Testing for qualitative interactions between treatment effects and patient subsets

Evaluation of evidence that treatment efficacy varies substantially among different subsets of patients is an important feature of the analysis of large clinical trials. Qualitative or crossover interactions are said to occur when one treatment is superior for some subsets of patients and the alternative treatment is superior for other subsets. A non-crossover interaction arises when there is variation in the magnitude, but not in the direction, of treatment effects among subsets. Some authors use the term quantitative interaction to mean non-crossover interaction. Non-crossover interactions are usually of less clinical importance than qualitative interactions, which often have major therapeutic significance. A likelihood ratio test is developed to test for qualitative interactions. Exact critical values are determined and tabulated.     

A priori contact preferences in molecular recognition

A molecular interaction library modeling favorable non-bonded interactions between atoms and molecular fragments is considered. In this paper, we represent the structure of the interaction library by a network diagram, which demonstrates that the underlying prediction model obtained for a molecular fragment is multi-layered. We clustered the molecular fragments into four groups by analyzing the pairwise distances between the molecular fragments. The distances are represented as an unrooted tree, in which the molecular fragments fall into four groups according to their function. For each fragment group, we modeled a group-specific a priori distribution with a Dirichlet distribution. The group-specific Dirichlet distributions enable us to derive a large population of similar molecular fragments that vary only in their contact preferences. Bayes' theorem then leads to a population distribution of the posterior probability vectors referred to as a "Dickey-Savage"-density. Two known methods for approximating multivariate integrals are applied to obtain marginal distributions of the Dickey-Savage density. The results of the numerical integration methods are compared with the simulated marginal distributions. By studying interactions between the protein structure of cyclohydrolase and its ligand guanosine-5'-triphosphate, we show that the marginal distributions of the posterior probabilities are more informative than the corresponding point estimates.     

Permutation Testing for Treatment–Covariate Interactions and Subgroup Identification

We consider the problem of using permutation-based methods to test for treatment–covariate interactions from randomized clinical trial data. Testing for interactions is common in the field of personalized medicine, as subgroups with enhanced treatment effects arise when treatment-by-covariate interactions exist. Asymptotic tests can often be performed for simple models, but in many cases, more complex methods are used to identify subgroups, and non-standard test statistics proposed, and asymptotic results may be difficult to obtain. In such cases, it is natural to consider permutation-based tests, which shuffle selected parts of the data in order to remove one or more associations of interest; however, in the case of interactions, it is generally not possible to remove only the associations of interest by simple permutations of the data. We propose a number of alternative permutation-based methods, designed to remove only the associations of interest, but preserving other associations. These methods estimate the interaction term in a model, then create data that “looks like” the original data except that the interaction term has been permuted. The proposed methods are shown to outperform traditional permutation methods in a simulation study. In addition, the proposed methods are illustrated using data from a randomized clinical trial of patients with hypertension.     

Press perturbation experiments and the indeterminacy of ecological interactions: effects of taxonomic resolution and experimental duration

The outcomes of press perturbation experiments on community dynamics are difficult to predict because there is a high degree of indeterminacy in the strength and direction of ecological interactions. Ecologists need to quantify uncertainties in estimates of interaction strength, by determining all the possible values a given interaction strength could take and the relative likelihood of each value. In this study, we assess the degree to which fish effects on zooplankton and phytoplankton are indeterminate in direction using a combination of experimental data and Monte Carlo simulations. Based on probability distributions of interaction strength (i.e. effect magnitude), we estimated the probability of each fish interaction being negative, positive or undetermined in direction. We then investigated how interaction strength and its predictability might vary with experimental duration and the taxonomic resolution of food web data. Results show that most effects of fish on phyto- and zooplankton were indeed indeterminate, and that the effects of fish were more predictable in direction as the taxonomic resolution of food web data decreased and the experimental duration increased. Results also show that most distributions of interaction strength were not normal, suggesting that normal based statistical procedures for testing hypothesis about interaction strength may be misleading, as well as predictions of food web models assuming normal distributions of interaction strength. By considering the probability distributions and confidence intervals of interaction parameters, ecologists would better understand the outcomes of species interactions and make more realistic predictions about our perturbations in natural food webs.     

Bayesian isotonic regression and trend analysis

In many applications, the mean of a response variable can be assumed to be a nondecreasing function of a continuous predictor, controlling for covariates. In such cases, interest often focuses on estimating the regression function, while also assessing evidence of an association. This article proposes a new framework for Bayesian isotonic regression and order-restricted inference. Approximating the regression function with a high-dimensional piecewise linear model, the nondecreasing constraint is incorporated through a prior distribution for the slopes consisting of a product mixture of point masses (accounting for flat regions) and truncated normal densities. To borrow information across the intervals and smooth the curve, the prior is formulated as a latent autoregressive normal process. This structure facilitates efficient posterior computation, since the full conditional distributions of the parameters have simple conjugate forms. Point and interval estimates of the regression function and posterior probabilities of an association for different regions of the predictor can be estimated from a single MCMC run. Generalizations to categorical outcomes and multiple predictors are described, and the approach is applied to an epidemiology application.     

Bias correction with a single null marker for population stratification in candidate gene association studies

Population stratification is a form of confounding by ethnicity that may cause bias to effect estimates and inflate test statistics in genetic association studies. Unlinked genetic markers have been used to adjust for test statistics, but their use in correcting biased effect estimates has not been addressed. We evaluated the potential of bias correction that could be achieved by a single null marker (M) in studies involving one candidate gene (G). When the distribution of M varied greatly across ethnicities, controlling for M in a logistic regression model substantially reduced biases on odds ratio estimates. When M had same distributions as G across ethnicities, biases were further reduced or eliminated by subtracting the regression coefficient of M from the coefficient of G in the model, which was fitted either with or without a multiplicative interaction term between M and G. Correction of bias due to population stratification depended specifically on the distributions of G and M, the difference between baseline disease risks across ethnicities, and whether G had an effect on disease risk or not. Our results suggested that marker choice and the specific treatment of that marker in analysis greatly influenced bias correction.     

Practical Use of MCMC Methods: Lessons from a Case Study

This paper uses the analysis of a data set to examine a number of issues in Bayesian statistics and the application of MCMC methods. The data concern the selectivity of fishing nets and logistic regression is used to relate the size of a fish to the probability it will be retained or escape from a trawl net. Hierarchical models relate information from different trawls and posterior distributions are determined using MCMC. Centring data is shown to radically reduce autocorrelation in chains and Rao‐Blackwellisation and chain‐thinning are found to have little effect on parameter estimates. The results of four convergence diagnostics are compared and the sensitivity of the posterior distribution to the prior distribution is examined using a novel method. Nested models are fitted to the data and compared using intrinsic Bayes factors, pseudo‐Bayes factors and credible intervals.     

Selecting factors predictive of heterogeneity in multivariate event time data

In multivariate survival analysis, investigators are often interested in testing for heterogeneity among clusters, both overall and within specific classes. We represent different hypotheses about the heterogeneity structure using a sequence of gamma frailty models, ranging from a null model with no random effects to a full model having random effects for each class. Following a Bayesian approach, we define prior distributions for the frailty variances consisting of mixtures of point masses at zero and inverse-gamma densities. Since frailties with zero variance effectively drop out of the model, this prior allocates probability to each model in the sequence, including the overall null hypothesis of homogeneity. Using a counting process formulation, the conditional posterior distributions of the frailties and proportional hazards regression coefficients have simple forms. Posterior computation proceeds via a data augmentation Gibbs sampling algorithm, a single run of which can be used to obtain model-averaged estimates of the population parameters and posterior model probabilities for testing hypotheses about the heterogeneity structure. The methods are illustrated using data from a lung cancer trial.     

Quantifying allosteric effects in proteins

In allosteric regulation, protein activity is altered when ligand binding causes changes in the protein conformational distribution. Little is known about which aspects of protein design lead to effective allosteric regulation, however. To increase understanding of the relation between protein structure and allosteric effects, we have developed theoretical tools to quantify the influence of protein-ligand interactions on probability distributions of reaction rates and protein conformations. We define the rate divergence, Dk, and the allosteric potential, Dx, as the Kullback-Leibler divergence between either the reaction-rate distributions or protein conformational distributions with and without the ligand bound. We then define Dx as the change in the conformational distribution of the combined protein/ligand system, derive Dx in the harmonic approximation, and identify contributions from 3 separate terms: the first term, D[stackxomega], results from changes in the eigenvalue spectrum; the second term, D[stackxDeltax], results from changes in the mean conformation; and the third term, Dxv, corresponds to changes in the eigenvectors. Using normal modes analysis, we have calculated these terms for a natural interaction between lysozyme and the ligand tri-N-acetyl-D-glucosamine, and compared them with calculations for a large number of simulated random interactions. The comparison shows that interactions in the known binding-site are associated with large values of Dxv. The results motivate using allosteric potential calculations to predict functional binding sites on proteins, and suggest the possibility that, in Nature, effective ligand interactions occur at intrinsic control points at which binding induces a relatively large change in the protein conformational distribution.     

Empirical Dynamic Material Flow Model for Tungsten in the USA

Multivariate polynomial regression (MPR) analysis was implemented to develop a nonlinear dynamic material flow model (DMFM) of tungsten in the United States for the years 1975–2000 without assumptions for lifetime distributions within reservoirs. Two external economic factors, the Consumer Price Index and the Industrial Production Index, were included as possible exogenous variables. Six types of vector time‐series models were developed using multilinear, simple interaction, and MPR models, each with and without the exogenous economic variables. The DMFMs developed in this work make one‐step‐ahead predictions. That is, the material flows in a given year were predicted using flows and exogenous variables from previous years. In contrast to approaches that utilize assumed lifetime distributions for material within reservoirs, such as the Weibull distribution, the approach used here is completely data driven. MPR models produced statistically better results than linear models for all 13 flows that were modeled. Four of these models used simple interaction terms (which we call linear interaction terms), and two of these incorporated exogenous variables. The other nine models utilized higher‐degree terms with interactions (called multivariate polynomial terms), and two of these incorporated exogenous variables. We conclude that nonlinear vector time series are capable of identifying complex relationships among material flows and exogenous variables. An understanding of these relationships has potential for managing, conserving, and/or forecasting the use of a resource.     

Bayesian LASSO,Scale Space and Decision Making in Association Genetics

Background

LASSO is a penalized regression method that facilitates model fitting in situations where there are as many, or even more explanatory variables than observations, and only a few variables are relevant in explaining the data. We focus on the Bayesian version of LASSO and consider four problems that need special attention: (i) controlling false positives, (ii) multiple comparisons, (iii) collinearity among explanatory variables, and (iv) the choice of the tuning parameter that controls the amount of shrinkage and the sparsity of the estimates. The particular application considered is association genetics, where LASSO regression can be used to find links between chromosome locations and phenotypic traits in a biological organism. However, the proposed techniques are relevant also in other contexts where LASSO is used for variable selection.

Results

We separate the true associations from false positives using the posterior distribution of the effects (regression coefficients) provided by Bayesian LASSO. We propose to solve the multiple comparisons problem by using simultaneous inference based on the joint posterior distribution of the effects. Bayesian LASSO also tends to distribute an effect among collinear variables, making detection of an association difficult. We propose to solve this problem by considering not only individual effects but also their functionals (i.e. sums and differences). Finally, whereas in Bayesian LASSO the tuning parameter is often regarded as a random variable, we adopt a scale space view and consider a whole range of fixed tuning parameters, instead. The effect estimates and the associated inference are considered for all tuning parameters in the selected range and the results are visualized with color maps that provide useful insights into data and the association problem considered. The methods are illustrated using two sets of artificial data and one real data set, all representing typical settings in association genetics.     

A Bayesian nonparametric model for zero‐inflated outcomes: Prediction,clustering, and causal estimation

Researchers are often interested in predicting outcomes, detecting distinct subgroups of their data, or estimating causal treatment effects. Pathological data distributions that exhibit skewness and zero‐inflation complicate these tasks—requiring highly flexible, data‐adaptive modeling. In this paper, we present a multipurpose Bayesian nonparametric model for continuous, zero‐inflated outcomes that simultaneously predicts structural zeros, captures skewness, and clusters patients with similar joint data distributions. The flexibility of our approach yields predictions that capture the joint data distribution better than commonly used zero‐inflated methods. Moreover, we demonstrate that our model can be coherently incorporated into a standardization procedure for computing causal effect estimates that are robust to such data pathologies. Uncertainty at all levels of this model flow through to the causal effect estimates of interest—allowing easy point estimation, interval estimation, and posterior predictive checks verifying positivity, a required causal identification assumption. Our simulation results show point estimates to have low bias and interval estimates to have close to nominal coverage under complicated data settings. Under simpler settings, these results hold while incurring lower efficiency loss than comparator methods. We use our proposed method to analyze zero‐inflated inpatient medical costs among endometrial cancer patients receiving either chemotherapy or radiation therapy in the SEER‐Medicare database.     

Correcting for bias in estimation of quantitative trait loci effects

Estimates of quantitative trait loci (QTL) effects derived from complete genome scans are biased, if no assumptions are made about the distribution of QTL effects. Bias should be reduced if estimates are derived by maximum likelihood, with the QTL effects sampled from a known distribution. The parameters of the distributions of QTL effects for nine economic traits in dairy cattle were estimated from a daughter design analysis of the Israeli Holstein population including 490 marker-by-sire contrasts. A separate gamma distribution was derived for each trait. Estimates for both the α and β parameters and their SE decreased as a function of heritability. The maximum likelihood estimates derived for the individual QTL effects using the gamma distributions for each trait were regressed relative to the least squares estimates, but the regression factor decreased as a function of the least squares estimate. On simulated data, the mean of least squares estimates for effects with nominal 1% significance was more than twice the simulated values, while the mean of the maximum likelihood estimates was slightly lower than the mean of the simulated values. The coefficient of determination for the maximum likelihood estimates was five-fold the corresponding value for the least squares estimates.     

Derivation of the shrinkage estimates of quantitative trait locus effects

The shrinkage estimate of a quantitative trait locus (QTL) effect is the posterior mean of the QTL effect when a normal prior distribution is assigned to the QTL. This note gives the derivation of the shrinkage estimate under the multivariate linear model. An important lemma regarding the posterior mean of a normal likelihood combined with a normal prior is introduced. The lemma is then used to derive the Bayesian shrinkage estimates of the QTL effects.     

Blinded sample size recalculation in multiple composite population designs with normal data and baseline adjustments

The increasing interest in subpopulation analysis has led to the development of various new trial designs and analysis methods in the fields of personalized medicine and targeted therapies. In this paper, subpopulations are defined in terms of an accumulation of disjoint population subsets and will therefore be called composite populations. The proposed trial design is applicable to any set of composite populations, considering normally distributed endpoints and random baseline covariates. Treatment effects for composite populations are tested by combining p-values, calculated on the subset levels, using the inverse normal combination function to generate test statistics for those composite populations while the closed testing procedure accounts for multiple testing. Critical boundaries for intersection hypothesis tests are derived using multivariate normal distributions, reflecting the joint distribution of composite population test statistics given no treatment effect exists. For sample size calculation and sample size, recalculation multivariate normal distributions are derived which describe the joint distribution of composite population test statistics under an assumed alternative hypothesis. Simulations demonstrate the absence of any practical relevant inflation of the type I error rate. The target power after sample size recalculation is typically met or close to being met.     

Effects of Telemetry Location Error on Space-Use Estimates Using a Fixed-Kernel Density Estimator

ABSTRACT Fixed-kernel density estimates using radiotelemetry locations are frequently used to quantify home ranges of animals, interactions, and resource selection. However, all telemetry data have location error and no studies have reported the effects of error on utilization distribution and area estimates using fixed-kernel density estimators. We simulated different home range sizes and shapes by mixing bivariate-normal distributions and then drawing random samples of various sizes from these distributions. We compared fixed-kernel density estimates with and without error to the true underlying distributions. The effects of telemetry error on fixed-kernel density estimates were related to sample size, distribution complexity, and ratio of median Circular Error Probable to home range size. We suggest a metric to assess the adequacy of the telemetry system being used to estimate an animal's space use before a study is undertaken. Telemetry location error is unlikely to significantly affect fixed-kernel density estimates for most wildlife telemetry studies with adequate sample sizes.     

Effects of residual smoothing on the posterior of the fixed effects in disease-mapping models

Disease-mapping models for areal data often have fixed effects to measure the effect of spatially varying covariates and random effects with a conditionally autoregressive (CAR) prior to account for spatial clustering. In such spatial regressions, the objective may be to estimate the fixed effects while accounting for the spatial correlation. But adding the CAR random effects can cause large changes in the posterior mean and variance of fixed effects compared to the nonspatial regression model. This article explores the impact of adding spatial random effects on fixed effect estimates and posterior variance. Diagnostics are proposed to measure posterior variance inflation from collinearity between the fixed effect covariates and the CAR random effects and to measure each region's influence on the change in the fixed effect's estimates by adding the CAR random effects. A new model that alleviates the collinearity between the fixed effect covariates and the CAR random effects is developed and extensions of these methods to point-referenced data models are discussed.     

Linking macroecology and community ecology: refining predictions of species distributions using biotic interaction networks

Macroecological models for predicting species distributions usually only include abiotic environmental conditions as explanatory variables, despite knowledge from community ecology that all species are linked to other species through biotic interactions. This disconnect is largely due to the different spatial scales considered by the two sub‐disciplines: macroecologists study patterns at large extents and coarse resolutions, while community ecologists focus on small extents and fine resolutions. A general framework for including biotic interactions in macroecological models would help bridge this divide, as it would allow for rigorous testing of the role that biotic interactions play in determining species ranges. Here, we present an approach that combines species distribution models with Bayesian networks, which enables the direct and indirect effects of biotic interactions to be modelled as propagating conditional dependencies among species’ presences. We show that including biotic interactions in distribution models for species from a California grassland community results in better range predictions across the western USA. This new approach will be important for improving estimates of species distributions and their dynamics under environmental change.     

Bayesian estimation of concordance among gene trees

Multigene sequence data have great potential for elucidating important and interesting evolutionary processes, but statistical methods for extracting information from such data remain limited. Although various biological processes may cause different genes to have different genealogical histories (and hence different tree topologies), we also may expect that the number of distinct topologies among a set of genes is relatively small compared with the number of possible topologies. Therefore evidence about the tree topology for one gene should influence our inferences of the tree topology on a different gene, but to what extent? In this paper, we present a new approach for modeling and estimating concordance among a set of gene trees given aligned molecular sequence data. Our approach introduces a one-parameter probability distribution to describe the prior distribution of concordance among gene trees. We describe a novel 2-stage Markov chain Monte Carlo (MCMC) method that first obtains independent Bayesian posterior probability distributions for individual genes using standard methods. These posterior distributions are then used as input for a second MCMC procedure that estimates a posterior distribution of gene-to-tree maps (GTMs). The posterior distribution of GTMs can then be summarized to provide revised posterior probability distributions for each gene (taking account of concordance) and to allow estimation of the proportion of the sampled genes for which any given clade is true (the sample-wide concordance factor). Further, under the assumption that the sampled genes are drawn randomly from a genome of known size, we show how one can obtain an estimate, with credibility intervals, on the proportion of the entire genome for which a clade is true (the genome-wide concordance factor). We demonstrate the method on a set of 106 genes from 8 yeast species.     

Adjustment for exploratory cut-off selection in randomized clinical trials with survival endpoint

Defining the target population based on predictive biomarkers plays an important role during clinical development. After establishing a relationship between a biomarker candidate and response to treatment in exploratory phases, a subsequent confirmatory trial ideally involves only subjects with high potential of benefiting from the new compound. In order to identify those subjects in case of a continuous biomarker, a cut-off is needed. Usually, a cut-off is chosen that resulted in a subgroup with a large observed treatment effect in an exploratory trial. However, such a data-driven selection may lead to overoptimistic expectations for the subsequent confirmatory trial. Treatment effect estimates, probability of success, and posterior probabilities are useful measures for deciding whether or not to conduct a confirmatory trial enrolling the biomarker-defined population. These measures need to be adjusted for selection bias. We extend previously introduced Approximate Bayesian Computation techniques for adjustment of subgroup selection bias to a time-to-event setting with cut-off selection. Challenges in this setting are that treatment effects become time-dependent and that subsets are defined by the biomarker distribution. Simulation studies show that the proposed method provides adjusted statistical measures which are superior to naïve Maximum Likelihood estimators as well as simple shrinkage estimators.