Linear regression is a widely used statistical approach to model the relationship between a dependent variable and one or more independent variables. Regression parameters are often estimated using the method of ordinary least squares (OLS). Unfortunately, OLS estimates are very sensitive to outliers. Tabatabai et. al. [30] introduced TELBS robust linear regression method. TELBS estimates have high asymptotic efficiency and high breakdown point. In this study we use simulation to assess the performance of TELBS robust technique in comparison with other methods such as M estimate, MM estimate, S estimate, and Least Trimmed Square (LTS) estimate. We examine the presence of outliers in the direction of response variable, covariates direction, and in both the response and covariates direction. In addition, two real data sets are used to illustrate these methods. Some diagnostic measures are introduced and computed to identify the outliers. Results indicate that as the percentage of outliers increases, TELBS method outperforms other methods considered in this study.

In this paper, innovative models for analysis of survival data, indexed by two shape parameters, which encompasses a wide range of situations are discussed. These models are called hypertabastic survival models. The models are shown to provide different shaped hazard rates. One of the unique strengths of the hypertabastic models is the flexibility of the hazard function that permits the data to determine the nature of the hazard function without its being inadvertently imposed through the selection of improper models. The consideration of the different hazard shapes explains the different mechanisms of the outcome behavior. This helps researchers understand the outcome status over time.

Modeling of the relationship between diameter and height of trees generally employs one of several possible nonlinear growth models, with Richards and Schnute among the most commonly used in the literature. The recently defined hyperbolastic models are nonlinear models of sigmoidal growth noted for their flexibility in representation of biological phenomena. We apply these models to a set of data representing the relationship between diameter and height for a stand of Gmelina arborea in Ibadan, Nigeria and compare the performance with several other nonlinear models. We also compare these nonlinear models with a standard allometric relation, as well as an expanded version. In addition we display the use of the multivariable form of these hyperbolastic models for use in models representing height as a function of diameter as well as other additional explanatory variables. (C) 2017 Published by Elsevier B.V.

In this paper we introduce new robust estimators for the logistic and probit regressions for binary, multinomial, nominal and ordinal data and apply these models to estimate the parameters when outliers or inluential observations are present. Maximum likelihood estimates don't behave well when outliers or inluential observations are present. One remedy is to remove inluential observations from the data and then apply the maximum likelihood technique on the deleted data. Another approach is to employ a robust technique that can handle outliers and inluential observations without removing any observations from the data sets. The robustness of the method is tested using real and simulated data sets.

Background: When outliers are present, the least squares method of nonlinear regression performs poorly.The main purpose of this paper is to provide a robust alternative technique to the Ordinary Least Squares nonlinear regression method.This new robust nonlinear regression method can provide accurate parameter estimates when outliers and/or influential observations are present.Method: Real and simulated data for drug concentration and tumor size-metastasis are used to assess the performance of this new estimator.Monte Carlo simulations are performed to evaluate the robustness of our new method in comparison with the Ordinary Least Squares method. Results:In simulated data with outliers, this new estimator of regression parameters seems to outperform the Ordinary Least Squares with respect to bias, mean squared errors, and mean estimated parameters.Two algorithms have been proposed.Additionally and for the sake of computational ease and illustration, a Mathematica program has been provided in the Appendix. Conclusion:The accuracy of our robust technique is superior to that of the Ordinary Least Squares.The robustness and simplicity of computations make this new technique more appropriate and useful tool for the analysis of nonlinear regressions.

Background The main purpose of this study was to model and analyze the dynamics of cervical cancer mortality rates for African American (Black) and White women residing in 13 states located in the eastern half of the United States of America from 1975 through 2010. Methods The cervical cancer mortality rates of the Surveillance, Epidemiology, and End Results (SEER) were used to model and analyze the dynamics of cervical cancer mortality. A longitudinal hyperbolastic mixed-effects type II model was used to model the cervical cancer mortality data and SAS PROC NLMIXED and Mathematica were utilized to perform the computations. Results Despite decreasing trends in cervical cancer mortality rates for both races, racial disparities in mortality rates still exist. In all 13 states, Black women had higher mortality rates at all times. The degree of disparities and pace of decline in mortality rates over time differed among these states. Determining the paces of decline over 36 years showed that Tennessee had the most rapid decline in cervical cancer mortality for Black women, and Mississippi had the most rapid decline for White Women. In contrast, slow declines in cervical cancer mortality were noted for Black women in Florida and for White women in Maryland. Conclusions In all 13 states, cervical cancer mortality rates for both racial groups have fallen. Disparities in the pace of decline in mortality rates in these states may be due to differences in the rates of screening for cervical cancers. Of note, the gap in cervical cancer mortality rates between Black women and White women is narrowing.

We introduce a new multivariable model to be used to study the growth dynamics of phytoplankton as a function of both time and the concentration of nutrients. This model is applied to a set of experimental data which describes the rate of growth as a function of these two variables. The form of the model allows easy extension to additional variables. Thus, the model can be used to analyze experimental data regarding the effects of various factors on phytoplankton growth rate. Such a model will also be useful in analysis of the role of concentration of various nutrients or trace elements, temperature, and light intensity, or other important explanatory variables, or combinations of such variables, in analyzing phytoplankton growth dynamics.

We review some of the important results on deconvolution, particularly the multi-channel deconvolution problem, for the setting of Euclidean space, focusing on the central role of the Hormander strongly coprime condition in this area of analysis. We then address the problem of deconvolution in the Heisenberg group setting, beginning with the results of [16]. We also extend the results of [16] to three solid tori, a higher dimensional analogue of the three squares considered in [9]. The work of [16] on multi-channel deconvolution is ongoing research, with several important issues still to be fully explored. We address some of these issues, with particular attention to extension of the strongly coprime condition. We also recall the related result of [16] providing a means to extend a deconvolution from a complex space to the Heisenberg group setting and consider a few extensions of this result.

As the field of stem cell research advances, there will be an ongoing and increasing need for mathematical models and other quantitative tools which facilitate research and discovery. Our main concern in this chapter is the description of the applications for several available models to this field of stem cell research. Two important examples of such models include the use of hyperbolastic growth models in stem cell proliferation and the hypertabastic survival model in analysis of time until differentiation. In addition we describe a number of other cases where development of mathematical models or quantitative tools may facilitate stem cell research. One additional area of particular concern is in mathematical modeling related to the cancer stem cell hypothesis. The targeting of cancer stem cells has the potential to transform cancer treatment, and we discuss several related issues and directions for mathematical modeling.

In this paper we introduce a new growth model called T growth model. This model is capable of representing sigmoidal growth as well as biphasic growth. This dual capability is achieved without introducing additional parameters. The T model is useful in modeling cellular proliferation or regression of cancer cells, stem cells, bacterial growth and drug dose-response relationships. We recommend usage of the T growth model for the growth of tumors as part of any system of differential equations. Use of this model within a system will allow more flexibility in representing the natural rate of tumor growth. For illustration, we examine some systems of tumor-immune interaction in which the T growth rate is applied. We also apply the model to a set of tumor growth data.

Keywords: robust linear regression, least squares estimator, M and MM estimators, magnetic resonance imaging, Cook's distance, detection of influential observations, Studentized residual

We consider variations on the Pompeiu transform for the Heisenberg group H n and focus on cases where the transform is known to be injective; in particular the cases of averages over a sphere and a ball, or two balls of appropriate radii. In these cases we develop a method which provides for the reconstruction of the function f from its integrals.

Background We explore the benefits of applying a new proportional hazard model to analyze survival of breast cancer patients. As a parametric model, the hypertabastic survival model offers a closer fit to experimental data than Cox regression, and furthermore provides explicit survival and hazard functions which can be used as additional tools in the survival analysis. In addition, one of our main concerns is utilization of multiple gene expression variables. Our analysis treats the important issue of interaction of different gene signatures in the survival analysis. Methods The hypertabastic proportional hazards model was applied in survival analysis of breast cancer patients. This model was compared, using statistical measures of goodness of fit, with models based on the semi-parametric Cox proportional hazards model and the parametric log-logistic and Weibull models. The explicit functions for hazard and survival were then used to analyze the dynamic behavior of hazard and survival functions. Results The hypertabastic model provided the best fit among all the models considered. Use of multiple gene expression variables also provided a considerable improvement in the goodness of fit of the model, as compared to use of only one. By utilizing the explicit survival and hazard functions provided by the model, we were able to determine the magnitude of the maximum rate of increase in hazard, and the maximum rate of decrease in survival, as well as the times when these occurred. We explore the influence of each gene expression variable on these extrema. Furthermore, in the cases of continuous gene expression variables, represented by a measure of correlation, we were able to investigate the dynamics with respect to changes in gene expression. Conclusions We observed that use of three different gene signatures in the model provided a greater combined effect and allowed us to assess the relative importance of each in determination of outcome in this data set. These results point to the potential to combine gene signatures to a greater effect in cases where each gene signature represents some distinct aspect of the cancer biology. Furthermore we conclude that the hypertabastic survival models can be an effective survival analysis tool for breast cancer patients.

This paper introduces a new dynamical model, called the oscillabolastic model, to analyze the dynamical behavior of biomedical data when one observes oscillatory behavior. The proposed oscillabolastic model is sufficiently flexible to represent various types of oscillatory behavior. The oscillabolastic model is applied to two sets of data. The first data set deals with the oscillabolastic modeling of Ehrlich ascites tumor cells and the second one is the oscillabolastic modeling of the mean signal intensity of Hes1 gene expression in response to serum stimulation. A generalized oscillabolastic model is also suggested to accommodate cases in which predictor variables other than time are also involved.

The standard Pompeiu results for complex balls in the setting of the Heisenberg group H(n) are also shown to carry over to Heisenberg balls. This extension is important because it allows for integration over sets of the same dimension as the ambient space H(n). Several different concepts of the Heisenberg ball are considered, using differing definitions for the metric on H(n). For purposes of this work, none of these is more natural than the others. The results for each of the spaces L(2), L(p) for 1 <= p < infinity, and for L(infinity), are directly comparable to the Pompeiu results for complex balls in H(n), as in [2, 3, 4]. The natural expression for the Pompeiu problem in H(n) is integration over complex balls, and the extension to the Heisenberg ball builds upon the methods for this case. The extra dimension primarily leads to extra complexity in the integrals. At the level of L(infinity), where the results require balls of two radii satisfying appropriate conditions, the additional dimension adds to the complexity in the functions defining the conditions for the radii. The different concepts of the Heisenberg ball lead to different forms for these arithmetic conditions defining the radii. The differences between these balls can also be seen when they are rotated with H(n), an issue to be further considered in a later work. The standard Pompeiu results have been extended to all of the cases considered here.

A new mathematical model for wound healing is introduced and applied to three sets of experimental data. The model is easy to implement but can accommodate a wide range of factors affecting the wound healing process. The data sets represent the areas of trace elements, diabetic wounds, growth factors, and nutrition within the field of wound healing. The model produces an explicit function accurately representing the time course of healing wounds from a given data set. Such a function is used to study variations in the healing velocity among different types of wounds and at different stages in the healing process. A new multivariable model of wound healing capable of analyzing the effects of several variables on accelerating the wound healing process is also introduced. Such a model can help to formulate appropriate strategies to treat wounds. It also would enable us to evaluate the efficacy of different treatment modalities during the inflammatory, proliferative, and tissue remodeling phases.

The mathematical models prevalently used to represent stem cell proliferation do not have the level of accuracy that might be desired. The hyperbolastic growth models promise a greater degree of precision in representing data of stem cell proliferation. The hyperbolastic growth model H3 is applied to experimental data in both embryonic stem cells and adult mesenchymal stem cells. In the embryonic stem cells the results are compared with other popular models, including the Deasy model, which is used prevalently for stem cell growth. In the case of modelling adult mesenchymal stem cells, H3 is also successfully applied to describe the proliferative index. We demonstrated that H3 can accurately represent the dynamics of stem cell proliferation for both embryonic and adult mesenchymal stem cells. We also recognize the importance of additional factors, such as cytokines, in determining the rate of growth. We propose the question of how to extend H3 to a multivariable model that can include the influence of growth factors.

BACKGROUND:An understanding of growth dynamics of tumors is important in understanding progression of cancer and designing appropriate treatment strategies. We perform a comparative study of the hyperbolastic growth models with the Weibull and Gompertz models, which are prevalently used in the field of tumor growth.METHODS:The hyperbolastic growth models H1, H2, and H3 are applied to growth of solid Ehrlich carcinoma under several different treatments. These are compared with results from Gompertz and Weibull models for the combined treatment.RESULTS:The growth dynamics of the solid Ehrlich carcinoma with the combined treatment are studied using models H1, H2, and H3, and the models are highly accurate in representing the growth. The growth dynamics are also compared with the untreated tumor, the tumor treated with only iodoacetate, and the tumor treated with only dimethylsulfoxide, and the combined treatment.CONCLUSIONS:The hyperbolastic models prove to be effective in representing and analyzing the growth dynamics of the solid Ehrlich carcinoma. These models are more precise than Gompertz and Weibull and show less error for this data set. The precision of H3 allows for its use in a comparative analysis of tumor growth rates between the various treatments.

In this survey we present some recent results in Euclidean and Heisenberg spaces for Pompeiu problem with sets of higher codimension. We prove a theorem which demonstrates a higher codimension set, together with a full complement of rotations, will possess the Pompeiu property. We also consider some aspects of the Morera side of the problem.