Robert Bartoszyński

Modeling cancer detection: tumor size as a source of information on unobservable stages of carcinogenesis

This paper is concerned with modern approaches to mechanistic modeling of the process of cancer detection. Measurements of tumor size at diagnosis represent a valuable source of information to enrich statistical inference on the processes underlying tumor latency. One possible way of utilizing this information is to model cancer detection as a quantal response variable. In doing so, one relates the chance of detecting a tumor to its current size. We present various theoretical results emerging from this approach and illustrate their usefulness with numerical examples and analyses of epidemiological data. An alternative approach based on a threshold type mechanism of tumor detection is briefly described.

A Multidimensional Goodness-of-Fit Test Based on Interpoint Distances

Abstract Distributional assumptions can be examined with multidimensional goodness-of-fit tests. We propose a conceptually simple test with an appealing logic and accessible asymptotic properties, which is generalizable to a variety of problems and appears to work well against diverse alternatives. To test whether a k-dimensional random sample X1, …, X n follows the distribution G, consider a triangle formed by two randomly selected data points X i and X j and a variable Y ∼ G. Our statistic estimates the likelihood that the side formed by the line from X i to X j is the smallest, the middle, or the largest side of the triangle.

A stochastic model for simulation of interactions between phytophagous spider mites and their phytoseiid predators

A stochastic predator/prey model describing the interaction betweenTetranychus urticae andPhytoseiulus persimilis in investigated via computer simulations and pilot experiments on Lima beans in a greenhouse. Most demographic events, including predation, death due to unknown causes, dispersal, and oviposition, are modelled as stochastic processes. Transitions from eggs to nymphs and from nymphs to adults are deterministic, as are management decisions (release of predators and application of miticide). Computer simulations provide adequate and realistic representations of biological processes, and the model shows stability over a range of inputs. Experimental validation of the model continues. Predictions of the model for optimal predator release or optimal timing of acaricide application have yet to be tested experimentally.

Linear birth and death processes under the influence of disasters with time-dependent killing probabilities

Supercritical linear birth-and-death processes are considered under the influence of disasters that arrive as a renewal process independently of the population size. The novelty of this paper lies in assuming that the killing probability in a disaster is a function of the time that has elapsed since the last disaster. A necessary and sufficient condition for a.s. extinction is found. When catastrophes form a Poisson process, formulas for the Laplace transforms of the expectation and variance of the population size as a function of time as well as moments of the odds of extinction are derived (these odds are random since they depend on the intercatastrophe times). Finally, we study numerical techniques leading to plots of the density of the probability of extinction.

Some stochastic models op cancer metastases

One can show that the popular view, according to which probability of a metastasis in a short time interval is proportional to tumor size, is not compatible with empirical data, and that tumors must be more prone to metastases when they are small. A model is proposed where the metastatic proneness of a tumor is inversely related to its heterogeneity. That is, we assume that initially a tumor consists of a single cell type, and that as it grows, cells may mutate to one of a finite number of different cell types. The mutation rate is either constant or a decreasing function of the number of cell types present. A metastatic rate is a function of the number of cell types present, and is again either a constant or a decreasing function. For each combination of the mutation rate and metastatic proneness the distribution of the time when the first metastasis would be shed is found. The resulting distributions are consistent with empirical data on metastases. The paper also provides the foundations for testing th...

High-dimensional simulation-based estimation

This paper describes a simulation-based technique for estimating the parameters of a high-dimensional stochastic model. The central idea is to find parameters which make the distribution of simulated multidimensional points Y identical to the distribution of the multidimensional points X observed in experiments. To do this, we minimize a criterion based on the heuristic that the univariate distribution of distances between the Ys and the Xs should be the same as the univariate distribution of distances among the replicated Xs themselves. The direction of random local searches in the parameter space for the minimizing value are guided by 1.(i) the degree of success of recent searches, and 2.(ii) a multiple regression fit of the recently investigated portion of the criterions response surface to a deterministic approximation of the stochastic model which can be rapidly investigated. This approximation is most likely to be used when it is most valid, that is when R^2 is close to one. To guard against entrapment at a local minimum, the algorithm at random times will search the global parameter space to look for promising other portions to investigate. Unlike simulated annealing, where the criterion function can be evaluated exactly, our algorithm must deal with the fact that the observed value of the criterion for a given set of parameters is itself based on simulation and thus subject to variability. This difficulty is handled through a cross-validation procedure which examines the distribution of the criterion at the last successful point. The methodology is applied to a detailed stochastic predator-prey model originally described in [1].