Bayesian Adaptive Designs for Early Phase Clinical Trials

My dissertation focuses mainly on Bayesian adaptive designs for phase I and phase II clinical trials. It includes three specific topics: (1) proposing a novel two-dimensional dose-finding algorithm for biological agents, (2) developing Bayesian adaptive screening designs to provide more efficient and ethical clinical trials, and (3) incorporating missing late-onset responses to make an early stopping decision. Treating patients with novel biological agents is becoming a leading trend in oncology. Unlike cytotoxic agents, for which toxicity and efficacy monotonically increase with dose, biological agents may exhibit non-monotonic patterns in their dose-response relationships. Using a trial with two biological agents as an example, we propose a phase I/II trial design to identify the biologically optimal dose combination (BODC), which is defined as the dose combination of the two agents with the highest efficacy and tolerable toxicity. A change-point model is used to reflect the fact that the dose-toxicity surface of the combinational agents may plateau at higher dose levels, and a flexible logistic model is proposed to accommodate the possible non-monotonic pattern for the dose-efficacy relationship. During the trial, we continuously update the posterior estimates of toxicity and efficacy and assign patients to the most appropriate dose combination. We propose a novel dose-finding algorithm to encourage sufficient exploration of untried dose combinations in the two-dimensional space. Extensive simulation studies show that the proposed design has desirable operating characteristics in identifying the BODC under various patterns of dose-toxicity and dose-efficacy relationships. Trials of combination therapies for the treatment of cancer are playing an increasingly important role in the battle against this disease. To more efficiently handle the large number of combination therapies that must be tested, we propose a novel Bayesian phase II adaptive screening design to simultaneously select among possible treatment combinations involving multiple agents. Our design is based on formulating the selection procedure as a Bayesian hypothesis testing problem in which the superiority of each treatment combination is equated to a single hypothesis. During the trial conduct, we use the current values of the posterior probabilities of all hypotheses to adaptively allocate patients to treatment combinations. Simulation studies show that the proposed design substantially outperforms the conventional multi-arm balanced factorial trial design. The proposed design yields a significantly higher probability for selecting the best treatment while at the same time allocating substantially more patients to efficacious treatments. The proposed design is most appropriate for the trials combining multiple agents and screening out the efficacious combination to be further investigated. The proposed Bayesian adaptive phase II screening design substantially outperformed the conventional complete factorial design. Our design allocates more patients to better treatments while at the same time providing higher power to identify the best treatment at the end of the trial. Phase II trial studies usually are single-arm trials which are conducted to test the efficacy of experimental agents and decide whether agents are promising to be sent to phase III trials. Interim monitoring is employed to stop the trial early for futility to avoid assigning unacceptable number of patients to inferior treatments. We propose a Bayesian single-arm phase II design with continuous monitoring for estimating the response rate of the experimental drug. To address the issue of late-onset responses, we use a piece-wise exponential model to estimate the hazard function of time to response data and handle the missing responses using the multiple imputation approach. We evaluate the operating characteristics of the proposed method through extensive simulation studies. We show that the proposed method reduces the total length of the trial duration and yields desirable operating characteristics for different physician-specified lower bounds of response rate with different true response rates.

Computer simulation of glioma growth and morphology.

Despite major advances in the study of glioma, the quantitative links between intra-tumor molecular/cellular properties, clinically observable properties such as morphology, and critical tumor behaviors such as growth and invasiveness remain unclear, hampering more effective coupling of tumor physical characteristics with implications for prognosis and therapy. Although molecular biology, histopathology, and radiological imaging are employed in this endeavor, studies are severely challenged by the multitude of different physical scales involved in tumor growth, i.e., from molecular nanoscale to cell microscale and finally to tissue centimeter scale. Consequently, it is often difficult to determine the underlying dynamics across dimensions. New techniques are needed to tackle these issues. Here, we address this multi-scalar problem by employing a novel predictive three-dimensional mathematical and computational model based on first-principle equations (conservation laws of physics) that describe mathematically the diffusion of cell substrates and other processes determining tumor mass growth and invasion. The model uses conserved variables to represent known determinants of glioma behavior, e.g., cell density and oxygen concentration, as well as biological functional relationships and parameters linking phenomena at different scales whose specific forms and values are hypothesized and calculated based on in vitro and in vivo experiments and from histopathology of tissue specimens from human gliomas. This model enables correlation of glioma morphology to tumor growth by quantifying interdependence of tumor mass on the microenvironment (e.g., hypoxia, tissue disruption) and on the cellular phenotypes (e.g., mitosis and apoptosis rates, cell adhesion strength). Once functional relationships between variables and associated parameter values have been informed, e.g., from histopathology or intra-operative analysis, this model can be used for disease diagnosis/prognosis, hypothesis testing, and to guide surgery and therapy. In particular, this tool identifies and quantifies the effects of vascularization and other cell-scale glioma morphological characteristics as predictors of tumor-scale growth and invasion.

Computer simulation of glioma growth and morphology.

Despite major advances in the study of glioma, the quantitative links between intra-tumor molecular/cellular properties, clinically observable properties such as morphology, and critical tumor behaviors such as growth and invasiveness remain unclear, hampering more effective coupling of tumor physical characteristics with implications for prognosis and therapy. Although molecular biology, histopathology, and radiological imaging are employed in this endeavor, studies are severely challenged by the multitude of different physical scales involved in tumor growth, i.e., from molecular nanoscale to cell microscale and finally to tissue centimeter scale. Consequently, it is often difficult to determine the underlying dynamics across dimensions. New techniques are needed to tackle these issues. Here, we address this multi-scalar problem by employing a novel predictive three-dimensional mathematical and computational model based on first-principle equations (conservation laws of physics) that describe mathematically the diffusion of cell substrates and other processes determining tumor mass growth and invasion. The model uses conserved variables to represent known determinants of glioma behavior, e.g., cell density and oxygen concentration, as well as biological functional relationships and parameters linking phenomena at different scales whose specific forms and values are hypothesized and calculated based on in vitro and in vivo experiments and from histopathology of tissue specimens from human gliomas. This model enables correlation of glioma morphology to tumor growth by quantifying interdependence of tumor mass on the microenvironment (e.g., hypoxia, tissue disruption) and on the cellular phenotypes (e.g., mitosis and apoptosis rates, cell adhesion strength). Once functional relationships between variables and associated parameter values have been informed, e.g., from histopathology or intra-operative analysis, this model can be used for disease diagnosis/prognosis, hypothesis testing, and to guide surgery and therapy. In particular, this tool identifies and quantifies the effects of vascularization and other cell-scale glioma morphological characteristics as predictors of tumor-scale growth and invasion.

Outcome-based adaptive randomization for binary data in clinical trials---Compare and contrast

Monte Carlo simulation has been conducted to investigate parameter estimation and hypothesis testing in some well known adaptive randomization procedures. The four urn models studied are Randomized Play-the-Winner (RPW), Randomized Pôlya Urn (RPU), Birth and Death Urn with Immigration (BDUI), and Drop-the-Loses Urn (DL). Two sequential estimation methods, the sequential maximum likelihood estimation (SMLE) and the doubly adaptive biased coin design (DABC), are simulated at three optimal allocation targets that minimize the expected number of failures under the assumption of constant variance of simple difference (RSIHR), relative risk (ORR), and odds ratio (OOR) respectively. Log likelihood ratio test and three Wald-type tests (simple difference, log of relative risk, log of odds ratio) are compared in different adaptive procedures. ^ Simulation results indicates that although RPW is slightly better in assigning more patients to the superior treatment, the DL method is considerably less variable and the test statistics have better normality. When compared with SMLE, DABC has slightly higher overall response rate with lower variance, but has larger bias and variance in parameter estimation. Additionally, the test statistics in SMLE have better normality and lower type I error rate, and the power of hypothesis testing is more comparable with the equal randomization. Usually, RSIHR has the highest power among the 3 optimal allocation ratios. However, the ORR allocation has better power and lower type I error rate when the log of relative risk is the test statistics. The number of expected failures in ORR is smaller than RSIHR. It is also shown that the simple difference of response rates has the worst normality among all 4 test statistics. The power of hypothesis test is always inflated when simple difference is used. On the other hand, the normality of the log likelihood ratio test statistics is robust against the change of adaptive randomization procedures. ^

Interim analysis of clinical trials: Simulation studies of fractional Brownian motion

Interim clinical trial monitoring procedures were motivated by ethical and economic considerations. Classical Brownian motion (Bm) techniques for statistical monitoring of clinical trials were widely used. Conditional power argument and α-spending function based boundary crossing probabilities are popular statistical hypothesis testing procedures under the assumption of Brownian motion. However, it is not rare that the assumptions of Brownian motion are only partially met for trial data. Therefore, I used a more generalized form of stochastic process, called fractional Brownian motion (fBm), to model the test statistics. Fractional Brownian motion does not hold Markov property and future observations depend not only on the present observations but also on the past ones. In this dissertation, we simulated a wide range of fBm data, e.g., H = 0.5 (that is, classical Bm) vs. 0.5< H <1, with treatment effects vs. without treatment effects. Then the performance of conditional power and boundary-crossing based interim analyses were compared by assuming that the data follow Bm or fBm. Our simulation study suggested that the conditional power or boundaries under fBm assumptions are generally higher than those under Bm assumptions when H > 0.5 and also matches better with the empirical results. ^

Robust effect sizes and their confidence intervals for group difference between trajectories in hierarchical linear growth model

Hierarchical linear growth model (HLGM), as a flexible and powerful analytic method, has played an increased important role in psychology, public health and medical sciences in recent decades. Mostly, researchers who conduct HLGM are interested in the treatment effect on individual trajectories, which can be indicated by the cross-level interaction effects. However, the statistical hypothesis test for the effect of cross-level interaction in HLGM only show us whether there is a significant group difference in the average rate of change, rate of acceleration or higher polynomial effect; it fails to convey information about the magnitude of the difference between the group trajectories at specific time point. Thus, reporting and interpreting effect sizes have been increased emphases in HLGM in recent years, due to the limitations and increased criticisms for statistical hypothesis testing. However, most researchers fail to report these model-implied effect sizes for group trajectories comparison and their corresponding confidence intervals in HLGM analysis, since lack of appropriate and standard functions to estimate effect sizes associated with the model-implied difference between grouping trajectories in HLGM, and also lack of computing packages in the popular statistical software to automatically calculate them. ^ The present project is the first to establish the appropriate computing functions to assess the standard difference between grouping trajectories in HLGM. We proposed the two functions to estimate effect sizes on model-based grouping trajectories difference at specific time, we also suggested the robust effect sizes to reduce the bias of estimated effect sizes. Then, we applied the proposed functions to estimate the population effect sizes (d ) and robust effect sizes (du) on the cross-level interaction in HLGM by using the three simulated datasets, and also we compared the three methods of constructing confidence intervals around d and du recommended the best one for application. At the end, we constructed 95% confidence intervals with the suitable method for the effect sizes what we obtained with the three simulated datasets. ^ The effect sizes between grouping trajectories for the three simulated longitudinal datasets indicated that even though the statistical hypothesis test shows no significant difference between grouping trajectories, effect sizes between these grouping trajectories can still be large at some time points. Therefore, effect sizes between grouping trajectories in HLGM analysis provide us additional and meaningful information to assess group effect on individual trajectories. In addition, we also compared the three methods to construct 95% confident intervals around corresponding effect sizes in this project, which handled with the uncertainty of effect sizes to population parameter. We suggested the noncentral t-distribution based method when the assumptions held, and the bootstrap bias-corrected and accelerated method when the assumptions are not met.^

POLG DNA testing as an emerging standard of care before instituting valproic acid therapy for pediatric seizure disorders.

PURPOSE: To review our clinical experience and determine if there are appropriate signs and symptoms to consider POLG sequencing prior to valproic acid (VPA) dosing in patients with seizures. METHODS: Four patients who developed VPA-induced hepatotoxicity were examined for POLG sequence variations. A subsequent chart review was used to describe clinical course prior to and after VPA dosing. RESULTS: Four patients of multiple different ethnicities, age 3-18 years, developed VPA-induced hepatotoxicity. All were given VPA due to intractable partial seizures. Three of the patients had developed epilepsia partialis continua. The time from VPA exposure to liver failure was between 2 and 3 months. Liver failure was reversible in one patient. Molecular studies revealed homozygous p.R597W or p.A467T mutations in two patients. The other two patients showed compound heterozygous mutations, p.A467T/p.Q68X and p.L83P/p.G888S. Clinical findings and POLG mutations were diagnostic of Alpers-Huttenlocher syndrome. CONCLUSION: Our cases underscore several important findings: POLG mutations have been observed in every ethnic group studied to date; early predominance of epileptiform discharges over the occipital region is common in POLG-induced epilepsy; the EEG and MRI findings varying between patients and stages of the disease; and VPA dosing at any stage of Alpers-Huttenlocher syndrome can precipitate liver failure. Our data support an emerging proposal that POLG gene testing should be considered in any child or adolescent who presents or develops intractable seizures with or without status epilepticus or epilepsia partialis continua, particularly when there is a history of psychomotor regression.

BRAIN ACTIVATION AND CONNECTIVITY IN NON-DISABLED MULTIPLE SCLEROSIS PATIENTS

Multiple sclerosis (MS) is the most common demyelinating disease affecting the central nervous system. There is no cure for MS and current therapies have limited efficacy. While the majority of individuals with MS develop significant clinical disability, a subset experiences a disease course with minimal impairment even in the presence of significant apparent tissue damage on magnetic resonance imaging (MRI). The current studies combined functional MRI and diffusion tensor imaging (DTI) to elucidate brain mechanisms associated with lack of clinical disability in patients with MS. Recent evidence has implicated cortical reorganization as a mechanism to limit the clinical manifestation of the disease. Functional MRI was used to test the hypothesis that non-disabled MS patients (Expanded Disability Status Scale ≤ 1.5) show increased recruitment of cognitive control regions (dorsolateral prefrontal and anterior cingulate cortex) while performing sensory, motor and cognitive tasks. Compared to matched healthy controls, patients increased activation of cognitive control brain regions when performing non-dominant hand movements and the 2-back working memory task. Using dynamic causal modeling, we tested whether increased cognitive control recruitment is associated with alterations in connectivity in the working memory functional network. Patients exhibited similar network connectivity to that of control subjects when performing working memory tasks. We subsequently investigated the integrity of major white matter tracts to assess structural connectivity and its relation to activation and functional integration of the cognitive control system. Patients showed substantial alterations in callosal, inferior and posterior white matter tracts and less pronounced involvement of the corticospinal tracts and superior longitudinal fasciculi (SLF). Decreased structural integrity within the right SLF in patients was associated with decreased performance, and decreased activation and connectivity of the cognitive control system when performing working memory tasks. These studies suggest that patient with MS without clinical disability increase cognitive control system recruitment across functional domains and rely on preserved functional and structural connectivity of brain regions associated with this network. Moreover, the current studies show the usefulness of combining brain activation data from functional MRI and structural connectivity data from DTI to improve our understanding of brain adaptation mechanisms to neurological disease.

Development and testing of a procedure for systematically assigning gestational age for clinical and research use

A graphing method was developed and tested to estimate gestational ages pre-and postnatally in a consistent manner for epidemiological research and clinical purposes on feti/infants of women with few consistent prenatal estimators of gestational age. Each patient's available data was plotted on a single page graph to give a comprehensive overview of that patient. A hierarchical classification of gestational age determination was then applied in a systematic manner, and reasonable gestational age estimates were produced. The method was tested for validity and reliability on 50 women who had known dates for their last menstrual period or dates of conception, and multiple ultrasound examinations and other gestational age estimating measures. The feasibility of the procedure was then tested on 1223 low income women with few gestational age estimators. The graphing method proved to have high inter- and intrarater reliability. It was quick, easy to use, inexpensive, and did not require special equipment. The graphing method estimate of gestational age for each infant was tested against the last menstrual period gestational age estimate using paired t-Tests, F tests and the Kolmogorov-Smirnov test of similar populations, producing a 98 percent probability or better that the means and data populations were the same. Less than 5 percent of the infants' gestational ages were misclassified using the graphing method, much lower than the amount of misclassification produced by ultrasound or neonatal examination estimates. ^

PHENOTYPIC ASSOCIATION OF GENE AMPLIFICATION WITH MULTIPLE DRUG RESISTANCE IN VINCRISTINE RESISTANT CHINESE HAMSTER OVARY CELLS

Resistance of tumors to pharmacologic agents poses a significant problem in the treatment of human malignancies. This study overviews the scope of clinical resistance and focuses upon current research attempts toward investigation of the phenomenon of multidrug resistance (MDR).^ The objective of this investigation was to determine whether gene amplification had a role in the development of the MDR phenotype in Chinese hamster ovary cells (CHO) primarily selected for resistance to vincristine (VCR). A DNA fragment, previously shown to be amplified in two independently derived Chinese hamster cell lines exhibiting the MDR phenotype, was also amplified in VCR hamster lines. Sequences flanking this fragment were shown to contain coding information for a 4.3 kb transcript overproduced in VCR cells. These sequences were not enriched in double minute DNA preparations isolated from VCR cells. There was an approximately forty-fold increase in both the level of gene amplification and transcript overproduction in the VCR cell lines, independent of the level of primary resistance. This DNA amplification and overproduction of the 4.3 kb transcript was also demonstrated in CHO cells independently selected for resistance to Adriamycin and vinblastine.^ All the DNA sequences of two hamster cDNA clones containing 785 and 932 base pair inserts showed direct homology to the published mouse mdr sequences (about 90%). This sequence conservation held for only portions of the gene when the human mdr1 sequences were compared with those from either the mouse or hamster.^ Somatic cell hybrids, constructed between VCR CHO cells and sensitive murine cells, were used to determine whether there was a functional relationship between the chromosome bearing the amplified sequences and the MDR phenotype. Concordant segregation between vincristine resistance, the MDR phenotype, the presence of MDR-associated amplified sequences, overexpression of the mRNA encoded by these sequences, overexpression of the mRNA encoded by these sequences, and CHO chromosome Z1 was consistent with the hypothesis that there is an amplified gene on chromosome Z1 of the VCR CHO cells which is responsible for MDR in these cells. ^

Multiple retinal detachment surgery: A functional outcome study. The Houston Vision Assessment Test-Retina (HVAT-Retina)

Retinal detachment is a common ophthalmologic procedure, and outcome is typically measured by a single factor-improvement in visual acuity. Health related functional outcome testing, which quantifies patient's self-reported perception of impairment, can be integrated with objective clinical findings. Based on the patient's self-assessed lifestyle impairment, the physician and patient together can make an informed decision on the treatment that is most likely to benefit the patient. ^ A functional outcome test (the Houston Vision Assessment Test-Retina; HVAT-Retina) was developed and validated in patients with multiple retinal detachments in the same eye. The HVAT-Retina divides an estimated total impairment into subcomponents: contribution of visual disability (potentially correctable by retinal detachment surgery) and nonvisual physical disabilities (co-morbidities not affected by retinal detachment surgery. ^ Seventy-six patients participated in this prospective multicenter study. Seven patients were excluded from the analysis because they were not certain of their answers. Cronbach's alpha coefficient was 0.91 for presurgery HVAT-Retina and 0.94 post-surgery. The item-to-total correlation ranged from 0.50 to 0.88. Visual impairment score improved by 9 points from pre-surgery (p = 0.0003). Physical impairment score also improved from pre-surgery (p = 0.0002). ^ In conclusion, the results of this study demonstrate that the instrument is reliable and valid in patients presenting with recurrent retinal detachments. The HVAT-Retina is a simple instrument and does not burden the patient or the health professional in terms of time or cost. It may be self-administrated, not requiring an interviewer. Because the HVAT-Retina was designed to demonstrate outcomes perceivable by the patient, it has the potential to guide the decision making process between patient and physician. ^

Obesity and risk of multiple myeloma: A case-control study

Introduction. Several studies have reported a positive association of body mass index (BMI) with multiple myeloma; however, the period of adulthood where BMI is most important remains unclear. In addition, it is well known that body fat is associated with both sex-steroid hormone storage and with increasing insulin levels; therefore, it was hypothesized that the association between obesity and multiple myeloma may be attributed to increased aromatization of androgen in adipose tissue. Objective. The overall objective of this case-control study was to determine whether multiple myeloma cases had higher BMI and greater adult weight gain relative to healthy controls. In addition, we tested the hypothesis that hormone replacement therapy use among women will further increase the association between BMI and risk of multiple myeloma. This study used data from a pilot case-control study at M.D. Anderson Cancer Center (MDACC), entitled Etiology of multiple myeloma, directed by Dr. Sara Strom and Dr. Sergio Giralt. Methods. The pilot study recruited a total of 122 cases of histopathologically confirmed multiple myeloma from MDACC. Controls (n=183) were selected from a database of random digit dialing controls accrued in the Department of Epidemiology at MDACC and were frequency matched to the cases on age (±5 years), gender, and race/ethnicity. Demographic and risk factor information were obtained from all participants who completed a self-administered questionnaire. Items included in the questionnaire include demographic information, height and weight at age 25, 40 and current/diagnosis, medical history, family history of cancer, smoking and alcohol use. Statistical analysis. Initial descriptive analysis included Student's t-test and Pearson's chi-squared tests. Odds ratios and 95% confidence intervals were calculated to quantify the association between the variables of interest and multiple myeloma. A multivariable model will be developed using unconditional logistic regression. Results. MM cases were 1.79 times (95% CI=0.99-3.32) more likely to have been overweight or obese (BMI > 25 kg/m2) at age 25 relative to healthy controls after controlling for age, gender, race/ethnicty, education and family history of cancer. Being overweight or obese at age 40 was not significantly associated with mutliple myeloma risk (OR=1.42, 95% CI=0.86-2.34) nor was being overweight or obses at diagnosis (OR=1.43, 95% CI=0.78, 2.63). We observed a statistically significant 2-fold increased odds of multiple myeloma in individuals who gained more than 4.7 kg during between 25 and 40 years (OR=1.97, 95% CI=1.15-3.39). When assessing HRT as a modifier of the BMI and multiple myeloma association among women (N=123), no association between obesity and MM status was observed among women who have never used HRT (OR=0.60, 95% CI=0.23-1.61; n=73). Yet among women who have ever used HRT (n=50), being overweight or obese was associated with an increase in MM risk (OR=2. 93, 95% CI=0.81-10.6) after adjusting for age; however, the association was not statistically significant. Significance. This study provides further evidence that increased BMI increases the risk of multiple myeloma. Furthermore, among women, HRT use may modify risk of disease. ^

Club drug use and unprotected anal intercourse among HIV-seronegative men who have sex with men seeking HIV/STD testing at a local STD clinic in Houston

Objective. To explore (1) the association between "club drug" use and unprotected anal intercourse (UAI) and (2) the association between binge drug use and UAI among HIV seronegative men who have sex with men (MSM) seeking HIV/STD testing at a local clinic in Houston. ^ Study design. A sub-sample of 297 HIV seronegative MSM from a cross-sectional study of drug and sexual behavior in Houston was conducted in 2006. Patients who were seeking HIV/STD testing at a local MSM-identified STD clinic were recruited for an anonymous computer-assisted interview. Analysis of identified secondary data consisted of self-reported information about demographic characteristics, use of drugs, and sexual behaviors. ^ Results. With new and casual sex partners, there was a strong and statistically significant association between use of "club drugs" and UAI. No association between binge drug use and UAI was evident. Men aware of HIV seropositivity or unaware of the HIV serostatus of their primary partner were less likely to report UAI. ^ Conclusion. These data suggest that in the Houston area, HIV-negative MSM club drug users, particularly multiple drug users, are at higher risk of UAI than comparable MSMs who do not use club drugs. Episode-level data regarding binge use of these and other drugs, and UAI should be collected in future studies to explore their relationship. The 'new partner' category should be added to sex partner types to measure sex and drug use behaviors in future studies.^ Keywords. HIV-negative MSM; club drugs; unprotected anal intercourse; binge drug use. ^

A standard operation procedure and a pilot water testing for herbicides in northwest Harris County, Texas

Herbicides are used to control the growth of weeds along highways, power lines, and many other urban locations. Exposure to herbicides has been linked to adverse health outcomes. This study was initiated to pretest for the presence of herbicides in multiple water sources near intersections in a corridor in the Northwest Harris County (specifically in the Highway 6/FM 1960, North Freeway 45, US 290 and S 99 corridor). Roadside water and tap water samples were collected and analyzed for herbicides using the established Environmental Protection Agency (EPA) Method 515.4: "Determination of Chlorinated Acids in Drinking Water by Liquid-Liquid Micro-extraction, Derivatization, and Fast Gas Chromatography with Electron Capture Detection." A standard operating procedure (adapted from the US EPA Method 515.4) was developed for subsequent, larger studies of environmental fate of herbicides and non-occupational exposure risks. Preliminary testing of 16 water samples was performed to pretest the existence of trace herbicides; all concentrations that were greater than the minimum reporting limits of each analyte are reported with a 99 percent confidence. This study failed to find concentrations above the limits of detection of the method in any of the samples collected on June 15, 2008. However, this does not indicate that the waters around the NW Harris County are free of herbicides and metabolites. A larger and repeated sampling in the region would be necessary to make that claim. ^

Statistical Methods for Differential Expressions of Genes Detected in Multiple-Condition Experiment of Microarray

Most studies of differential gene-expressions have been conducted between two given conditions. The two-condition experimental (TCE) approach is simple in that all genes detected display a common differential expression pattern responsive to a common two-condition difference. Therefore, the genes that are differentially expressed under the other conditions other than the given two conditions are undetectable with the TCE approach. In order to address the problem, we propose a new approach called multiple-condition experiment (MCE) without replication and develop corresponding statistical methods including inference of pairs of conditions for genes, new t-statistics, and a generalized multiple-testing method for any multiple-testing procedure via a control parameter C. We applied these statistical methods to analyze our real MCE data from breast cancer cell lines and found that 85 percent of gene-expression variations were caused by genotypic effects and genotype-ANAX1 overexpression interactions, which agrees well with our expected results. We also applied our methods to the adenoma dataset of Notterman et al. and identified 93 differentially expressed genes that could not be found in TCE. The MCE approach is a conceptual breakthrough in many aspects: (a) many conditions of interests can be conducted simultaneously; (b) study of association between differential expressions of genes and conditions becomes easy; (c) it can provide more precise information for molecular classification and diagnosis of tumors; (d) it can save lot of experimental resources and time for investigators.^