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. ^

A BAYESIAN APPROACH TO DOSE-RESPONSE ASSESSMENT AND DRUG-DRUG INTERACTION ANALYSIS: APPLICATION TO IN VITRO STUDIES

The considerable search for synergistic agents in cancer research is motivated by the therapeutic benefits achieved by combining anti-cancer agents. Synergistic agents make it possible to reduce dosage while maintaining or enhancing a desired effect. Other favorable outcomes of synergistic agents include reduction in toxicity and minimizing or delaying drug resistance. Dose-response assessment and drug-drug interaction analysis play an important part in the drug discovery process, however analysis are often poorly done. This dissertation is an effort to notably improve dose-response assessment and drug-drug interaction analysis. The most commonly used method in published analysis is the Median-Effect Principle/Combination Index method (Chou and Talalay, 1984). The Median-Effect Principle/Combination Index method leads to inefficiency by ignoring important sources of variation inherent in dose-response data and discarding data points that do not fit the Median-Effect Principle. Previous work has shown that the conventional method yields a high rate of false positives (Boik, Boik, Newman, 2008; Hennessey, Rosner, Bast, Chen, 2010) and, in some cases, low power to detect synergy. There is a great need for improving the current methodology. We developed a Bayesian framework for dose-response modeling and drug-drug interaction analysis. First, we developed a hierarchical meta-regression dose-response model that accounts for various sources of variation and uncertainty and allows one to incorporate knowledge from prior studies into the current analysis, thus offering a more efficient and reliable inference. Second, in the case that parametric dose-response models do not fit the data, we developed a practical and flexible nonparametric regression method for meta-analysis of independently repeated dose-response experiments. Third, and lastly, we developed a method, based on Loewe additivity that allows one to quantitatively assess interaction between two agents combined at a fixed dose ratio. The proposed method makes a comprehensive and honest account of uncertainty within drug interaction assessment. Extensive simulation studies show that the novel methodology improves the screening process of effective/synergistic agents and reduces the incidence of type I error. We consider an ovarian cancer cell line study that investigates the combined effect of DNA methylation inhibitors and histone deacetylation inhibitors in human ovarian cancer cell lines. The hypothesis is that the combination of DNA methylation inhibitors and histone deacetylation inhibitors will enhance antiproliferative activity in human ovarian cancer cell lines compared to treatment with each inhibitor alone. By applying the proposed Bayesian methodology, in vitro synergy was declared for DNA methylation inhibitor, 5-AZA-2'-deoxycytidine combined with one histone deacetylation inhibitor, suberoylanilide hydroxamic acid or trichostatin A in the cell lines HEY and SKOV3. This suggests potential new epigenetic therapies in cell growth inhibition of ovarian cancer cells.

Nonlinear tests for genetic studies of complex disease

Linkage and association studies are major analytical tools to search for susceptibility genes for complex diseases. With the availability of large collection of single nucleotide polymorphisms (SNPs) and the rapid progresses for high throughput genotyping technologies, together with the ambitious goals of the International HapMap Project, genetic markers covering the whole genome will be available for genome-wide linkage and association studies. In order not to inflate the type I error rate in performing genome-wide linkage and association studies, multiple adjustment for the significant level for each independent linkage and/or association test is required, and this has led to the suggestion of genome-wide significant cut-off as low as 5 × 10 −7. Almost no linkage and/or association study can meet such a stringent threshold by the standard statistical methods. Developing new statistics with high power is urgently needed to tackle this problem. This dissertation proposes and explores a class of novel test statistics that can be used in both population-based and family-based genetic data by employing a completely new strategy, which uses nonlinear transformation of the sample means to construct test statistics for linkage and association studies. Extensive simulation studies are used to illustrate the properties of the nonlinear test statistics. Power calculations are performed using both analytical and empirical methods. Finally, real data sets are analyzed with the nonlinear test statistics. Results show that the nonlinear test statistics have correct type I error rates, and most of the studied nonlinear test statistics have higher power than the standard chi-square test. This dissertation introduces a new idea to design novel test statistics with high power and might open new ways to mapping susceptibility genes for complex diseases. ^

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 hybrid method in combining treatment effects from matched and unmatched studies

In the biomedical studies, the general data structures have been the matched (paired) and unmatched designs. Recently, many researchers are interested in Meta-Analysis to obtain a better understanding from several clinical data of a medical treatment. The hybrid design, which is combined two data structures, may create the fundamental question for statistical methods and the challenges for statistical inferences. The applied methods are depending on the underlying distribution. If the outcomes are normally distributed, we would use the classic paired and two independent sample T-tests on the matched and unmatched cases. If not, we can apply Wilcoxon signed rank and rank sum test on each case. ^ To assess an overall treatment effect on a hybrid design, we can apply the inverse variance weight method used in Meta-Analysis. On the nonparametric case, we can use a test statistic which is combined on two Wilcoxon test statistics. However, these two test statistics are not in same scale. We propose the Hybrid Test Statistic based on the Hodges-Lehmann estimates of the treatment effects, which are medians in the same scale.^ To compare the proposed method, we use the classic meta-analysis T-test statistic on the combined the estimates of the treatment effects from two T-test statistics. Theoretically, the efficiency of two unbiased estimators of a parameter is the ratio of their variances. With the concept of Asymptotic Relative Efficiency (ARE) developed by Pitman, we show ARE of the hybrid test statistic relative to classic meta-analysis T-test statistic using the Hodges-Lemann estimators associated with two test statistics.^ From several simulation studies, we calculate the empirical type I error rate and power of the test statistics. The proposed statistic would provide effective tool to evaluate and understand the treatment effect in various public health studies as well as clinical trials.^

BAYESIAN STATISTICAL METHODS IN GENE-ENVIRONMENT AND GENE-GENE INTERACTION STUDIES

Complex diseases such as cancer result from multiple genetic changes and environmental exposures. Due to the rapid development of genotyping and sequencing technologies, we are now able to more accurately assess causal effects of many genetic and environmental factors. Genome-wide association studies have been able to localize many causal genetic variants predisposing to certain diseases. However, these studies only explain a small portion of variations in the heritability of diseases. More advanced statistical models are urgently needed to identify and characterize some additional genetic and environmental factors and their interactions, which will enable us to better understand the causes of complex diseases. In the past decade, thanks to the increasing computational capabilities and novel statistical developments, Bayesian methods have been widely applied in the genetics/genomics researches and demonstrating superiority over some regular approaches in certain research areas. Gene-environment and gene-gene interaction studies are among the areas where Bayesian methods may fully exert its functionalities and advantages. This dissertation focuses on developing new Bayesian statistical methods for data analysis with complex gene-environment and gene-gene interactions, as well as extending some existing methods for gene-environment interactions to other related areas. It includes three sections: (1) Deriving the Bayesian variable selection framework for the hierarchical gene-environment and gene-gene interactions; (2) Developing the Bayesian Natural and Orthogonal Interaction (NOIA) models for gene-environment interactions; and (3) extending the applications of two Bayesian statistical methods which were developed for gene-environment interaction studies, to other related types of studies such as adaptive borrowing historical data. We propose a Bayesian hierarchical mixture model framework that allows us to investigate the genetic and environmental effects, gene by gene interactions (epistasis) and gene by environment interactions in the same model. It is well known that, in many practical situations, there exists a natural hierarchical structure between the main effects and interactions in the linear model. Here we propose a model that incorporates this hierarchical structure into the Bayesian mixture model, such that the irrelevant interaction effects can be removed more efficiently, resulting in more robust, parsimonious and powerful models. We evaluate both of the 'strong hierarchical' and 'weak hierarchical' models, which specify that both or one of the main effects between interacting factors must be present for the interactions to be included in the model. The extensive simulation results show that the proposed strong and weak hierarchical mixture models control the proportion of false positive discoveries and yield a powerful approach to identify the predisposing main effects and interactions in the studies with complex gene-environment and gene-gene interactions. We also compare these two models with the 'independent' model that does not impose this hierarchical constraint and observe their superior performances in most of the considered situations. The proposed models are implemented in the real data analysis of gene and environment interactions in the cases of lung cancer and cutaneous melanoma case-control studies. The Bayesian statistical models enjoy the properties of being allowed to incorporate useful prior information in the modeling process. Moreover, the Bayesian mixture model outperforms the multivariate logistic model in terms of the performances on the parameter estimation and variable selection in most cases. Our proposed models hold the hierarchical constraints, that further improve the Bayesian mixture model by reducing the proportion of false positive findings among the identified interactions and successfully identifying the reported associations. This is practically appealing for the study of investigating the causal factors from a moderate number of candidate genetic and environmental factors along with a relatively large number of interactions. The natural and orthogonal interaction (NOIA) models of genetic effects have previously been developed to provide an analysis framework, by which the estimates of effects for a quantitative trait are statistically orthogonal regardless of the existence of Hardy-Weinberg Equilibrium (HWE) within loci. Ma et al. (2012) recently developed a NOIA model for the gene-environment interaction studies and have shown the advantages of using the model for detecting the true main effects and interactions, compared with the usual functional model. In this project, we propose a novel Bayesian statistical model that combines the Bayesian hierarchical mixture model with the NOIA statistical model and the usual functional model. The proposed Bayesian NOIA model demonstrates more power at detecting the non-null effects with higher marginal posterior probabilities. Also, we review two Bayesian statistical models (Bayesian empirical shrinkage-type estimator and Bayesian model averaging), which were developed for the gene-environment interaction studies. Inspired by these Bayesian models, we develop two novel statistical methods that are able to handle the related problems such as borrowing data from historical studies. The proposed methods are analogous to the methods for the gene-environment interactions on behalf of the success on balancing the statistical efficiency and bias in a unified model. By extensive simulation studies, we compare the operating characteristics of the proposed models with the existing models including the hierarchical meta-analysis model. The results show that the proposed approaches adaptively borrow the historical data in a data-driven way. These novel models may have a broad range of statistical applications in both of genetic/genomic and clinical studies.

Mutual Information for Testing Gene-Environment Interaction

Despite current enthusiasm for investigation of gene-gene interactions and gene-environment interactions, the essential issue of how to define and detect gene-environment interactions remains unresolved. In this report, we define gene-environment interactions as a stochastic dependence in the context of the effects of the genetic and environmental risk factors on the cause of phenotypic variation among individuals. We use mutual information that is widely used in communication and complex system analysis to measure gene-environment interactions. We investigate how gene-environment interactions generate the large difference in the information measure of gene-environment interactions between the general population and a diseased population, which motives us to develop mutual information-based statistics for testing gene-environment interactions. We validated the null distribution and calculated the type 1 error rates for the mutual information-based statistics to test gene-environment interactions using extensive simulation studies. We found that the new test statistics were more powerful than the traditional logistic regression under several disease models. Finally, in order to further evaluate the performance of our new method, we applied the mutual information-based statistics to three real examples. Our results showed that P-values for the mutual information-based statistics were much smaller than that obtained by other approaches including logistic regression models.

RANGE ADAPTIVE PROTON THERAPY FOR PROSTATE CANCER

Purpose: The rapid distal falloff of a proton beam allows for sparing of normal tissues distal to the target. However proton beams that aim directly towards critical structures are avoided due to concerns of range uncertainties, such as CT number conversion and anatomy variations. We propose to eliminate range uncertainty and enable prostate treatment with a single anterior beam by detecting the proton’s range at the prostate-rectal interface and adaptively adjusting the range in vivo and in real-time. Materials and Methods: A prototype device, consisting of an endorectal liquid scintillation detector and dual-inverted Lucite wedges for range compensation, was designed to test the feasibility and accuracy of the technique. Liquid scintillation filled volume was fitted with optical fiber and placed inside the rectum of an anthropomorphic pelvic phantom. Photodiode-generated current signal was generated as a function of proton beam distal depth, and the spatial resolution of this technique was calculated by relating the variance in detecting proton spills to its maximum penetration depth. The relative water-equivalent thickness of the wedges was measured in a water phantom and prospectively tested to determine the accuracy of range corrections. Treatment simulation studies were performed to test the potential dosimetric benefit in sparing the rectum. Results: The spatial resolution of the detector in phantom measurement was 0.5 mm. The precision of the range correction was 0.04 mm. The residual margin to ensure CTV coverage was 1.1 mm. The composite distal margin for 95% treatment confidence was 2.4 mm. Planning studies based on a previously estimated 2mm margin (90% treatment confidence) for 27 patients showed a rectal sparing up to 51% at 70 Gy and 57% at 40 Gy relative to IMRT and bilateral proton treatment. Conclusion: We demonstrated the feasibility of our design. Use of this technique allows for proton treatment using a single anterior beam, significantly reducing the rectal dose.

A cladistic analysis of apolipoprotein B gene variation and plasma lipid and apolipoprotein B levels, using familial data

Any functionally important mutation is embedded in an evolutionary matrix of other mutations. Cladistic analysis, based on this, is a method of investigating gene effects using a haplotype phylogeny to define a set of tests which localize causal mutations to branches of the phylogeny. Previous implementations of cladistic analysis have not addressed the issue of analyzing data from related individuals, though in human studies, family data are usually needed to obtain unambiguous haplotypes. In this study, a method of cladistic analysis is described in which haplotype effects are parameterized in a linear model which accounts for familial correlations. The method was used to study the effect of apolipoprotein (Apo) B gene variation on total-, LDL-, and HDL-cholesterol, triglyceride, and Apo B levels in 121 French families. Five polymorphisms defined Apo B haplotypes: the signal peptide Insertion/deletion, Bsp 1286I, XbaI, MspI, and EcoRI. Eleven haplotypes were found, and a haplotype phylogeny was constructed and used to define a set of tests of haplotype effects on lipid and apo B levels.^ This new method of cladistic analysis, the parametric method, found significant effects for single haplotypes for all variables. For HDL-cholesterol, 3 clusters of evolutionarily-related haplotypes affecting levels were found. Haplotype effects accounted for about 10% of the genetic variance of triglyceride and HDL-cholesterol levels. The results of the parametric method were compared to those of a method of cladistic analysis based on permutational testing. The permutational method detected fewer haplotype effects, even when modified to account for correlations within families. Simulation studies exploring these differences found evidence of systematic errors in the permutational method due to the process by which haplotype groups were selected for testing.^ The applicability of cladistic analysis to human data was shown. The parametric method is suggested as an improvement over the permutational method. This study has identified candidate haplotypes for sequence comparisons in order to locate the functional mutations in the Apo B gene which may influence plasma lipid levels. ^

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. ^

Evaluation of the causal pathways through which subjective norms influence mammography intention and stage of change in a national sample of women veterans

Few studies have investigated causal pathways linking psychosocial factors to each other and to screening mammography. Conflicting hypotheses exist in the theoretic literature regarding the role and importance of subjective norms, a person's perceived social pressure to perform the behavior and his/her motivation to comply. The Theory of Reasoned Action (TRA) hypothesizes that subjective norms directly affect intention; while the Transtheoretical Model (TTM) hypothesizes that attitudes mediate the influence of subjective norms on stage of change. No one has examined which hypothesis best predicts the effect of subjective norms on mammography intention and stage of change. Two statistical methods are available for testing mediation, sequential regression analysis (SRA) and latent variable structural equation modeling (LVSEM); however, software to apply LVSEM to dichotomous variables like intention has only recently become available. No one has compared the methods to determine whether or not they yield similar results for dichotomous variables. ^ Study objectives were to: (1) determine whether the effect of subjective norms on mammography intention and stage of change are mediated by pros and cons; and (2) compare mediation results from the SRA and LVSEM approaches when the outcome is dichotomous. We conducted a secondary analysis of data from a national sample of women veterans enrolled in Project H.O.M.E. (H&barbelow;ealthy O&barbelow;utlook on the M&barbelow;ammography E&barbelow;xperience), a behavioral intervention trial. ^ Results showed that the TTM model described the causal pathways better than the TRA one; however, we found support for only one of the TTM causal mechanisms. Cons was the sole mediator. The mediated effect of subjective norms on intention and stage of change by cons was very small. These findings suggest that interventionists focus their efforts on reducing negative attitudes toward mammography when resources are limited. ^ Both the SRA and LVSEM methods provided evidence for complete mediation, and the direction, magnitude, and standard errors of the parameter estimates were very similar. Because SRA parameter estimates were not biased toward the null, we can probably assume negligible measurement error in the independent and mediator variables. Simulation studies are needed to further our understanding of how these two methods perform under different data conditions. ^

Bayesian generalized linear models for meta-analysis of diagnostic tests

With the recognition of the importance of evidence-based medicine, there is an emerging need for methods to systematically synthesize available data. Specifically, methods to provide accurate estimates of test characteristics for diagnostic tests are needed to help physicians make better clinical decisions. To provide more flexible approaches for meta-analysis of diagnostic tests, we developed three Bayesian generalized linear models. Two of these models, a bivariate normal and a binomial model, analyzed pairs of sensitivity and specificity values while incorporating the correlation between these two outcome variables. Noninformative independent uniform priors were used for the variance of sensitivity, specificity and correlation. We also applied an inverse Wishart prior to check the sensitivity of the results. The third model was a multinomial model where the test results were modeled as multinomial random variables. All three models can include specific imaging techniques as covariates in order to compare performance. Vague normal priors were assigned to the coefficients of the covariates. The computations were carried out using the 'Bayesian inference using Gibbs sampling' implementation of Markov chain Monte Carlo techniques. We investigated the properties of the three proposed models through extensive simulation studies. We also applied these models to a previously published meta-analysis dataset on cervical cancer as well as to an unpublished melanoma dataset. In general, our findings show that the point estimates of sensitivity and specificity were consistent among Bayesian and frequentist bivariate normal and binomial models. However, in the simulation studies, the estimates of the correlation coefficient from Bayesian bivariate models are not as good as those obtained from frequentist estimation regardless of which prior distribution was used for the covariance matrix. The Bayesian multinomial model consistently underestimated the sensitivity and specificity regardless of the sample size and correlation coefficient. In conclusion, the Bayesian bivariate binomial model provides the most flexible framework for future applications because of its following strengths: (1) it facilitates direct comparison between different tests; (2) it captures the variability in both sensitivity and specificity simultaneously as well as the intercorrelation between the two; and (3) it can be directly applied to sparse data without ad hoc correction. ^

A multivariate frailty model for disease recurrences and survival

A multivariate frailty hazard model is developed for joint-modeling of three correlated time-to-event outcomes: (1) local recurrence, (2) distant recurrence, and (3) overall survival. The term frailty is introduced to model population heterogeneity. The dependence is modeled by conditioning on a shared frailty that is included in the three hazard functions. Independent variables can be included in the model as covariates. The Markov chain Monte Carlo methods are used to estimate the posterior distributions of model parameters. The algorithm used in present application is the hybrid Metropolis-Hastings algorithm, which simultaneously updates all parameters with evaluations of gradient of log posterior density. The performance of this approach is examined based on simulation studies using Exponential and Weibull distributions. We apply the proposed methods to a study of patients with soft tissue sarcoma, which motivated this research. Our results indicate that patients with chemotherapy had better overall survival with hazard ratio of 0.242 (95% CI: 0.094 - 0.564) and lower risk of distant recurrence with hazard ratio of 0.636 (95% CI: 0.487 - 0.860), but not significantly better in local recurrence with hazard ratio of 0.799 (95% CI: 0.575 - 1.054). The advantages and limitations of the proposed models, and future research directions are discussed. ^

Systematic review of genetic risk score in coronary heart disease and other diseases

In order to better take advantage of the abundant results from large-scale genomic association studies, investigators are turning to a genetic risk score (GRS) method in order to combine the information from common modest-effect risk alleles into an efficient risk assessment statistic. The statistical properties of these GRSs are poorly understood. As a first step toward a better understanding of GRSs, a systematic analysis of recent investigations using a GRS was undertaken. GRS studies were searched in the areas of coronary heart disease (CHD), cancer, and other common diseases using bibliographic databases and by hand-searching reference lists and journals. Twenty-one independent case-control studies, cohort studies, and simulation studies (12 in CHD, 9 in other diseases) were identified. The underlying statistical assumptions of the GRS using the experience of the Framingham risk score were investigated. Improvements in the construction of a GRS guided by the concept of composite indicators are discussed. The GRS will be a promising risk assessment tool to improve prediction and diagnosis of common diseases.^

Comparison of multiple comparison methods for identifying differential gene expression in simulated and real papillary thyroid cancer microarray data

The difficulty of detecting differential gene expression in microarray data has existed for many years. Several correction procedures try to avoid the family-wise error rate in multiple comparison process, including the Bonferroni and Sidak single-step p-value adjustments, Holm's step-down correction method, and Benjamini and Hochberg's false discovery rate (FDR) correction procedure. Each multiple comparison technique has its advantages and weaknesses. We studied each multiple comparison method through numerical studies (simulations) and applied the methods to the real exploratory DNA microarray data, which detect of molecular signatures in papillary thyroid cancer (PTC) patients. According to our results of simulation studies, Benjamini and Hochberg step-up FDR controlling procedure is the best process among these multiple comparison methods and we discovered 1277 potential biomarkers among 54675 probe sets after applying the Benjamini and Hochberg's method to PTC microarray data.^