A Two-stage Design for Comparing Clinical Trials

A two-stage design is proposed to choose among several experimental treatments and a standard treatment in clinical trials. The first stage employs a selection procedure to select the best treatment, provided it is better than the standard. The second stage tests the hypothesis between the best treatment selected at the first stage (if any) and the standard treatment. All the treatments are assumed to follow normal distributions and the best treatment is the one with the largest population mean. The level and the power are defined and they are used to set up equations to solve unknown first stage sample size, second stage sample size, and procedure parameters. The optimal design is the one that gives the smallest average sample size. Numerical results are presented to illustrate the improvement of one design as compared to existing one stage design.     

Two-stage designs for experiments with a large number of hypotheses

MOTIVATION: When a large number of hypotheses are investigated the false discovery rate (FDR) is commonly applied in gene expression analysis or gene association studies. Conventional single-stage designs may lack power due to low sample sizes for the individual hypotheses. We propose two-stage designs where the first stage is used to screen the 'promising' hypotheses which are further investigated at the second stage with an increased sample size. A multiple test procedure based on sequential individual P-values is proposed to control the FDR for the case of independent normal distributions with known variance. RESULTS: The power of optimal two-stage designs is impressively larger than the power of the corresponding single-stage design with equal costs. Extensions to the case of unknown variances and correlated test statistics are investigated by simulations. Moreover, it is shown that the simple multiple test procedure using first stage data for screening purposes and deriving the test decisions only from second stage data is a very powerful option.     

Paradoxical evidence integration in rapid decision processes

Decisions about noisy stimuli require evidence integration over time. Traditionally, evidence integration and decision making are described as a one-stage process: a decision is made when evidence for the presence of a stimulus crosses a threshold. Here, we show that one-stage models cannot explain psychophysical experiments on feature fusion, where two visual stimuli are presented in rapid succession. Paradoxically, the second stimulus biases decisions more strongly than the first one, contrary to predictions of one-stage models and intuition. We present a two-stage model where sensory information is integrated and buffered before it is fed into a drift diffusion process. The model is tested in a series of psychophysical experiments and explains both accuracy and reaction time distributions.     

Low-dose-rate extrapolation using the multistage model

The distribution of the maximum likelihood estimates of virtually safe levels of exposure to environmental chemicals is derived by using large-sample theory and Monte Carlo simulation according to the Armitage-Doll multistage model. Using historical dose-response we develop a set of 33 two-stage models upon which we base our conclusions. The large-sample distributions of the virtually safe dose are normal for cases in which the multistage-model parameters have nonzero expectation, and are skewed in other cases. The large-sample theory does not provide a good approximation of the distribution observed for small bioassays when Monte Carlo simulation is used. The constrained nature of the multistage-model parameters leads to bimodal distributions for small bioassays. The two modes are the direct result of estimating the linear parameter in the multistage model; the lower mode results from estimating this parameter to be nonzero, and the upper mode from estimating it to be zero. The results of this research emphasize the need for incorporation of the biological theory in the model-selection process.     

A general approach for two-stage analysis of multilevel clustered non-Gaussian data

In this article, we propose a two-stage approach to modeling multilevel clustered non-Gaussian data with sufficiently large numbers of continuous measures per cluster. Such data are common in biological and medical studies utilizing monitoring or image-processing equipment. We consider a general class of hierarchical models that generalizes the model in the global two-stage (GTS) method for nonlinear mixed effects models by using any square-root-n-consistent and asymptotically normal estimators from stage 1 as pseudodata in the stage 2 model, and by extending the stage 2 model to accommodate random effects from multiple levels of clustering. The second-stage model is a standard linear mixed effects model with normal random effects, but the cluster-specific distributions, conditional on random effects, can be non-Gaussian. This methodology provides a flexible framework for modeling not only a location parameter but also other characteristics of conditional distributions that may be of specific interest. For estimation of the population parameters, we propose a conditional restricted maximum likelihood (CREML) approach and establish the asymptotic properties of the CREML estimators. The proposed general approach is illustrated using quartiles as cluster-specific parameters estimated in the first stage, and applied to the data example from a collagen fibril development study. We demonstrate using simulations that in samples with small numbers of independent clusters, the CREML estimators may perform better than conditional maximum likelihood estimators, which are a direct extension of the estimators from the GTS method.     

Adaptive enrichment designs with a continuous biomarker

A popular design for clinical trials assessing targeted therapies is the two-stage adaptive enrichment design with recruitment in stage 2 limited to a biomarker-defined subgroup chosen based on data from stage 1. The data-dependent selection leads to statistical challenges if data from both stages are used to draw inference on treatment effects in the selected subgroup. If subgroups considered are nested, as when defined by a continuous biomarker, treatment effect estimates in different subgroups follow the same distribution as estimates in a group-sequential trial. This result is used to obtain tests controlling the familywise type I error rate (FWER) for six simple subgroup selection rules, one of which also controls the FWER for any selection rule. Two approaches are proposed: one based on multivariate normal distributions suitable if the number of possible subgroups, k, is small, and one based on Brownian motion approximations suitable for large k. The methods, applicable in the wide range of settings with asymptotically normal test statistics, are illustrated using survival data from a breast cancer trial.     

An adaptive trial design to optimize dose-schedule regimes with delayed outcomes

This paper proposes a two-stage phase I-II clinical trial design to optimize dose-schedule regimes of an experimental agent within ordered disease subgroups in terms of the toxicity-efficacy trade-off. The design is motivated by settings where prior biological information indicates it is certain that efficacy will improve with ordinal subgroup level. We formulate a flexible Bayesian hierarchical model to account for associations among subgroups and regimes, and to characterize ordered subgroup effects. Sequentially adaptive decision-making is complicated by the problem, arising from the motivating application, that efficacy is scored on day 90 and toxicity is evaluated within 30 days from the start of therapy, while the patient accrual rate is fast relative to these outcome evaluation intervals. To deal with this in a practical manner, we take a likelihood-based approach that treats unobserved toxicity and efficacy outcomes as missing values, and use elicited utilities that quantify the efficacy-toxicity trade-off as a decision criterion. Adaptive randomization is used to assign patients to regimes while accounting for subgroups, with randomization probabilities depending on the posterior predictive distributions of utilities. A simulation study is presented to evaluate the design's performance under a variety of scenarios, and to assess its sensitivity to the amount of missing data, the prior, and model misspecification.     

High throughput screening of co-expressed gene pairs with controlled false discovery rate (FDR) and minimum acceptable strength (MAS).

Many exploratory microarray data analysis tools such as gene clustering and relevance networks rely on detecting pairwise gene co-expression. Traditional screening of pairwise co-expression either controls biological significance or statistical significance, but not both. The former approach does not provide stochastic error control, and the later approach screens many co-expressions with excessively low correlation. We have designed and implemented a statistically sound two-stage co-expression detection algorithm that controls both statistical significance (false discovery rate, FDR) and biological significance (minimum acceptable strength, MAS) of the discovered co-expressions. Based on estimation of pairwise gene correlation, the algorithm provides an initial co-expression discovery that controls only FDR, which is then followed by a second stage co-expression discovery which controls both FDR and MAS. It also computes and thresholds the set of FDR p-values for each correlation that satisfied the MAS criterion. Using simulated data, we validated asymptotic null distributions of the Pearson and Kendall correlation coefficients and the two-stage error-control procedure; we also compared our two-stage test procedure with another two-stage test procedure using the receiver operating characteristic (ROC) curve. We then used yeast galactose metabolism data to illustrate the advantage of our method for clustering genes and constructing a relevance network. The method has been implemented in an R package "GeneNT" that is freely available from the Comprehensive R Archive Network (CRAN): www.cran.r-project.org/.     

Graphical Method to Detect Departure from Normality with Special Application to Tukey's g and h Distributions

Given a sample, the problem of choosing one of a variety of theoretical distributions for data analysis has been a major concern to both theoretical and applied statisticians. On the basis of a graph drawn using a normal characterization, a method is given to detect departure from normality. Special mathematical study is presented for the proposed graphs. Tukey's g and h family of distributions are used as an alternate to normality.     

The impact of species-neutral stage structure on macroecological patterns

Despite its radical assumption of ecological equivalence between species, neutral biodiversity theory can often provide good fits to species abundance distributions observed in nature. Major criticisms of neutral theory have focused on interspecific differences, which are in conflict with ecological equivalence. However, neutrality in nature is also broken by differences between conspecific individuals at different life stages, which in many communities may vastly exceed interspecific differences between individuals at similar stages. These within-species asymmetries have not been fully explored in species-neutral models, and it is not known whether demographic stage structure affects macroecological patterns in neutral theory. Here, we present a two-stage neutral model where fecundity and mortality change as an individual transitions from one stage to the other. We explore several qualitatively different scenarios, and compare numerically obtained species abundance distributions to the predictions of unstructured neutral theory. We find that abundance distributions are generally robust to this kind of stage structure, but significant departures from unstructured predictions occur if adults have sufficiently low fecundity and mortality. In addition, we show that the cumulative number of births per species, which is distributed as a power law with a 3/2 exponent, is invariant even when the abundance distribution departs from unstructured model predictions. Our findings potentially explain power law-like abundance distributions in organisms with strong demographic structure, such as eusocial insects and humans, and partially rehabilitate species abundance distributions from past criticisms as to their inability to distinguish between biological mechanisms.     

Dose-finding with two agents in Phase I oncology trials

We propose an adaptive two-stage Bayesian design for finding one or more acceptable dose combinations of two cytotoxic agents used together in a Phase I clinical trial. The method requires that each of the two agents has been studied previously as a single agent, which is almost invariably the case in practice. A parametric model is assumed for the probability of toxicity as a function of the two doses. Informative priors for parameters characterizing the single-agent toxicity probability curves are either elicited from the physician(s) planning the trial or obtained from historical data, and vague priors are assumed for parameters characterizing two-agent interactions. A method for eliciting the single-agent parameter priors is described. The design is applied to a trial of gemcitabine and cyclophosphamide, and a simulation study is presented.     

The determination of density and molecular weight distributions of lipoproteins by sedimentation equilibrium.

A method is presented by which an experimental record of total concentration as a function of radial distance, obtained in a sedimentation equilibrium experiment conducted with a noninteracting mixture in the absence of a density gradient, may be analyzed to obtain the unimodal distributions of molecular weight and of partial molar volume when these vary concomitantly and continuously. Particular attention is given to the caracterization of classes of lipoproteins exhibiting Gaussian distributions of these quantities, although the analysis is applicable to other types of unimodal distribution. Equations are also formulated permitting the definition of the corresponding distributions of partial specific volume and of density. The analysis procedure is based on a method (employing Laplace transforms) developed previously, but differs from it in that it avoids the necessity of differentiating experimental results, which introduces error. The method offers certain advantages over other procedures used to characterize and compare lipoprotein samples (exhibiting unimodal distributions) with regard to the duration of the experiment, economy of the sample, and, particularly, the ability to define in principle all of the relevant distributions from one sedimentation equilibrium experiment and an external measurement of the weight average partial specific volume. These points and the steps in the analysis procedure are illustrated with experimental results obtained in the sedimentation equilibrium of a sample of human serum low density lipoprotein. The experimental parameters (such as solution density, column height, and angular velocity) used in the conduction of these experiments were selected on the basis of computer-simulated examples, which are also presented. These provide a guide for other workers interested in characterizing lipoproteins of this class.     

Cricetodontini from the Calatayud–Daroca Basin (Spain): A taxonomical description and update of their stratigraphical distributions

The most updated stratigraphical distributions of the Cricetodontini from the Calatayud–Daroca Basin are presented. Cricetodon sansaniensis (local biozone F, MN6, middle Aragonian) and Cricetodon jotae (local biozone G1, G2 and G3, MN6-MN7/8, Middle–Upper Aragonian) from the middle Miocene are described and discussed. Generally, the genera of Cricetodontini were used to define large time intervals, whereas the species were neglected in the local biostratigraphical studies. The results presented herein show that, for the Calatayud–Daroca Basin, most of the stratigraphical distributions of the species of Cricetodontini are strongly linked with the local biozonation. That reveals that they could be used as good biostratigraphical indicators. Moreover, the unusually long local biozone G3 could be redefined and subdivided based on the distributions of the four species of Cricetodontini present in it. However, it is recommended to complete the study of the whole rodent fauna before proposing a new biozonation.     

Analysis of a two-stage case-control study with cluster sampling of controls: application to nonmelanoma skin cancer

We present a pseudolikelihood approach for analyzing a two-stage population-based case-control study with cluster sampling of controls. These methods were developed to analyze data from a study of nonmelanoma skin cancer (NMSC). This study was designed to evaluate the role of ultraviolet radiation (UVB) on NMSC risk while adjusting for age group, which is known for all subjects, and for other individual-level factors, such as susceptibility to sunburn, which are known only for participants in the case-control study. The methods presented yield estimates of relative and absolute risk, with standard errors, while accounting naturally for the two-stage sampling of the cohort and cluster sampling of controls.     

A simple method to fit geometric series and broken stick models in community ecology and island biogeography

Species abundance distributions are widely used in explaining natural communities, their natural evolution and the impacts of environmental disturbance. A commonly used approach is that of rank-abundance distributions. Favored, biologically founded models are the geometric series (GS) and the broken stick (BS) model. Comparing observed abundance distributions with those predicted by models is an extremely time-consuming task. Also, using goodness-of-fit tests for frequency distributions (like Chi-square or Kolmogorov–Smirnov tests) to compare observed with expected frequencies is problematic because the best way to calculate expected frequencies may be controversial. More important, the Chi-square test may prove if an observed distribution statistically differs from a model, but does not allow the investigator to choose among competing models from which the observed distribution does not differ. Both models can be easily tested by regression analysis. In GS, if a log scale is used for abundance, the species exactly fall along a straight line. The BS distribution shows up as nearly linear when a log scale is used for the rank axis. Regression analysis is proposed here as a simpler and more efficient method to fit the GS and BS models. Also, regression analysis (1) does not suffer from assumptions related to Chi-square tests; (2) obviates the need to establish expected frequencies, and (3) offers the possibility to choose the best fit among competing models. A possible extension of abundance-rank analysis to species richness on islands is also proposed as a method to discriminate between relict and equilibrial models. Examples of application to field data are also presented.     

A simple method to fit geometric series and broken stick models in community ecology and island biogeography

Species abundance distributions are widely used in explaining natural communities, their natural evolution and the impacts of environmental disturbance. A commonly used approach is that of rank-abundance distributions. Favored, biologically founded models are the geometric series (GS) and the broken stick (BS) model. Comparing observed abundance distributions with those predicted by models is an extremely time-consuming task. Also, using goodness-of-fit tests for frequency distributions (like Chi-square or Kolmogorov–Smirnov tests) to compare observed with expected frequencies is problematic because the best way to calculate expected frequencies may be controversial. More important, the Chi-square test may prove if an observed distribution statistically differs from a model, but does not allow the investigator to choose among competing models from which the observed distribution does not differ. Both models can be easily tested by regression analysis. In GS, if a log scale is used for abundance, the species exactly fall along a straight line. The BS distribution shows up as nearly linear when a log scale is used for the rank axis. Regression analysis is proposed here as a simpler and more efficient method to fit the GS and BS models. Also, regression analysis (1) does not suffer from assumptions related to Chi-square tests; (2) obviates the need to establish expected frequencies, and (3) offers the possibility to choose the best fit among competing models. A possible extension of abundance-rank analysis to species richness on islands is also proposed as a method to discriminate between relict and equilibrial models. Examples of application to field data are also presented.     

The effect of superspreading on epidemic outbreak size distributions

Recently, evidence has been presented to suggest that there are significant heterogeneities in the transmission of communicable diseases. Here, a stochastic simulation model of an epidemic process that allows for these heterogeneities is used to demonstrate the potentially considerable effect that heterogeneity of transmission will have on epidemic outbreak size distributions. Our simulation results agree well with approximations gained from the theory of branching processes. Outbreak size distributions have previously been used to infer basic epidemiological parameters. We show that if superspreading does occur then such distributions must be interpreted with care. The simulation results are discussed in relation to measles epidemics in isolated populations and in predominantly urban scenarios. The effect of three different disease control policies on outbreak size distributions are shown for varying levels of heterogeneity and disease control effort.     

Statistical Inference Methods for Two Crossing Survival Curves: A Comparison of Methods

A common problem that is encountered in medical applications is the overall homogeneity of survival distributions when two survival curves cross each other. A survey demonstrated that under this condition, which was an obvious violation of the assumption of proportional hazard rates, the log-rank test was still used in 70% of studies. Several statistical methods have been proposed to solve this problem. However, in many applications, it is difficult to specify the types of survival differences and choose an appropriate method prior to analysis. Thus, we conducted an extensive series of Monte Carlo simulations to investigate the power and type I error rate of these procedures under various patterns of crossing survival curves with different censoring rates and distribution parameters. Our objective was to evaluate the strengths and weaknesses of tests in different situations and for various censoring rates and to recommend an appropriate test that will not fail for a wide range of applications. Simulation studies demonstrated that adaptive Neyman’s smooth tests and the two-stage procedure offer higher power and greater stability than other methods when the survival distributions cross at early, middle or late times. Even for proportional hazards, both methods maintain acceptable power compared with the log-rank test. In terms of the type I error rate, Renyi and Cramér—von Mises tests are relatively conservative, whereas the statistics of the Lin-Xu test exhibit apparent inflation as the censoring rate increases. Other tests produce results close to the nominal 0.05 level. In conclusion, adaptive Neyman’s smooth tests and the two-stage procedure are found to be the most stable and feasible approaches for a variety of situations and censoring rates. Therefore, they are applicable to a wider spectrum of alternatives compared with other tests.     

Models for circular-linear and circular-circular data constructed from circular distributions based on nonnegative trigonometric sums

Johnson and Wehrly (1978, Journal of the American Statistical Association 73, 602-606) and Wehrly and Johnson (1980, Biometrika 67, 255-256) show one way to construct the joint distribution of a circular and a linear random variable, or the joint distribution of a pair of circular random variables from their marginal distributions and the density of a circular random variable, which in this article is referred to as joining circular density. To construct flexible models, it is necessary that the joining circular density be able to present multimodality and/or skewness in order to model different dependence patterns. Fernández-Durán (2004, Biometrics 60, 499-503) constructed circular distributions based on nonnegative trigonometric sums that can present multimodality and/or skewness. Furthermore, they can be conveniently used as a model for circular-linear or circular-circular joint distributions. In the current work, joint distributions for circular-linear and circular-circular data constructed from circular distributions based on nonnegative trigonometric sums are presented and applied to two data sets, one for circular-linear data related to the air pollution patterns in Mexico City and the other for circular-circular data related to the pair of dihedral angles between consecutive amino acids in a protein.