soil nitrogen.

Assessment of Expert Opinion: Seasonal Sheep Preference and Plant Response to Grazing

Abstract Expert opinion was sought on 2 issues relating to herbivory: seasonal sheep preferences for plant species and seasonal plant response to grazing. Expert opinion is commonly used to parameterize models: it is therefore important to assess its quality. Understanding the limitations of expert knowledge can allow prioritization of future research. Nine experts in plant or grazing ecology from Scotland/Northern England were individually interviewed. The experts ranked sheep preferences for species in 4 rangeland vegetation types and provided categorical information on plant response to grazing. For both issues, seasonal information was collected. Uncertainty (unanswered questions) on plant responses was much higher than uncertainty on sheep preferences. Uncertainty on sheep preference was significantly negatively correlated with plant species commonness, but not with quantity of scientific literature. Uncertainty on plant responses was significantly negatively correlated with both plant commonness and literature. There was agreement among experts on sheep preferences; standardized seasonal information for selected plant species is presented. In general, experts considered graminoids to be preferred over dwarf shrubs, with forbs and other species groups intermediate. Seasonal variation in sheep preference was greater for heath and mire than for grasslands. There was limited agreement among experts on seasonal plant responses. Some experts considered grazing in summer to affect growth more than grazing in winter, whereas others thought season had little effect. Sufficient agreement was found at the species level to present results on plant responses. Experts considered graminoids more resilient to grazing than dwarf shrubs. Experts agreed on sheep preference at different times of year, and on the overall resilience of plant species to grazing. However, the experts held 2 paradigms on the impact of seasonal grazing. Further research is required to explore this, because seasonal grazing regimes are currently promoted as conservation management tools.

Exposure Assessment for Pesticide Intake from Multiple Food Products: A Bayesian Latent‐Variable Approach

Pesticide risk assessment for food products involves combining information from consumption and concentration data sets to estimate a distribution for the pesticide intake in a human population. Using this distribution one can obtain probabilities of individuals exceeding specified levels of pesticide intake. In this article, we present a probabilistic, Bayesian approach to modeling the daily consumptions of the pesticide Iprodione though multiple food products. Modeling data on food consumption and pesticide concentration poses a variety of problems, such as the large proportions of consumptions and concentrations that are recorded as zero, and correlation between the consumptions of different foods. We consider daily food consumption data from the Netherlands National Food Consumption Survey and concentration data collected by the Netherlands Ministry of Agriculture. We develop a multivariate latent-Gaussian model for the consumption data that allows for correlated intakes between products. For the concentration data, we propose a univariate latent-t model. We then combine predicted consumptions and concentrations from these models to obtain a distribution for individual daily Iprodione exposure. The latent-variable models allow for both skewness and large numbers of zeros in the consumption and concentration data. The use of a probabilistic approach is intended to yield more robust estimates of high percentiles of the exposure distribution than an empirical approach. Bayesian inference is used to facilitate the treatment of data with a complex structure.

Estimation of economically optimum seed rates for winter wheat from series of trials

The results of recent trials for winter wheat (Triticum aestivum L.) have influenced farming practice in the UK by encouraging the use of lower seed rates. Spink et al. (2000) have demonstrated that, particularly if sown early, wheat can compensate for reduced plant populations by increased tiller production. Results from seed-rate trials are usually analysed separately for each environment or each combination of environment and variety, and not combined into a single model. They therefore address the question of what the best seed rate would have been for each combination, rather than answer the more relevant question of what rate to choose for a future site. The current paper presents a Bayesian method for combining data from seed-rate trials and choosing optimum seed rates: this method can incorporate information on seed and treatment costs, crop value and covariates. More importantly. for use as an advisory tool, it allows incorporation of expert knowledge of the crop and of the target site. The method is illustrated using two series of trials: the first, carried out at two sites in 1997-99, investigated the effects of sowing date and variety in addition to seed rate. The second was conducted at seven sites in 2001-03 and included latitude and certain management factors. Recommended seed rates based on these series vary substantially with sowing date and latitude. Two non-linear dose-response functions are fitted to the data, the widely used exponential-plus-linear function and the inverse-quadratic function (Nelder 1966). The inverse-quadratic function is found to provide a better fit to the data than the exponential-plus-linear and the latter function gives estimated optimum rates which are as much as 40% lower. The economic consequences of using one function rather than the other are not great in these circumstances. The method is found to be robust to changes in the prior distribution and to other changes in the model used for dependence of yield on sowing date, latitude, variety and management factors.

Calibration of quantitative PCR assays

Quantitative real-time PCR (polymerase chain reaction) assays are increasingly used to measure quantities of nucleic acids in samples. They may be used to provide a high-throughput alternative to more traditional biological assays. In this case, a calibration process may be required to convert the PCR measurements onto a more relevant scale. This is most commonly undertaken using simple linear regression. However, such calibration models are usually unrealistic since they ignore the various sources of variation associated with the PCR and conventional assays. Taking account of these various sources is necessary if the errors associated with predictions based on the calibration model are to be well estimated. In this article, we demonstrate a more complete approach to calibration of quantitative PCR. As an example, we develop a Bayesian calibration model for measuring the quantity of the fungus common bunt (Tilletia caries) on wheat seed, based on our understanding of the properties of the assays. As well as illustrating the steps in developing such a model, we show how the fit of the model might be assessed.

Bayesian selection of fertilizer level when crop price depends on quality

Abstract In studies of the most economical level of fertilizer to be applied to a crop, it is usual to assume that the price which the crop will command is a known constant. This assumption may not be realistic where the price structure is related to the quality of the crop; for example, where a premium is payable according to a measurable quality characteristic, and this characteristic is influenced by the fertilizer dose. Previous work developed a Bayesian approach to finding the optimum fertilizer level when the crop price is constant. That approach incorporates expert knowledge of the crop and soil conditions, and can encompass a choice of variety and fertilizer level. The procedure is extended to model the distribution of the quality characteristic, and hence that of the crop value, at a target location. An example is given in which a premium is offered for barley grain with low levels of nitrogen. The choices of fertilizer level and variety are found to be robust to changes in the dose–response functions for yield and grain nitrogen concentration and in the prior distribution of variance components.

Group Testing, the Pooled Hypergeometric Distribution, and Estimating the Number of Defectives in Small Populations

The testing of combined bacteriological samples – or “group testing” – was introduced to reduce the cost of identifying defective individuals in populations containing small proportions of defectives. It may also be applied to plants, animals, or food samples to estimate proportions infected, or to accept or reject populations. Given the proportion defective in the population, the number of positive combined samples is approximately binomial when the population is large: we find the exact distribution when groups include the same number of samples. We derive some properties of this distribution, and consider maximum-likelihood and Bayesian estimation of the number defective.

A hierarchical Bayesian mixture model for repeated dietary records

Many dietary consumption variables show strong positive skewness or large proportions of zeros. Attempts to normalize such data using transformations such as powers and logarithms can be unsuccessful: this results in poor estimates of their probability distributions, and hence of the proportions of the population whose consumption is beyond recommended limits. As an alternative to such transformations, the use of finite mixtures of standard distributions offers flexible modeling of data having skewed or multi-modal distributions, such as data on dietary consumption. In many dietary studies, individuals are asked to report their consumptions on several days. The use of finite-mixture models for such repeated data requires generalization to take account of the resulting hierarchical structure in the data. We first consider how finite mixture models might be extended to data with repeated records, and then apply a Bayesian version of one such extension to data on the consumption of retinol (Vitamin A) by British adults over 7 consecutive days. We also illustrate how factors such as sex and age may be included in the model. The mixture-model approach is found to provide better estimates than alternative methods of the probability distributions of daily consumptions and of maximum consumption over 7days.