N-mixture models for estimating population size from spatially replicated counts

Spatial replication is a common theme in count surveys of animals. Such surveys often generate sparse count data from which it is difficult to estimate population size while formally accounting for detection probability. In this article, I describe a class of models (N-mixture models) which allow for estimation of population size from such data. The key idea is to view site-specific population sizes, N, as independent random variables distributed according to some mixing distribution (e.g., Poisson). Prior parameters are estimated from the marginal likelihood of the data, having integrated over the prior distribution for N. Carroll and Lombard (1985, Journal of American Statistical Association 80, 423-426) proposed a class of estimators based on mixing over a prior distribution for detection probability. Their estimator can be applied in limited settings, but is sensitive to prior parameter values that are fixed a priori. Spatial replication provides additional information regarding the parameters of the prior distribution on N that is exploited by the N-mixture models and which leads to reasonable estimates of abundance from sparse data. A simulation study demonstrates superior operating characteristics (bias, confidence interval coverage) of the N-mixture estimator compared to the Caroll and Lombard estimator. Both estimators are applied to point count data on six species of birds illustrating the sensitivity to choice of prior on p and substantially different estimates of abundance as a consequence.     

Distinguishing effects on tumor multiplicity and growth rate in chemoprevention experiments

In some types of cancer chemoprevention experiments and short-term carcinogenicity bioassays, the data consist of the number of observed tumors per animal and the times at which these tumors were first detected. In such studies, there is interest in distinguishing between treatment effects on the number of tumors induced by a known carcinogen and treatment effects on the tumor growth rate. Since animals may die before all induced tumors reach a detectable size, separation of these effects can be difficult. This paper describes a flexible parametric model for data of this type. Under our model, the tumor detection times are realizations of a delayed Poisson process that is characterized by the age-specific tumor induction rate and a random latency interval between tumor induction and detection. The model accommodates distinct treatment and animal-specific effects on the number of induced tumors (multiplicity) and the time to tumor detection (growth rate). A Gibbs sampler is developed for estimation of the posterior distributions of the parameters. The methods are illustrated through application to data from a breast cancer chemoprevention experiment.     

A mathematical algorithm that computes breast cancer sizes and doubling times detected by screening

This paper presents a mathematical algorithm that computes the sizes and growth rates of breast cancer detected in a hypothetical population that is screened for the disease. The algorithm works by simulating the outcomes of the hypothetical population twice, first without screening and then with screening. The simulation without screening relies on an underlying model of the natural history of the disease. The simulation with screening uses this natural history model to track the growth of breast tumors backwards in the time starting from the time they would have been detected without screening. The method of tracking tumor growth backward in time is different from methods that track tumor growth forward in time by starting from an estimated time of tumor onset. The screening algorithm combines the natural history model, the method tracking of tumor growth backward in time, the age group, the interval between screening exams, and the detection threshold of the screening exam to compute the joint distribution of tumor size and growth rate among screen-detected and interval patients. The algorithm also computes the sensitivity and leadtime distribution. It allows for arbitrary age groups, detection thresholds and screening intervals and may contribute to the design of future screening trials.     

An approximate likelihood estimator for the prevalence of infections in vectors using pools of varying sizes

There are several arthropods that can transmit disease to humans. To make inferences about the rate of infection of these arthropods, it is common to collect a large sample of vectors, divide them into groups (called pools), and apply a test to detect infection. This paper presents an approximate likelihood point estimator to rate of infection for pools of different sizes, when the variability of these sizes is small and the infection rate is low. The performance of this estimator was evaluated in four simulated scenarios, created from real experiments selected in the literature. The new estimator performed well in three of these scenarios. As expected, the new estimator performed poorly in the scenario with great variability in the size of the pools for some values of the parameter space.     

A growth rate distribution model for the age dependence of human cancer incidence: a proposed role for promotion in cancer of the lung and breast

A biological model is proposed to account for the steep rise of human cancer incidence with age. The model casts in mathematical terms the assumptions that each clone destined to give rise to a detectable tumor displays a characteristic net growth rate and that the assembly of such clones displays a distribution of growth rates. Incidence is introduced as the rate of appearance of clones whose size permits detection. While the cancer formation process may involve a series of stages, we assume that the overall kinetics of tumor detection reflect one stage of development whose duration spans the major portion of the latent period between initial cell alteration and final detection. We further assume that the net growth rates of tumor-forming clones increase in the presence of promotors and resume their original growth rates when the promoting substance is removed. Assuming that cigarette smoke has promoting activity, we show how the model could account for the abrupt impact of cessation of smoking on subsequent lung cancer incidence. If we assume that clones destined to be detected as breast cancers experience promotional activity during the period of a woman's fertility, the model predicts that as a consequence of the slowing down of the clones a discontinuous decline in incidence would follow menopause. Since women experience menopause over a range of ages, we show how aggregating the contributions from these menopausal ages results in an overall age dependence of incidence with no discontinuities and with the observed change in the incidence rate for breast cancer near the age range of menopause.     

Fitting and using growth curves

Summary A technique is presented for fitting and analyzing growth patterns using Gompertz, power, and exponential curves. Data collection involves measuring growth rate as a function of size. This is useful because growth rates at many different sizes can be measured at the same time, which removes the effect of environmental change from the observed growth pattern. Using size instead of age as the independent variable is important because size is usually more closely related to growth rate than is age. The particular technique presented here yields estimates of the variance of the curve parameters so that growth curves for different populations can be compared.     

Optimal individual growth and reproduction in perennial species with indeterminate growth

Summary A model predicting optimal timing of growth and reproduction in perennial species with indeterminate growth living in a seasonal environment, is presented. According to the model, the optimal fraction of growing season devoted to growth decreases with increasing individual age and size, which leads to S-shaped growth curves. Winter mortality seems to be a crucial factor affecting the timing of growth and reproduction, under the same function describing the dependence of growth rate and reproductive rate on body size. When winter mortality is heavy, it is often optimal to start reproducing in the first year, and to devote a large proportion of the subsequent years to reproduction, thus leading to small adult body sizes.The model has been applied to two species of mollusc and one species of fish. The model predictions fit well to the field data for these three species.     

Time-dependent ROC curves for censored survival data and a diagnostic marker

ROC curves are a popular method for displaying sensitivity and specificity of a continuous diagnostic marker, X, for a binary disease variable, D. However, many disease outcomes are time dependent, D(t), and ROC curves that vary as a function of time may be more appropriate. A common example of a time-dependent variable is vital status, where D(t) = 1 if a patient has died prior to time t and zero otherwise. We propose summarizing the discrimination potential of a marker X, measured at baseline (t = 0), by calculating ROC curves for cumulative disease or death incidence by time t, which we denote as ROC(t). A typical complexity with survival data is that observations may be censored. Two ROC curve estimators are proposed that can accommodate censored data. A simple estimator is based on using the Kaplan-Meier estimator for each possible subset X > c. However, this estimator does not guarantee the necessary condition that sensitivity and specificity are monotone in X. An alternative estimator that does guarantee monotonicity is based on a nearest neighbor estimator for the bivariate distribution function of (X, T), where T represents survival time (Akritas, M. J., 1994, Annals of Statistics 22, 1299-1327). We present an example where ROC(t) is used to compare a standard and a modified flow cytometry measurement for predicting survival after detection of breast cancer and an example where the ROC(t) curve displays the impact of modifying eligibility criteria for sample size and power in HIV prevention trials.     

Direct inference of SNP heterozygosity rates and resolution of LOH detection

Single nucleotide polymorphisms (SNPs) have been increasingly utilized to investigate somatic genetic abnormalities in premalignancy and cancer. LOH is a common alteration observed during cancer development, and SNP assays have been used to identify LOH at specific chromosomal regions. The design of such studies requires consideration of the resolution for detecting LOH throughout the genome and identification of the number and location of SNPs required to detect genetic alterations in specific genomic regions. Our study evaluated SNP distribution patterns and used probability models, Monte Carlo simulation, and real human subject genotype data to investigate the relationships between the number of SNPs, SNP HET rates, and the sensitivity (resolution) for detecting LOH. We report that variances of SNP heterozygosity rate in dbSNP are high for a large proportion of SNPs. Two statistical methods proposed for directly inferring SNP heterozygosity rates require much smaller sample sizes (intermediate sizes) and are feasible for practical use in SNP selection or verification. Using HapMap data, we showed that a region of LOH greater than 200 kb can be reliably detected, with losses smaller than 50 kb having a substantially lower detection probability when using all SNPs currently in the HapMap database. Higher densities of SNPs may exist in certain local chromosomal regions that provide some opportunities for reliably detecting LOH of segment sizes smaller than 50 kb. These results suggest that the interpretation of the results from genome-wide scans for LOH using commercial arrays need to consider the relationships among inter-SNP distance, detection probability, and sample size for a specific study. New experimental designs for LOH studies would also benefit from considering the power of detection and sample sizes required to accomplish the proposed aims.     

Analysis of Detection Processes for Breast Cancer

Most existing records of detection processes for breast cancer are in the form of cancer registries or are results of large clinical trials. Statistical modelling can be applied to these data sets to study various properties of breast cancer. In particular we estimate the probability of cure given the size of the tumour at detection, the distribution of tumour growth rates and the distribution of the size of the tumour at detection. There has been a strong recent interest in early detection methods. These consist of giving regular examinations, called screenings. The effect of screening design on the probability of cure is considered. The results of an existing screening trial are used to derive another estimate of the tumour growth rate distribution which agrees well both with our earlier estimate and the most widely used empirical estimate in the literature. The calculation of lead time, which is the time gained in detection when screenings are given, is also discussed.     

On the use of star-shaped genealogies in inference of coalescence times

Genealogies from rapidly growing populations have approximate "star" shapes. We study the degree to which this approximation holds in the context of estimating the time to the most recent common ancestor (T(MRCA)) of a set of lineages. In an exponential growth scenario, we find that unless the product of population size (N) and growth rate (r) is at least approximately 10(5), the "pairwise comparison estimator" of T(MRCA) that derives from the star genealogy assumption has bias of 10-50%. Thus, the estimator is appropriate only for large populations that have grown very rapidly. The "tree-length estimator" of T(MRCA) is more biased than the pairwise comparison estimator, having low bias only for extremely large values of Nr.     

Optimal reproductive strategies in age-structured populations of zooplankton

SUMMARY. We describe a model of zooplankton population dynamics that accounts for differences in mortality and physiology among animals of different ages or sizes. The model follows changes in numbers of individuals and changes in individual and egg biomass through time and it expresses mortality and net assimilation as functions of animal size.
We investigated the effect of egg size, age at first reproduction, and size at first reproduction on the per capita growth rates of populations growing under different conditions. In the absence of predation or when exposed to vertebrate predators that prefer large prey, populations achieve maximum growth rates when animals hatch from small eggs and reach maturity quickly at small sizes. Populations exposed to invertebrate predators that concentrate on small animals may increase r in two different ways. One way is for animals to increase juvenile survivorship by hatching from large eggs and by shortening the juvenile period. An alternative strategy is for animals to hatch from small eggs and to postpone maturity until they grow beyond the range of sizes available to their predators. Certain life history strategies maximize r if animals continue to grow after they reach maturity. By growing larger, non-primiparous females are able to hatch larger clutches and thereby increase the overall rate of population growth.
The model analysis shows how to assess age-dependent mortality rates from field data. The net rate of population increase and the age distribution of eggs together provide specific, quantitative information about mortality.     

A fast and reliable computational method for estimating population genetic parameters

The estimation of ancestral and current effective population sizes in expanding populations is a fundamental problem in population genetics. Recently it has become possible to scan entire genomes of several individuals within a population. These genomic data sets can be used to estimate basic population parameters such as the effective population size and population growth rate. Full-data-likelihood methods potentially offer a powerful statistical framework for inferring population genetic parameters. However, for large data sets, computationally intensive methods based upon full-likelihood estimates may encounter difficulties. First, the computational method may be prohibitively slow or difficult to implement for large data. Second, estimation bias may markedly affect the accuracy and reliability of parameter estimates, as suggested from past work on coalescent methods. To address these problems, a fast and computationally efficient least-squares method for estimating population parameters from genomic data is presented here. Instead of modeling genomic data using a full likelihood, this new approach uses an analogous function, in which the full data are replaced with a vector of summary statistics. Furthermore, these least-squares estimators may show significantly less estimation bias for growth rate and genetic diversity than a corresponding maximum-likelihood estimator for the same coalescent process. The least-squares statistics also scale up to genome-sized data sets with many nucleotides and loci. These results demonstrate that least-squares statistics will likely prove useful for nonlinear parameter estimation when the underlying population genomic processes have complex evolutionary dynamics involving interactions between mutation, selection, demography, and recombination.     

Identity by descent mapping of founder mutations in cancer using high-resolution tumor SNP data

Dense genotype data can be used to detect chromosome fragments inherited from a common ancestor in apparently unrelated individuals. A disease-causing mutation inherited from a common founder may thus be detected by searching for a common haplotype signature in a sample population of patients. We present here FounderTracker, a computational method for the genome-wide detection of founder mutations in cancer using dense tumor SNP profiles. Our method is based on two assumptions. First, the wild-type allele frequently undergoes loss of heterozygosity (LOH) in the tumors of germline mutation carriers. Second, the overlap between the ancestral chromosome fragments inherited from a common founder will define a minimal haplotype conserved in each patient carrying the founder mutation. Our approach thus relies on the detection of haplotypes with significant identity by descent (IBD) sharing within recurrent regions of LOH to highlight genomic loci likely to harbor a founder mutation. We validated this approach by analyzing two real cancer data sets in which we successfully identified founder mutations of well-characterized tumor suppressor genes. We then used simulated data to evaluate the ability of our method to detect IBD tracts as a function of their size and frequency. We show that FounderTracker can detect haplotypes of low prevalence with high power and specificity, significantly outperforming existing methods. FounderTracker is thus a powerful tool for discovering unknown founder mutations that may explain part of the "missing" heritability in cancer. This method is freely available and can be used online at the FounderTracker website.     

Multifactorial analysis of the results of integrated radiation study in breast cancer

Breast cancer can be successfully diagnosed at its early stage on the basis on the detailed analysis of its clinical and radiation signs, by keeping in mind the significance of multiple factors that influence the development of the tumor. Among these factors, the endogenous causes associated with female menstrual status in particular, and the states caused by the cytohistological features of the occurred cancer are of particular importance. To assess the significance of these factors, the authors examined a set of clinical, mammographic, and ultrasound data from 120 women of different ages with the breast cancer established and verified by cytohistological examination. The findings were compared in relation to the menstrual status (pre- and postmenopausal age) at the first stage and to the sizes of a cancer nodule detected by mammography at the second stage. A number of clinical and X-ray signs of cancer were noted to be associated with premenopausal age (the uneven pattern of a skin-matted nodule, enlarged lymph nodes, limited skin thickening, oval mottled tumor shadow with prominences and frequent calcifications) and others are substantially more frequently seen in menopause (the outlines of a tumor nodule with prominences, as well as altered adjacent tissues as prominences and strands). Assessing the pattern of changes in relation to the sizes of a tumor revealed ascertained that the development of the signs of invasive tumor growth was associated with the altered low-grade cancer/ moderate-grade tumor extent ratio.     

Some properties of an estimator for the basic reproduction number of the general epidemic model.

In this paper some properties of a convenient estimator, derived from a martingale estimating function, for the basic reproduction number of the general epidemic model are given for both finite and large samples. These properties give some guidelines for using this convenient estimator. It is shown that it underestimates the parameter and that the bias tends to zero when the population size and the initial number of infectives are increased simultaneously. The bias cannot be removed for a fixed number of introductory infectives. However, the estimator is asymptotically unbiased, conditional on a major outbreak. A simulation study shows that the central limit theorem applies for moderate population sizes.     

Evolution of marine invertebrate reproductive patterns

A simple model of the evolution of reproductive patterns in marine benthic invertebrates is presented. The aim is to discuss the dichotomous distribution of forms into those which produce a large number of small eggs with planktotrophic development, and those which produce a small number of large eggs with direct or lecithotrophic development. The fecundity of adult individuals is assumed to be inversely proportional to egg size, and the mortality of planktonic larvae is assumed to be density independent and size dependent. The growth of planktonic larvae is assumed to be sigmoid with metamorphosis occurring at a given size. The model concludes that there are at most two evolutionarily stable egg sizes. Depending on larval growth rate and death rate, the metamorphosis size, a smaller egg size, or both may be evolutionarily stable. The predictions of the model are compared to patterns observed in nature. Illustrative data are supplied mainly from prosobranch molluscs.     

Evolution of Size-Dependent Flowering in Onopordum illyricum: A Quantitative Assessment of the Role of Stochastic Selection Pressures

We explore the evolution of delayed, size-dependent reproduction in the monocarpic perennial Onopordum illyricum, using a range of mathematical models, parameterized with long-term field data. Analysis of the long-term data indicated that mortality, flowering, and growth were age and size dependent. Using mixed models, we estimated the variance about each of these relationships and also individual-specific effects. For the field populations, recruitment was the main density-dependent process, although there were weak effects of local density on growth and mortality. Using parameterized growth models, which assume plants grow along a deterministic trajectory, we predict plants should flower at sizes approximately 50% smaller than observed in the field. We then develop a simple criterion, termed the "1-yr look-ahead criterion," based on equating seed production now with that of next year, allowing for mortality and growth, to determine at what size a plant should flower. This model allows the incorporation of variance about the growth function and individual-specific effects. The model predicts flowering at sizes approximately double that observed, indicating that variance about the growth curve selects for larger sizes at flowering. The 1-yr look-ahead approach is approximate because it ignores growth opportunities more than 1 yr ahead. To assess the accuracy of this approach, we develop a more complicated dynamic state variable model. Both models give similar results indicating the utility of the 1-yr look-ahead criterion. To allow for temporal variation in the model parameters, we used an individual-based model with a genetic algorithm. This gave very accurate prediction of the observed flowering strategies. Sensitivity analysis of the model suggested that temporal variation in the parameters of the growth equation made waiting to flower more risky, so selected for smaller sizes at flowering. The models clearly indicate the need to incorporate stochastic variation in life-history analyses.     

Information dynamics in carcinogenesis and tumor growth

The storage and transmission of information is vital to the function of normal and transformed cells. We use methods from information theory and Monte Carlo theory to analyze the role of information in carcinogenesis. Our analysis demonstrates that, during somatic evolution of the malignant phenotype, the accumulation of genomic mutations degrades intracellular information. However, the degradation is constrained by the Darwinian somatic ecology in which mutant clones proliferate only when the mutation confers a selective growth advantage. In that environment, genes that normally decrease cellular proliferation, such as tumor suppressor or differentiation genes, suffer maximum information degradation. Conversely, those that increase proliferation, such as oncogenes, are conserved or exhibit only gain of function mutations. These constraints shield most cellular populations from catastrophic mutator-induced loss of the transmembrane entropy gradient and, therefore, cell death. The dynamics of constrained information degradation during carcinogenesis cause the tumor genome to asymptotically approach a minimum information state that is manifested clinically as dedifferentiation and unconstrained proliferation. Extreme physical information (EPI) theory demonstrates that altered information flow from cancer cells to their environment will manifest in-vivo as power law tumor growth with an exponent of size 1.62. This prediction is based only on the assumption that tumor cells are at an absolute information minimum and are capable of "free field" growth that is, they are unconstrained by external biological parameters. The prediction agrees remarkably well with several studies demonstrating power law growth in small human breast cancers with an exponent of 1.72+/-0.24. This successful derivation of an analytic expression for cancer growth from EPI alone supports the conceptual model that carcinogenesis is a process of constrained information degradation and that malignant cells are minimum information systems. EPI theory also predicts that the estimated age of a clinically observed tumor is subject to a root-mean square error of about 30%. This is due to information loss and tissue disorganization and probably manifests as a randomly variable lag phase in the growth pattern that has been observed experimentally. This difference between tumor size and age may impose a fundamental limit on the efficacy of screening based on early detection of small tumors. Independent of the EPI analysis, Monte Carlo methods are applied to predict statistical tumor growth due to perturbed information flow from the environment into transformed cells. A "simplest" Monte Carlo model is suggested by the findings in the EPI approach that tumor growth arises out of a minimally complex mechanism. The outputs of large numbers of simulations show that (a) about 40% of the populations do not survive the first two-generations due to mutations in critical gene segments; but (b) those that do survive will experience power law growth identical to the predicted rate obtained from the independent EPI approach. The agreement between these two very different approaches to the problem strongly supports the idea that tumor cells regress to a state of minimum information during carcinogenesis, and that information dynamics are integrally related to tumor development and growth.     

Modeling the Natural History and Detection of Lung Cancer Based on Smoking Behavior

In this study, we developed a method for modeling the progression and detection of lung cancer based on the smoking behavior at an individual level. The model allows obtaining the characteristics of lung cancer in a population at the time of diagnosis. Lung cancer data from Surveillance, Epidemiology and End Results (SEER) database collected between 2004 and 2008 were used to fit the lung cancer progression and detection model. The fitted model combined with a smoking based carcinogenesis model was used to predict the distribution of age, gender, tumor size, disease stage and smoking status at diagnosis and the results were validated against independent data from the SEER database collected from 1988 to 1999. The model accurately predicted the gender distribution and median age of LC patients of diagnosis, and reasonably predicted the joint tumor size and disease stage distribution.