Statistical Modeling for Biological Systems

The aim of this memorial survey paper is to present some joint work with Andrei Yu. Yakovlev (http://​www.​biology-direct.​com/​content/​3/​1/​10) focused on new ideas for the theory of branching processes arising in cell proliferation modeling. The following topics are considered: some basic characteristics of cell cycle temporal organization, distributions of pulse-labeled discrete markers in branching cell populations, distributions of a continuous label in proliferating cell populations, limiting age and residual lifetime distributions for continuous-time branching processes, limit theorems and estimation theory for multitype branching populations and relative frequencies with a large number of ancestor, age-dependent branching populations with randomly chosen paths of evolution. Some of the presented results have not been published yet.

This article considers age-dependent branching processes with non-homogeneous Poisson immigration as models of cell proliferation kinetics. Asymptotic approximations for the expectation and variance–covariance functions of the process are developed. Estimators relying on the proposed asymptotic approximations are constructed, and their performance investigated using simulations.

Sample correlations between gene pairs within expression profiles are potentially informative regarding gene regulatory pathway structure. However, as is the case with other statistical summaries, observed correlation may be induced or suppressed by factors which are unrelated to gene functionality. In this paper, we consider the effect of heterogeneity on observed correlations, both at the tissue and subject level. Using gene expression profiles from highly enriched samples of three distinct embryonic glial cell types of the rodent neural tube, the effect of tissue heterogeneity on correlations is directly estimated for a simple two component model. Then, a stochastic model of cell population kinetics is used to assess correlation effects for more complex mixtures. Finally, a mathematical model for correlation effects of subject-level heterogeneity is developed. Although decomposition of correlation into functional and nonfunctional sources will generally not be possible, since this depends on nonobservable parameters, reasonable bounds on the size of such effects can be made using the methods proposed here.

Testing multiple hypotheses when observations are positively correlated is very common in practice. The dependence between observations can induce dependence between test statistics and distort the joint distribution of the true and false positives. It has a profound impact on the performance of common multiple testing procedures. While the marginal statistical properties of the true and false discoveries such as their means and variances have been extensively studied in the past, their correlation remains unknown.

By conducting a thorough simulation study, we find that the true and false positives are likely to be negatively correlated if testing power is high and the opposite holds true—they are likely to be positively correlated if testing power is low. The fact that positive dependence between observations can induce negative correlation between the true and false discoveries may assist researchers in designing multiple testing procedures for dependent tests in the future.

We review some results concerning the levels at which multiple testing procedures (MTPs) control certain type I error rates under a general and unknown dependence structure of the p-values on which the MTP is based. The type I error rates we deal with are (1) the classical family-wise error rate (FWER); (2) its immediate generalization: the probability of k or more false rejections (the generalized FWER); (3) the per-family error rate—the expected number of false rejections (PFER). The procedures considered are those satisfying the condition of monotonicity: reduction in some (or all) of the p-values used as input for the MTP can only increase the number of rejected hypotheses. It turns out that this natural condition, either by itself or combined with a property of being a step-down or step-up MTP (where the terms “step-down” and “step-up” are understood in their most general sense), has powerful consequences. Those include optimality results, inequalities, and identities involving different numerical characteristics of a procedure, and computational formulas.

While such methods have been proposed in the literature, for example by Hall in 1983, by Besag and Clifford in 1991 and by Lock in 1991, they have not been widely applied in multiple testing applications generated by high dimensional data sets, where they would likely be of some benefit. In this article related methods will first be reviewed. It will then be shown how commonly used multiple testing procedures may be modified so as to introduce sequential procedures while preserving the validity of reported error rates. A number of examples will show how such procedures can reduce computation time by an order of magnitude with little loss in power.

Traditional approaches to the analysis of epidemiologic data are focused on estimation of the relative risk and are based on the proportional hazards model. Proportionality of hazards in epidemiologic data is a strong assumption that is often violated but seldom checked. Risk often depends on detailed patterns of exposure to environmental agents, but detailed exposure histories are difficult to incorporate in the traditional approaches to analyses of epidemiologic data. For epidemiologic data on cancer, an alternative approach to analysis can be based on ideas of multistage carcinogenesis. The process of carcinogenesis is characterized by mutation accumulation and clonal expansion of partially altered cells on the pathway to cancer. Although this paradigm is now firmly established, most epidemiologic studies of cancer incorporate ideas of multistage carcinogenesis neither in their design nor in their analyses. In this paper we will briefly discuss stochastic multistage models of carcinogenesis and the construction of the appropriate likelihoods for analyses of epidemiologic data using these models. Statistical analyses based on multistage models can quite explicitly incorporate detailed exposure histories in the construction of the likelihood. We will give examples to show that using ideas of multistage carcinogenesis can help reconcile seemingly contradictory findings, and yield insights into epidemiologic studies of cancer that would be difficult or impossible to get from conventional methods. Finally, multistage cancer models provide a unified framework for analyses of data from diverse sources.

Epidemiological research, such as the identification of disease risks attributable to environmental chemical exposures, is often hampered by small population effects, large measurement error, and limited a priori knowledge regarding the complex relationships between the many chemicals under study. However, even an ideal study design does not preclude the possibility of reported false positive exposure effects due to inappropriate statistical methodology. Three issues often overlooked include (1) definition of a meaningful measure of association; (2) use of model estimation strategies (such as machine-learning) that acknowledge that the true data-generating model is unknown; (3) accounting for multiple testing. In this paper, we propose an algorithm designed to address each of these limitations in turn by combining recent advances in the causal inference and multiple-testing literature along with modifications to traditional nonparametric inference methods.

Many popular statistical methods for survival data analysis do not allow the possibility of cure or at least of considerable lengthening of survival for breast cancer patients. Increasingly prolonged follow-up provides new data about the post-treatment outcome, revealing situations with the possibility of high cure rates for early stage breast cancer patients. Then the exclusive use of “classical” statistical models of survival analysis can lead to conclusions not completely reflecting clinical reality. It therefore appears preferable to perform additional analyses based on survival models with cured fraction to study long-term survival data, especially in oncology. This approach allows statistical methods to be adapted to the biological process of tumor growth and dissemination. After presenting the Cox model with time-dependent covariates and the Yakovlev parametric model, we study the prognostic role of age in young women (≤50 years) in Institut Curie breast cancer data. Age is a prognostic factor within all three models, but the interpretation is not the same. With the Cox model the younger women have a bad prognosis (HR = 1.86) comparing to the older one. But the HR does not verify the proportional hazard hypothesis. So the Cox model with time-dependent covariates gives a better interpretation: the age-effect decreases significantly with time. With the Yakovlev model we find that the decreasing age-effect can be viewed through the proportion of cured patients. More, there is an effect of age on survival (palliative effect) and also on the cure rates (curative effect). So the cure rate models demonstrate their utility in analyzing long-term survival data.

Yang (Ann Stat, 6:112–116, 1978) considered an empirical estimate of the mean residual life function on a fixed finite interval. She proved it to be strongly uniformly consistent and (when appropriately standardized) weakly convergent to a Gaussian process. These results are extended to the whole half line, and the variance of the limiting process is studied. Also, nonparametric simultaneous confidence bands for the mean residual life function are obtained by transforming the limiting process to Brownian motion.

In this paper we consider the generalized self-consistency approach to maximum likelihood estimation (MLE). The idea is to represent a given likelihood as a marginal one based on artificial missing data. The computational advantage is sought in the likelihood simplification at the complete-data level. Semiparametric survival models and models for categorical data are used as an example. Justifications for the approach are outlined when the model at the complete-data level is not a legitimate probability model or if it does not exist at all.

In search for better treatment, biomedical researchers have defined an increasing number of new anticancer compounds attacking the tumour disease with drugs targeted to specific molecular structure and acting very differently from standard cytotoxic drugs. This has put high pressure on early clinical drug testing since drugs may need to be tested in parallel when only a limited number of patients—e.g., in rare diseases—or limited funding for a single compound is available. Furthermore, at planning stage, basic information to define an adequate design may be rudimentary. Therefore, flexibility in design and conduct of clinical studies has become one of the methodological challenges in the search for better anticancer treatments. Using the example of a comparative phase II study in patients with rare non-clear cell renal cell carcinoma and high uncertainty about effective treatment options, three flexible design options are explored for two-stage two-armed survival trials. Whereas the two considered classical group sequential approaches integrate early stopping for futility in two-sided hypothesis tests, the presented adaptive group sequential design enlarges these methods by sample size recalculation after the interim analysis if the study has not been stopped for futility. Simulation studies compare the characteristics of the different design approaches.

Statistical methods have been proposed to estimate parameters in multivariate stochastic differential equations (SDEs) from discrete observations. In this paper, we propose a method to improve the performance of the local linearization method proposed by Shoji and Ozaki (Biometrika 85:240–243, 1998), i.e., to avoid the ill-conditioned problem in the computational algorithm. Simulation studies are performed to compare the new method to three other methods, the benchmark Euler method and methods due to Pedersen (1995) and to Hurn et al. (2003). Our results show that the new method performs the best when the sample size is large and the methods proposed by Pedersen and Hurn et al. perform better when the sample size is small. These results provide useful guidance for practitioners.

Definitions and basic properties of bivariate Archimedean copula models for survival data are reviewed with an emphasis on their motivation via frailties. I present some new characterization results for Archimedean copula models based on a notion I call semi-invariance under truncation.

Genetic microarrays have been the primary technology for quantitative transcriptome analysis since the mid-1990s. Via statistical testing methodology developed for microarray data, researchers can study genes and gene pathways involved in a disease. Recently a new technology known as RNA-seq has been developed to quantitatively study the transcriptome. This new technology can also study genes and gene pathways, although the statistical methodology used for microarrays must be adapted to this new platform. In this manuscript, we discuss methods of gene pathway analysis in microarrays and next generation sequencing and their advantages over standard “gene by gene” testing schemes.

This article is a slightly edited and updated version of an evening talk during the random trees week at the Mathematisches Forschungsinstitut Oberwolfach, January 2009. It gives a—personally biased—sketch of the development of branching processes, from the mid nineteenth century to 2010, emphasizing relations to bioscience and demography, and to society and culture in general.

This article describes Dr. Andrei Yakovlev’s unique philosophy of mathematical modeling in biomedical sciences. Although he never formulated it in a systematic way, it has always been central to his work and manifested amply in the course of the author’s 22-year research collaboration with this visionary scholar. We address methodological tensions between mathematics and biomedical sciences, epistemological status of mathematical models, and various methodological questions of a more practical nature arising in mathematical modeling and statistical data analysis including model selection, model identifiability, and concordance between the model and the observables.