Bifurcation and singularity analysis of a molecular network for the induction of long-term memory.

Withdrawal reflexes of the mollusk Aplysia exhibit sensitization, a simple form of long-term memory (LTM). Sensitization is due, in part, to long-term facilitation (LTF) of sensorimotor neuron synapses. LTF is induced by the modulatory actions of serotonin (5-HT). Pettigrew et al. developed a computational model of the nonlinear intracellular signaling and gene network that underlies the induction of 5-HT-induced LTF. The model simulated empirical observations that repeated applications of 5-HT induce persistent activation of protein kinase A (PKA) and that this persistent activation requires a suprathreshold exposure of 5-HT. This study extends the analysis of the Pettigrew model by applying bifurcation analysis, singularity theory, and numerical simulation. Using singularity theory, classification diagrams of parameter space were constructed, identifying regions with qualitatively different steady-state behaviors. The graphical representation of these regions illustrates the robustness of these regions to changes in model parameters. Because persistent protein kinase A (PKA) activity correlates with Aplysia LTM, the analysis focuses on a positive feedback loop in the model that tends to maintain PKA activity. In this loop, PKA phosphorylates a transcription factor (TF-1), thereby increasing the expression of an ubiquitin hydrolase (Ap-Uch). Ap-Uch then acts to increase PKA activity, closing the loop. This positive feedback loop manifests multiple, coexisting steady states, or multiplicity, which provides a mechanism for a bistable switch in PKA activity. After the removal of 5-HT, the PKA activity either returns to its basal level (reversible switch) or remains at a high level (irreversible switch). Such an irreversible switch might be a mechanism that contributes to the persistence of LTM. The classification diagrams also identify parameters and processes that might be manipulated, perhaps pharmacologically, to enhance the induction of memory. Rational drug design, to affect complex processes such as memory formation, can benefit from this type of analysis.

Development of a Bayesian Joint Logistic Model to Better Study the Association between Haplotypes and Disease

In 2011, there will be an estimated 1,596,670 new cancer cases and 571,950 cancer-related deaths in the US. With the ever-increasing applications of cancer genetics in epidemiology, there is great potential to identify genetic risk factors that would help identify individuals with increased genetic susceptibility to cancer, which could be used to develop interventions or targeted therapies that could hopefully reduce cancer risk and mortality. In this dissertation, I propose to develop a new statistical method to evaluate the role of haplotypes in cancer susceptibility and development. This model will be flexible enough to handle not only haplotypes of any size, but also a variety of covariates. I will then apply this method to three cancer-related data sets (Hodgkin Disease, Glioma, and Lung Cancer). I hypothesize that there is substantial improvement in the estimation of association between haplotypes and disease, with the use of a Bayesian mathematical method to infer haplotypes that uses prior information from known genetics sources. Analysis based on haplotypes using information from publically available genetic sources generally show increased odds ratios and smaller p-values in both the Hodgkin, Glioma, and Lung data sets. For instance, the Bayesian Joint Logistic Model (BJLM) inferred haplotype TC had a substantially higher estimated effect size (OR=12.16, 95% CI = 2.47-90.1 vs. 9.24, 95% CI = 1.81-47.2) and more significant p-value (0.00044 vs. 0.008) for Hodgkin Disease compared to a traditional logistic regression approach. Also, the effect sizes of haplotypes modeled with recessive genetic effects were higher (and had more significant p-values) when analyzed with the BJLM. Full genetic models with haplotype information developed with the BJLM resulted in significantly higher discriminatory power and a significantly higher Net Reclassification Index compared to those developed with haplo.stats for lung cancer. Future analysis for this work could be to incorporate the 1000 Genomes project, which offers a larger selection of SNPs can be incorporated into the information from known genetic sources as well. Other future analysis include testing non-binary outcomes, like the levels of biomarkers that are present in lung cancer (NNK), and extending this analysis to full GWAS studies.

Extended kalman filter for estimation of parameters in nonlinear state-space models of biochemical networks.

It is system dynamics that determines the function of cells, tissues and organisms. To develop mathematical models and estimate their parameters are an essential issue for studying dynamic behaviors of biological systems which include metabolic networks, genetic regulatory networks and signal transduction pathways, under perturbation of external stimuli. In general, biological dynamic systems are partially observed. Therefore, a natural way to model dynamic biological systems is to employ nonlinear state-space equations. Although statistical methods for parameter estimation of linear models in biological dynamic systems have been developed intensively in the recent years, the estimation of both states and parameters of nonlinear dynamic systems remains a challenging task. In this report, we apply extended Kalman Filter (EKF) to the estimation of both states and parameters of nonlinear state-space models. To evaluate the performance of the EKF for parameter estimation, we apply the EKF to a simulation dataset and two real datasets: JAK-STAT signal transduction pathway and Ras/Raf/MEK/ERK signaling transduction pathways datasets. The preliminary results show that EKF can accurately estimate the parameters and predict states in nonlinear state-space equations for modeling dynamic biochemical networks.

Evaluating statistical methods used to estimate the number of postsynaptic receptors.

Calcium levels in spines play a significant role in determining the sign and magnitude of synaptic plasticity. The magnitude of calcium influx into spines is highly dependent on influx through N-methyl D-aspartate (NMDA) receptors, and therefore depends on the number of postsynaptic NMDA receptors in each spine. We have calculated previously how the number of postsynaptic NMDA receptors determines the mean and variance of calcium transients in the postsynaptic density, and how this alters the shape of plasticity curves. However, the number of postsynaptic NMDA receptors in the postsynaptic density is not well known. Anatomical methods for estimating the number of NMDA receptors produce estimates that are very different than those produced by physiological techniques. The physiological techniques are based on the statistics of synaptic transmission and it is difficult to experimentally estimate their precision. In this paper we use stochastic simulations in order to test the validity of a physiological estimation technique based on failure analysis. We find that the method is likely to underestimate the number of postsynaptic NMDA receptors, explain the source of the error, and re-derive a more precise estimation technique. We also show that the original failure analysis as well as our improved formulas are not robust to small estimation errors in key parameters.

Models of DNA sequence evolution

Models of DNA sequence evolution and methods for estimating evolutionary distances are needed for studying the rate and pattern of molecular evolution and for inferring the evolutionary relationships of organisms or genes. In this dissertation, several new models and methods are developed.^ The rate variation among nucleotide sites: To obtain unbiased estimates of evolutionary distances, the rate heterogeneity among nucleotide sites of a gene should be considered. Commonly, it is assumed that the substitution rate varies among sites according to a gamma distribution (gamma model) or, more generally, an invariant+gamma model which includes some invariable sites. A maximum likelihood (ML) approach was developed for estimating the shape parameter of the gamma distribution $(\alpha)$ and/or the proportion of invariable sites $(\theta).$ Computer simulation showed that (1) under the gamma model, $\alpha$ can be well estimated from 3 or 4 sequences if the sequence length is long; and (2) the distance estimate is unbiased and robust against violations of the assumptions of the invariant+gamma model.^ However, this ML method requires a huge amount of computational time and is useful only for less than 6 sequences. Therefore, I developed a fast method for estimating $\alpha,$ which is easy to implement and requires no knowledge of tree. A computer program was developed for estimating $\alpha$ and evolutionary distances, which can handle the number of sequences as large as 30.^ Evolutionary distances under the stationary, time-reversible (SR) model: The SR model is a general model of nucleotide substitution, which assumes (i) stationary nucleotide frequencies and (ii) time-reversibility. It can be extended to SRV model which allows rate variation among sites. I developed a method for estimating the distance under the SR or SRV model, as well as the variance-covariance matrix of distances. Computer simulation showed that the SR method is better than a simpler method when the sequence length $L>1,000$ bp and is robust against deviations from time-reversibility. As expected, when the rate varies among sites, the SRV method is much better than the SR method.^ The evolutionary distances under nonstationary nucleotide frequencies: The statistical properties of the paralinear and LogDet distances under nonstationary nucleotide frequencies were studied. First, I developed formulas for correcting the estimation biases of the paralinear and LogDet distances. The performances of these formulas and the formulas for sampling variances were examined by computer simulation. Second, I developed a method for estimating the variance-covariance matrix of the paralinear distance, so that statistical tests of phylogenies can be conducted when the nucleotide frequencies are nonstationary. Third, a new method for testing the molecular clock hypothesis was developed in the nonstationary case. ^

Statistical characterization and development of summary measures of non-normal distributions of nuclear morphometry data from prostate cancer cells for use as prognostic indicators

Nuclear morphometry (NM) uses image analysis to measure features of the cell nucleus which are classified as: bulk properties, shape or form, and DNA distribution. Studies have used these measurements as diagnostic and prognostic indicators of disease with inconclusive results. The distributional properties of these variables have not been systematically investigated although much of the medical data exhibit nonnormal distributions. Measurements are done on several hundred cells per patient so summary measurements reflecting the underlying distribution are needed.^ Distributional characteristics of 34 NM variables from prostate cancer cells were investigated using graphical and analytical techniques. Cells per sample ranged from 52 to 458. A small sample of patients with benign prostatic hyperplasia (BPH), representing non-cancer cells, was used for general comparison with the cancer cells.^ Data transformations such as log, square root and 1/x did not yield normality as measured by the Shapiro-Wilks test for normality. A modulus transformation, used for distributions having abnormal kurtosis values, also did not produce normality.^ Kernel density histograms of the 34 variables exhibited non-normality and 18 variables also exhibited bimodality. A bimodality coefficient was calculated and 3 variables: DNA concentration, shape and elongation, showed the strongest evidence of bimodality and were studied further.^ Two analytical approaches were used to obtain a summary measure for each variable for each patient: cluster analysis to determine significant clusters and a mixture model analysis using a two component model having a Gaussian distribution with equal variances. The mixture component parameters were used to bootstrap the log likelihood ratio to determine the significant number of components, 1 or 2. These summary measures were used as predictors of disease severity in several proportional odds logistic regression models. The disease severity scale had 5 levels and was constructed of 3 components: extracapsulary penetration (ECP), lymph node involvement (LN+) and seminal vesicle involvement (SV+) which represent surrogate measures of prognosis. The summary measures were not strong predictors of disease severity. There was some indication from the mixture model results that there were changes in mean levels and proportions of the components in the lower severity levels. ^

Sequential updating via dynamic hierarchical models: A Bayesian approach to longitudinal data analysis

Most statistical analysis, theory and practice, is concerned with static models; models with a proposed set of parameters whose values are fixed across observational units. Static models implicitly assume that the quantified relationships remain the same across the design space of the data. While this is reasonable under many circumstances this can be a dangerous assumption when dealing with sequentially ordered data. The mere passage of time always brings fresh considerations and the interrelationships among parameters, or subsets of parameters, may need to be continually revised. ^ When data are gathered sequentially dynamic interim monitoring may be useful as new subject-specific parameters are introduced with each new observational unit. Sequential imputation via dynamic hierarchical models is an efficient strategy for handling missing data and analyzing longitudinal studies. Dynamic conditional independence models offers a flexible framework that exploits the Bayesian updating scheme for capturing the evolution of both the population and individual effects over time. While static models often describe aggregate information well they often do not reflect conflicts in the information at the individual level. Dynamic models prove advantageous over static models in capturing both individual and aggregate trends. Computations for such models can be carried out via the Gibbs sampler. An application using a small sample repeated measures normally distributed growth curve data is presented. ^

A full pedigree based method for the statistical assessment of genetic anticipation

Genetic anticipation is defined as a decrease in age of onset or increase in severity as the disorder is transmitted through subsequent generations. Anticipation has been noted in the literature for over a century. Recently, anticipation in several diseases including Huntington's Disease, Myotonic Dystrophy and Fragile X Syndrome were shown to be caused by expansion of triplet repeats. Anticipation effects have also been observed in numerous mental disorders (e.g. Schizophrenia, Bipolar Disorder), cancers (Li-Fraumeni Syndrome, Leukemia) and other complex diseases. ^ Several statistical methods have been applied to determine whether anticipation is a true phenomenon in a particular disorder, including standard statistical tests and newly developed affected parent/affected child pair methods. These methods have been shown to be inappropriate for assessing anticipation for a variety of reasons, including familial correlation and low power. Therefore, we have developed family-based likelihood modeling approaches to model the underlying transmission of the disease gene and penetrance function and hence detect anticipation. These methods can be applied in extended families, thus improving the power to detect anticipation compared with existing methods based only upon parents and children. The first method we have proposed is based on the regressive logistic hazard model. This approach models anticipation by a generational covariate. The second method allows alleles to mutate as they are transmitted from parents to offspring and is appropriate for modeling the known triplet repeat diseases in which the disease alleles can become more deleterious as they are transmitted across generations. ^ To evaluate the new methods, we performed extensive simulation studies for data simulated under different conditions to evaluate the effectiveness of the algorithms to detect genetic anticipation. Results from analysis by the first method yielded empirical power greater than 87% based on the 5% type I error critical value identified in each simulation depending on the method of data generation and current age criteria. Analysis by the second method was not possible due to the current formulation of the software. The application of this method to Huntington's Disease and Li-Fraumeni Syndrome data sets revealed evidence for a generation effect in both cases. ^

Estimating parameters in Markov models for longitudinal studies with missing data or surrogate outcomes

The discrete-time Markov chain is commonly used in describing changes of health states for chronic diseases in a longitudinal study. Statistical inferences on comparing treatment effects or on finding determinants of disease progression usually require estimation of transition probabilities. In many situations when the outcome data have some missing observations or the variable of interest (called a latent variable) can not be measured directly, the estimation of transition probabilities becomes more complicated. In the latter case, a surrogate variable that is easier to access and can gauge the characteristics of the latent one is usually used for data analysis. ^ This dissertation research proposes methods to analyze longitudinal data (1) that have categorical outcome with missing observations or (2) that use complete or incomplete surrogate observations to analyze the categorical latent outcome. For (1), different missing mechanisms were considered for empirical studies using methods that include EM algorithm, Monte Carlo EM and a procedure that is not a data augmentation method. For (2), the hidden Markov model with the forward-backward procedure was applied for parameter estimation. This method was also extended to cover the computation of standard errors. The proposed methods were demonstrated by the Schizophrenia example. The relevance of public health, the strength and limitations, and possible future research were also discussed. ^

Statistical and methodological challenges for disaster preparedness and medical needs assessment in Rio Grande Valley of Texas

In recent years, disaster preparedness through assessment of medical and special needs persons (MSNP) has taken a center place in public eye in effect of frequent natural disasters such as hurricanes, storm surge or tsunami due to climate change and increased human activity on our planet. Statistical methods complex survey design and analysis have equally gained significance as a consequence. However, there exist many challenges still, to infer such assessments over the target population for policy level advocacy and implementation. ^ Objective. This study discusses the use of some of the statistical methods for disaster preparedness and medical needs assessment to facilitate local and state governments for its policy level decision making and logistic support to avoid any loss of life and property in future calamities. ^ Methods. In order to obtain precise and unbiased estimates for Medical Special Needs Persons (MSNP) and disaster preparedness for evacuation in Rio Grande Valley (RGV) of Texas, a stratified and cluster-randomized multi-stage sampling design was implemented. US School of Public Health, Brownsville surveyed 3088 households in three counties namely Cameron, Hidalgo, and Willacy. Multiple statistical methods were implemented and estimates were obtained taking into count probability of selection and clustering effects. Statistical methods for data analysis discussed were Multivariate Linear Regression (MLR), Survey Linear Regression (Svy-Reg), Generalized Estimation Equation (GEE) and Multilevel Mixed Models (MLM) all with and without sampling weights. ^ Results. Estimated population for RGV was 1,146,796. There were 51.5% female, 90% Hispanic, 73% married, 56% unemployed and 37% with their personal transport. 40% people attained education up to elementary school, another 42% reaching high school and only 18% went to college. Median household income is less than $15,000/year. MSNP estimated to be 44,196 (3.98%) [95% CI: 39,029; 51,123]. All statistical models are in concordance with MSNP estimates ranging from 44,000 to 48,000. MSNP estimates for statistical methods are: MLR (47,707; 95% CI: 42,462; 52,999), MLR with weights (45,882; 95% CI: 39,792; 51,972), Bootstrap Regression (47,730; 95% CI: 41,629; 53,785), GEE (47,649; 95% CI: 41,629; 53,670), GEE with weights (45,076; 95% CI: 39,029; 51,123), Svy-Reg (44,196; 95% CI: 40,004; 48,390) and MLM (46,513; 95% CI: 39,869; 53,157). ^ Conclusion. RGV is a flood zone, most susceptible to hurricanes and other natural disasters. People in the region are mostly Hispanic, under-educated with least income levels in the U.S. In case of any disaster people in large are incapacitated with only 37% have their personal transport to take care of MSNP. Local and state government’s intervention in terms of planning, preparation and support for evacuation is necessary in any such disaster to avoid loss of precious human life. ^ Key words: Complex Surveys, statistical methods, multilevel models, cluster randomized, sampling weights, raking, survey regression, generalized estimation equations (GEE), random effects, Intracluster correlation coefficient (ICC).^

A computerized statistical framework for coalescent analysis

Coalescent theory represents the most significant progress in theoretical population genetics in the past three decades. The coalescent theory states that all genes or alleles in a given population are ultimately inherited from a single ancestor shared by all members of the population, known as the most recent common ancestor. It is now widely recognized as a cornerstone for rigorous statistical analyses of molecular data from population [1]. The scientists have developed a large number of coalescent models and methods[2,3,4,5,6], which are not only applied in coalescent analysis and process, but also in today’s population genetics and genome studies, even public health. The thesis aims at completing a statistical framework based on computers for coalescent analysis. This framework provides a large number of coalescent models and statistic methods to assist students and researchers in coalescent analysis, whose results are presented in various formats as texts, graphics and printed pages. In particular, it also supports to create new coalescent models and statistical methods. ^

Assessment of the effect on statistical power of regression model misspecification by using techniques of mathematical statistics and simulation study

Objectives. This paper seeks to assess the effect on statistical power of regression model misspecification in a variety of situations. ^ Methods and results. The effect of misspecification in regression can be approximated by evaluating the correlation between the correct specification and the misspecification of the outcome variable (Harris 2010).In this paper, three misspecified models (linear, categorical and fractional polynomial) were considered. In the first section, the mathematical method of calculating the correlation between correct and misspecified models with simple mathematical forms was derived and demonstrated. In the second section, data from the National Health and Nutrition Examination Survey (NHANES 2007-2008) were used to examine such correlations. Our study shows that comparing to linear or categorical models, the fractional polynomial models, with the higher correlations, provided a better approximation of the true relationship, which was illustrated by LOESS regression. In the third section, we present the results of simulation studies that demonstrate overall misspecification in regression can produce marked decreases in power with small sample sizes. However, the categorical model had greatest power, ranging from 0.877 to 0.936 depending on sample size and outcome variable used. The power of fractional polynomial model was close to that of linear model, which ranged from 0.69 to 0.83, and appeared to be affected by the increased degrees of freedom of this model.^ Conclusion. Correlations between alternative model specifications can be used to provide a good approximation of the effect on statistical power of misspecification when the sample size is large. When model specifications have known simple mathematical forms, such correlations can be calculated mathematically. Actual public health data from NHANES 2007-2008 were used as examples to demonstrate the situations with unknown or complex correct model specification. Simulation of power for misspecified models confirmed the results based on correlation methods but also illustrated the effect of model degrees of freedom on power.^

A STUDY OF INFLUENZA AND OF STATISTICAL METHODS USED TO REPORT INFLUENZA

This paper defines and compares several models for describing excess influenza pneumonia mortality in Houston. First, the methodology used by the Center for Disease Control is examined and several variations of this methodology are studied. All of the models examined emphasize the difficulty of omitting epidemic weeks.^ In an attempt to find a better method of describing expected and epidemic mortality, time series methods are examined. Grouping in four-week periods, truncating the data series to adjust epidemic periods, and seasonally-adjusting the series y(,t), by:^ (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)^ is the best method examined. This new series w(,t) is stationary and a moving average model MA(1) gives a good fit for forecasting influenza and pneumonia mortality in Houston.^ Influenza morbidity, other causes of death, sex, race, age, climate variables, environmental factors, and school absenteeism are all examined in terms of their relationship to influenza and pneumonia mortality. Both influenza morbidity and ischemic heart disease mortality show a very high relationship that remains when seasonal trends are removed from the data. However, when jointly modeling the three series it is obvious that the simple time series MA(1) model of truncated, seasonally-adjusted four-week data gives a better forecast.^

Using spatial linear models with SAR and CAR structure to examine Texas lung cancer incidence rates

Scholars have found that socioeconomic status was one of the key factors that influenced early-stage lung cancer incidence rates in a variety of regions. This thesis examined the association between median household income and lung cancer incidence rates in Texas counties. A total of 254 individual counties in Texas with corresponding lung cancer incidence rates from 2004 to 2008 and median household incomes in 2006 were collected from the National Cancer Institute Surveillance System. A simple linear model and spatial linear models with two structures, Simultaneous Autoregressive Structure (SAR) and Conditional Autoregressive Structure (CAR), were used to link median household income and lung cancer incidence rates in Texas. The residuals of the spatial linear models were analyzed with Moran's I and Geary's C statistics, and the statistical results were used to detect similar lung cancer incidence rate clusters and disease patterns in Texas.^

THE NATURAL AND ORTHOGONAL INTERACTION (NOIA) MODELS FOR QUANTITATIVE TRAITS (QTs) AND COMPLEX DISEASES

My dissertation focuses on developing methods for gene-gene/environment interactions and imprinting effect detections for human complex diseases and quantitative traits. It includes three sections: (1) generalizing the Natural and Orthogonal interaction (NOIA) model for the coding technique originally developed for gene-gene (GxG) interaction and also to reduced models; (2) developing a novel statistical approach that allows for modeling gene-environment (GxE) interactions influencing disease risk, and (3) developing a statistical approach for modeling genetic variants displaying parent-of-origin effects (POEs), such as imprinting. In the past decade, genetic researchers have identified a large number of causal variants for human genetic diseases and traits by single-locus analysis, and interaction has now become a hot topic in the effort to search for the complex network between multiple genes or environmental exposures contributing to the outcome. Epistasis, also known as gene-gene interaction is the departure from additive genetic effects from several genes to a trait, which means that the same alleles of one gene could display different genetic effects under different genetic backgrounds. In this study, we propose to implement the NOIA model for association studies along with interaction for human complex traits and diseases. We compare the performance of the new statistical models we developed and the usual functional model by both simulation study and real data analysis. Both simulation and real data analysis revealed higher power of the NOIA GxG interaction model for detecting both main genetic effects and interaction effects. Through application on a melanoma dataset, we confirmed the previously identified significant regions for melanoma risk at 15q13.1, 16q24.3 and 9p21.3. We also identified potential interactions with these significant regions that contribute to melanoma risk. Based on the NOIA model, we developed a novel statistical approach that allows us to model effects from a genetic factor and binary environmental exposure that are jointly influencing disease risk. Both simulation and real data analyses revealed higher power of the NOIA model for detecting both main genetic effects and interaction effects for both quantitative and binary traits. We also found that estimates of the parameters from logistic regression for binary traits are no longer statistically uncorrelated under the alternative model when there is an association. Applying our novel approach to a lung cancer dataset, we confirmed four SNPs in 5p15 and 15q25 region to be significantly associated with lung cancer risk in Caucasians population: rs2736100, rs402710, rs16969968 and rs8034191. We also validated that rs16969968 and rs8034191 in 15q25 region are significantly interacting with smoking in Caucasian population. Our approach identified the potential interactions of SNP rs2256543 in 6p21 with smoking on contributing to lung cancer risk. Genetic imprinting is the most well-known cause for parent-of-origin effect (POE) whereby a gene is differentially expressed depending on the parental origin of the same alleles. Genetic imprinting affects several human disorders, including diabetes, breast cancer, alcoholism, and obesity. This phenomenon has been shown to be important for normal embryonic development in mammals. Traditional association approaches ignore this important genetic phenomenon. In this study, we propose a NOIA framework for a single locus association study that estimates both main allelic effects and POEs. We develop statistical (Stat-POE) and functional (Func-POE) models, and demonstrate conditions for orthogonality of the Stat-POE model. We conducted simulations for both quantitative and qualitative traits to evaluate the performance of the statistical and functional models with different levels of POEs. Our results showed that the newly proposed Stat-POE model, which ensures orthogonality of variance components if Hardy-Weinberg Equilibrium (HWE) or equal minor and major allele frequencies is satisfied, had greater power for detecting the main allelic additive effect than a Func-POE model, which codes according to allelic substitutions, for both quantitative and qualitative traits. The power for detecting the POE was the same for the Stat-POE and Func-POE models under HWE for quantitative traits.