Bayesian inferences in the Cox model for order-restricted hypotheses

In studying the relationship between an ordered categorical predictor and an event time, it is standard practice to include dichotomous indicators of the different levels of the predictor in a Cox model. One can then use a multiple degree-of-freedom score or partial likelihood ratio test for hypothesis testing. Often, interest focuses on comparing the null hypothesis of no difference to an order-restricted alternative, such as a monotone increase across levels of a predictor. This article proposes a Bayesian approach for addressing hypotheses of this type. We reparameterize the Cox model in terms of a cumulative product of parameters having conjugate prior densities, consisting of mixtures of point masses at one, and truncated gamma densities. Due to the structure of the model, posterior computation can proceed via a simple and efficient Gibbs sampling algorithm. Posterior probabilities for the global null hypothesis and subhypotheses, comparing the hazards for specific groups, can be calculated directly from the output of a single Gibbs chain. The approach allows for level sets across which a predictor has no effect. Generalizations to multiple predictors are described, and the method is applied to a study of emergency medical treatment for stroke.     

Monotonicity of interleukin-1 receptor-ligand binding with respect to antagonist in the presence of decoy receptor

We consider the interaction between interleukin-1 IL-1, its receptor IL-1RI, the receptor antagonist IL-1Ra and a decoy receptor (or trap) that binds both with the ligand and the antagonist. We study how the interaction between IL-1Ra and the decoy receptor influences the effect of either reagent on reducing the equilibrium concentration of the receptor-ligand complex. We obtain that, given a certain relationship among the equilibrium constants and the total concentrations of solutes, IL-1Ra can reverse the effect of the decoy receptor of decreasing the equilibrium concentration of the receptor-ligand complex. This finding derives from a mathematical result applicable to any reversible chemical reaction system comprising four species arranged in a square such that each species binds its two immediate neighbors. The result gives the monotonicity of the equilibrium concentrations of the complex species as functions of the total concentrations of the simple species.     

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.     

Bayesian inference on order-constrained parameters in generalized linear models

In biomedical studies, there is often interest in assessing the association between one or more ordered categorical predictors and an outcome variable, adjusting for covariates. For a k-level predictor, one typically uses either a k-1 degree of freedom (df) test or a single df trend test, which requires scores for the different levels of the predictor. In the absence of knowledge of a parametric form for the response function, one can incorporate monotonicity constraints to improve the efficiency of tests of association. This article proposes a general Bayesian approach for inference on order-constrained parameters in generalized linear models. Instead of choosing a prior distribution with support on the constrained space, which can result in major computational difficulties, we propose to map draws from an unconstrained posterior density using an isotonic regression transformation. This approach allows flat regions over which increases in the level of a predictor have no effect. Bayes factors for assessing ordered trends can be computed based on the output from a Gibbs sampling algorithm. Results from a simulation study are presented and the approach is applied to data from a time-to-pregnancy study.     

Mathematical models for some types of chemical information systems

Ant species on a high evolutionary level have evolved chemical recruitment systems such as mass recruitment or quality recruitment. The recruitment process from the nest to a food source may be damped by crowding effects at the source. For four patterns of behavior (mass/quality recruitment; with/without damping) we study mathematical models for the time development of the quantity of food at the source. Each of the models can be reduced to a second order time-delayed differential equation which will be studied in the equivalent form of a first order (nonlinear) functional differential equation. We discuss the complete exploitation of a given source. In case of mass recruitment there possibly remains a threshold quantity of food not worth exploiting. However, every source will be exploited completely (in finite time) provided that the volatility of the trail pheromone is small compared with the exploitation activities of the colony and the distance from the nest to the source. In addition, for the damped models the capacity of the crowded source must be large compared with the initial quantity of food offered. The efficiency of the exploitation activities of some species allows conclusions on their evolutionary development.     

Sensitivity analyses comparing time-to-event outcomes only existing in a subset selected postrandomization and relaxing monotonicity

In randomized studies researchers may be interested in the effect of treatment assignment on a time-to-event outcome that only exists in a subset selected after randomization. For example, in preventative HIV vaccine trials, it is of interest to determine whether randomization to vaccine affects the time from infection diagnosis until initiation of antiretroviral therapy. Earlier work assessed the effect of treatment on outcome among the principal stratum of individuals who would have been selected regardless of treatment assignment. These studies assumed monotonicity, that one of the principal strata was empty (e.g., every person infected in the vaccine arm would have been infected if randomized to placebo). Here, we present a sensitivity analysis approach for relaxing monotonicity with a time-to-event outcome. We also consider scenarios where selection is unknown for some subjects because of noninformative censoring (e.g., infection status k years after randomization is unknown for some because of staggered study entry). We illustrate our method using data from an HIV vaccine trial.     

Effects of dominance on the probability of fixation of a mutant allele

We consider whether the fixation probability of an allele in a two-allele diploid system is always a monotonic function of the selective advantage of the allele. We show that while this conjecture is correct for intermediate dominance, it is not correct in general for either overdominant or underdominant alleles, and that for some parameter ranges the fixation probability can initially decrease and then increase as a function of the amount of selection. We have partial results that characterize the ranges of parameters for which this happens.      

Two-stage designs for dose-finding trials with a biologic endpoint using stepwise tests

Summary.   We deal with the design problem of early phase dose-finding clinical trials with monotone biologic endpoints, such as biological measurements, laboratory values of serum level, and gene expression. A specific objective of this type of trial is to identify the minimum dose that exhibits adequate drug activity and shifts the mean of the endpoint from a zero dose to the so-called minimum effective dose. Stepwise test procedures for dose finding have been well studied in the context of nonhuman studies where the sampling plan is done in one stage. In this article, we extend the notion of stepwise testing to a two-stage enrollment plan in an attempt to reduce the potential sample size requirement by shutting down unpromising doses in a futility interim. In particular, we examine four two-stage designs and apply them to design a statin trial with four doses and a placebo in patients with Hodgkin's disease. We discuss the calibration of the design parameters and the implementation of these proposed methods. In the context of the statin trial, a calibrated two-stage design can reduce the average total sample size up to 38% (from 125 to 78) from a one-stage step-down test, while maintaining comparable error rates and probability of correct selection. The price for the reduction in the average sample size is the slight increase in the maximum total sample size from 125 to 130.