Benchmark dose risk analysis with mixed-factor quantal data in environmental risk assessment.

Benchmark analysis is a general risk estimation strategy for identifying the benchmark dose (BMD) past which the risk of exhibiting an adverse environmental response exceeds a fixed, target value of benchmark response (BMR). Estimation of BMD and of its lower confidence limit (BMDL) is well understood for the case of an adverse response to a single stimulus. In many environmental settings, however, one or more additional, secondary, qualitative factor(s) may collude to affect the adverse outcome, such that the risk changes with differential levels of the secondary factor. This paper extends the single-dose BMD paradigm to a mixed-factor setting with a secondary qualitative factor possessing two levels. With focus on quantal-response data and using a generalized linear model with a complementary-log link function, we derive expressions for BMD and BMDL. We study the operating characteristics of six different multiplicity-adjusted approaches to calculate the BMDL, using Monte Carlo evaluations. We illustrate the calculations via an example data set from environmental carcinogenicity testing.

Computing Accurate Probabilistic Estimates of One-D Entropy from Equiprobable Random Samples.

We develop a simple Quantile Spacing (QS) method for accurate probabilistic estimation of one-dimensional entropy from equiprobable random samples, and compare it with the popular Bin-Counting (BC) and Kernel Density (KD) methods. In contrast to BC, which uses equal-width bins with varying probability mass, the QS method uses estimates of the quantiles that divide the support of the data generating probability density function (pdf) into equal-probability-mass intervals. And, whereas BC and KD each require optimal tuning of a hyper-parameter whose value varies with sample size and shape of the pdf, QS only requires specification of the number of quantiles to be used. Results indicate, for the class of distributions tested, that the optimal number of quantiles is a fixed fraction of the sample size (empirically determined to be ~0.25-0.35), and that this value is relatively insensitive to distributional form or sample size. This provides a clear advantage over BC and KD since hyper-parameter tuning is not required. Further, unlike KD, there is no need to select an appropriate kernel-type, and so QS is applicable to pdfs of arbitrary shape, including those with discontinuous slope and/or magnitude. Bootstrapping is used to approximate the sampling variability distribution of the resulting entropy estimate, and is shown to accurately reflect the true uncertainty. For the four distributional forms studied (Gaussian, Log-Normal, Exponential and Bimodal Gaussian Mixture), expected estimation bias is less than 1% and uncertainty is low even for samples of as few as 100 data points; in contrast, for KD the small sample bias can be as large as -10% and for BC as large as -50%. We speculate that estimating quantile locations, rather than bin-probabilities, results in more efficient use of the information in the data to approximate the underlying shape of an unknown data generating pdf.

Staggered chromosomal hybrid zones in the house mouse: relevance to reticulate evolution and speciation.

In the house mouse there are numerous chromosomal races distinguished by different combinations of metacentric chromosomes. These may come into contact with each other and with the ancestral all-acrocentric race, and form hybrid zones. The chromosomal clines that make up these hybrid zones may be coincident or separated from each other (staggered). Such staggered hybrid zones are interesting because they may include populations of individuals homozygous for a mix of features of the hybridising races. We review the characteristics of four staggered hybrid zones in the house mouse and discuss whether they are examples of primary or secondary contact and whether they represent reticulate evolution or not. However, the most important aspect of staggered hybrid zones is that the homozygous populations within the zones have the potential to expand their distributions and become new races (a process termed 'zonal raciation'). In this way they can add to the total 'stock' of chromosomal races in the species concerned. Speciation is an infrequent phenomenon that may involve an unusual set of circumstances. Each one of the products of zonal raciation has the potential to become a new species and by having more races increases the chance of a speciation event.

Genome-wide patterns of gene flow across a house mouse hybrid zone.

Hybrid zones between closely related species or subspecies provide useful settings for studying the genetic architecture of speciation. Using markers distributed throughout the mouse genome, we use a hybrid zone between two recently diverged species of house mice (Mus musculus and Mus domesticus) as a natural mapping experiment to identify genomic regions that may be involved in reproductive isolation. Using cline analysis we document a nearly 50-fold variation in level of introgression among markers. Some markers have extremely narrow cline widths; these genomic regions may contribute to reproductive isolation. Biological processes associated with these narrow clines include physiological and immune responses to the environment as well as physiological and behavioral aspects of reproduction. Other autosomal markers exhibit asymmetrically broad clines, usually with high frequencies of M. domesticus alleles on the M. musculus side of the hybrid zone. These markers identify genome regions likely housing genes with alleles that are spreading from one species to the other. Biological processes associated with these wide clines include cell signaling, olfaction, and pheromone response. These processes play important roles in survival and reproduction, and associated genes are likely targets of selection. Patterns of linkage disequilibrium in the center of the hybrid zone suggest that isolation may be caused by multiple epistatic interactions between sets of genes. These data highlight the complex genetic architecture underlying speciation even at early stages of divergence and point to some of the biological processes that may govern this architecture.