Intermediate Monitoring,Sample Size Reassessment,and Multi-Treatment Optimal Response-Adaptive Designs for Phase III Clinical Trials with More Than One Constraint

Optimal response-adaptive designs in Phase III clinical trial set up are becoming more and more current interest. In the present article, an optimal response-adaptive design is introduced for more than two treatments at hand. We minimize an objective function subject to more than one inequality constraints. For this purpose, we propose an extensive computer search algorithm. The proposed procedure is illustrated with extensive numerical computation and simulations. Some real data set is used to illustrate the proposed methodology.     

Asymptotically Optimal Design Points for Rejection Algorithms

ABSTRACT

Very fast automatic rejection algorithms were developed recently which allow us to generate random variates from large classes of unimodal distributions. They require the choice of several design points which decompose the domain of the distribution into small sub-intervals. The optimal choice of these points is an important but unsolved problem. Therefore, we present an approach that allows us to characterize optimal design points in the asymptotic case (when their number tends to infinity) under mild regularity conditions. We describe a short algorithm to calculate these asymptotically optimal points in practice. Numerical experiments indicate that they are very close to optimal even when only six or seven design points are calculated.     

Simulation of Constrained Variables in Engineering Risk Analyses

The problem of sampling random variables with overlapping pdfs subject to inequality constraints is addressed. Often, the values of physical variables in an engineering model are interrelated. This mutual dependence imposes inequality constraints on the random variables representing these parameters. Ignoring the interdependencies and sampling the variables independently can lead to inconsistency/bias. We propose an algorithm to generate samples of constrained random variables that are characterized by typical continuous probability distributions and are subject to different kinds of inequality constraints. The sampling procedure is illustrated for various representative cases and one realistic application to simulation of structural natural frequencies.     

Optimal Designs for Random Coefficient Regression Models with Heteroscedastic Errors

The purpose of this article is to present the optimal designs based on D-, G-, A-, I-, and D _β-optimality criteria for random coefficient regression (RCR) models with heteroscedastic errors. A sufficient condition for the heteroscedastic structure is given to make sure that the search of optimal designs can be confined at extreme settings of the design region when the criteria satisfy the assumption of the real valued monotone design criteria. Analytical solutions of D-, G-, A-, I-, and D _β-optimal designs for the RCR models are derived. Two examples are presented for random slope models with specific heteroscedastic errors.     

A gradient approach to efficient design and analysis of multivariate EWMA control charts

Compared to the grid search approach to optimal design of control charts, the gradient-based approach is more computationally efficient as the gradient information indicates the direction to search the optimal design parameters. However, the optimal parameters of multivariate exponentially weighted moving average (MEWMA) control charts are often obtained by using grid search in the existing literature. Note that the average run length (ARL) performance of the MEWMA chart can be calculated based on a Markov chain model, making it feasible to estimate the ARL gradient from it. Motivated by this, this paper develops an ARL gradient-based approach for the optimal design and sensitivity analysis of MEWMA control charts. It is shown that the proposed method is able to provide a fast, accurate, and easy-to-implement algorithm for the design and analysis of MEWMA charts, as compared to the conventional design approach based on grid search.     

A multi-objective coordinate-exchange two-phase local search algorithm for multi-stratum experiments

A multi-stratum design is a useful tool for industrial experimentation, where factors that have levels which are harder to set than others, due to time or cost constraints, are frequently included. The number of different levels of hardness to set defines the number of strata that should be used. The simplest case is the split-plot design, which includes two strata and two sets of factors defined by their level of hardness-to-set. In this paper, we propose a novel computational algorithm which can be used to construct optimal multi-stratum designs for any number of strata and up to six optimality criteria simultaneously. Our algorithm allows the study of the entire Pareto front of the optimization problem and the selection of the designs representing the desired trade-off between the competing objectives. We apply our algorithm to several real case scenarios and we show that the efficiencies of the designs obtained present experimenters with several good options according to their objectives.     

ASYMPTOTIC AND SMALL SAMPLE STATISTICAL PROPERTIES OF RANDOM FRAILTY VARIANCE ESTIMATES FOR SHARED GAMMA FRAILTY MODELS

This paper concerns maximum likelihood estimation for the semiparametric shared gamma frailty model; that is the Cox proportional hazards model with the hazard function multiplied by a gamma random variable with mean 1 and variance θ. A hybrid ML-EM algorithm is applied to 26 400 simulated samples of 400 to 8000 observations with Weibull hazards. The hybrid algorithm is much faster than the standard EM algorithm, faster than standard direct maximum likelihood (ML, Newton Raphson) for large samples, and gives almost identical results to the penalised likelihood method in S-PLUS 2000. When the true value θ₀ of θ is zero, the estimates of θ are asymptotically distributed as a 50–50 mixture between a point mass at zero and a normal random variable on the positive axis. When θ₀ > 0, the asymptotic distribution is normal. However, for small samples, simulations suggest that the estimates of θ are approximately distributed as an x ? (100 ? x)% mixture, 0 ≤ x ≤ 50, between a point mass at zero and a normal random variable on the positive axis even for θ₀ > 0. In light of this, p-values and confidence intervals need to be adjusted accordingly. We indicate an approximate method for carrying out the adjustment.     

An economical acceptance–rejection algorithm for uniform random variate generation over constrained simplexes

This paper develops an algorithm for uniform random generation over a constrained simplex, which is the intersection of a standard simplex and a given set. Uniform sampling from constrained simplexes has numerous applications in different fields, such as portfolio optimization, stochastic multi-criteria decision analysis, experimental design with mixtures and decision problems involving discrete joint distributions with imprecise probabilities. The proposed algorithm is developed by combining the acceptance–rejection and conditional methods along with the use of optimization tools. The acceptance rate of the algorithm is analytically compared to that of a crude acceptance–rejection algorithm, which generates points over the simplex and then rejects any points falling outside the intersecting set. Finally, using convex optimization, the setup phase of the algorithm is detailed for the special cases where the intersecting set is a general convex set, a convex set defined by a finite number of convex constraints or a polyhedron.     

Generating pseudo-random variables from mixture models by exemplary sampling

Generating samples from a two-stage distribution is an important part of the study of mixture models. These samples are used to examine estimation procedures, and other properties of the mixture model. In this paper we present an exemplary sampling method for generating data from the mixed distribution. This method uses the order statistic spacings of the mixing distribution and random sampling from the distribution conditional on the mixing variable to produce samples from the mixed distribution. We show that this exemplary procedure often produces data with an empirical distribution function closer to the mixed distribution than the Method of Composition. We illustrate the method with an example.     

Finite mixture of regression models for a stratified sample

Despite the popularity and importance, there is limited work on modelling data which come from complex survey design using finite mixture models. In this work, we explored the use of finite mixture regression models when the samples were drawn using a complex survey design. In particular, we considered modelling data collected based on stratified sampling design. We developed a new design-based inference where we integrated sampling weights in the complete-data log-likelihood function. The expectation–maximisation algorithm was developed accordingly. A simulation study was conducted to compare the new methodology with the usual finite mixture of a regression model. The comparison was done using bias-variance components of mean square error. Additionally, a simulation study was conducted to assess the ability of the Bayesian information criterion to select the optimal number of components under the proposed modelling approach. The methodology was implemented on real data with good results.     

D-optimal minimax design criterion for two-level fractional factorial designs

A D-optimal minimax design criterion is proposed to construct two-level fractional factorial designs, which can be used to estimate a linear model with main effects and some specified interactions. D-optimal minimax designs are robust against model misspecification and have small biases if the linear model contains more interaction terms. When the D-optimal minimax criterion is compared with the D-optimal design criterion, we find that the D-optimal design criterion is quite robust against model misspecification. Lower and upper bounds derived for the loss functions of optimal designs can be used to estimate the efficiencies of any design and evaluate the effectiveness of a search algorithm. Four algorithms to search for optimal designs for any run size are discussed and compared through several examples. An annealing algorithm and a sequential algorithm are particularly effective to search for optimal designs.     

An iterative algorithm for optimal variable weighting in K-means clustering

The K-means clustering method is a widely adopted clustering algorithm in data mining and pattern recognition, where the partitions are made by minimizing the total within group sum of squares based on a given set of variables. Weighted K-means clustering is an extension of the K-means method by assigning nonnegative weights to the set of variables. In this paper, we aim to obtain more meaningful and interpretable clusters by deriving the optimal variable weights for weighted K-means clustering. Specifically, we improve the weighted k-means clustering method by introducing a new algorithm to obtain the globally optimal variable weights based on the Karush-Kuhn-Tucker conditions. We present the mathematical formulation for the clustering problem, derive the structural properties of the optimal weights, and implement an recursive algorithm to calculate the optimal weights. Numerical examples on simulated and real data indicate that our method is superior in both clustering accuracy and computational efficiency.     

Adaptive grid semidefinite programming for finding optimal designs

We find optimal designs for linear models using a novel algorithm that iteratively combines a semidefinite programming (SDP) approach with adaptive grid techniques. The proposed algorithm is also adapted to find locally optimal designs for nonlinear models. The search space is first discretized, and SDP is applied to find the optimal design based on the initial grid. The points in the next grid set are points that maximize the dispersion function of the SDP-generated optimal design using nonlinear programming. The procedure is repeated until a user-specified stopping rule is reached. The proposed algorithm is broadly applicable, and we demonstrate its flexibility using (i) models with one or more variables and (ii) differentiable design criteria, such as A-, D-optimality, and non-differentiable criterion like E-optimality, including the mathematically more challenging case when the minimum eigenvalue of the information matrix of the optimal design has geometric multiplicity larger than 1. Our algorithm is computationally efficient because it is based on mathematical programming tools and so optimality is assured at each stage; it also exploits the convexity of the problems whenever possible. Using several linear and nonlinear models with one or more factors, we show the proposed algorithm can efficiently find optimal designs.     

The particle filter based on random number searching algorithm for parameter estimation

This article addresses the issue of parameter estimation in linear system in the presence of Gaussian noises, under which the random number searching algorithm (LJ (Luus and Jaakola) algorithm) is combined with the Rao-Blackwellised particle filter (RBPF) algorithm. This yields the so-called RBPF algorithm based on LJ (RBPF-LJ). Unlike the mature alternatives of generic particle filter, the parameter particles of RBPF-LJ are set as random numbers that search in the parameter value scope, which is regulated based on the estimation result to track the changes of the unknown parameter. The contrasting simulations show that the proposed RBPF-LJ outperform the RBPF as well as the particle filter based on kernel smoothing contraction algorithm on the estimation of the dynamically linear or nonlinear parameter and it can obtain the similar estimation results on the static parameter if some coefficients are regulated.     

A combinatorial search problem on matroids

In this paper we consider the worst-case adaptive complexity of the search problem , where is also the set of independent sets of a matroid over S. We give a formula for the number of questions needed and an algorithm to find the optimal search algorithm for any matroid. This algorithm uses only O(|S|³) steps (i.e. questions to the independence oracle). This is also the length of Edmonds’ partitioning algorithm for matroids, which does not seem to be avoidable.     

Sequential Monte Carlo for counting vertex covers in general graphs

In this paper we describe a sequential importance sampling (SIS) procedure for counting the number of vertex covers in general graphs. The optimal SIS proposal distribution is the uniform over a suitably restricted set, but is not implementable. We will consider two proposal distributions as approximations to the optimal. Both proposals are based on randomization techniques. The first randomization is the classic probability model of random graphs, and in fact, the resulting SIS algorithm shows polynomial complexity for random graphs. The second randomization introduces a probabilistic relaxation technique that uses Dynamic Programming. The numerical experiments show that the resulting SIS algorithm enjoys excellent practical performance in comparison with existing methods. In particular the method is compared with cachet—an exact model counter, and the state of the art SampleSearch, which is based on Belief Networks and importance sampling.     

Fast EM-type implementations for mixed effects models

The mixed effects model, in its various forms, is a common model in applied statistics. A useful strategy for fitting this model implements EM-type algorithms by treating the random effects as missing data. Such implementations, however, can be painfully slow when the variances of the random effects are small relative to the residual variance. In this paper, we apply the 'working parameter' approach to derive alternative EM-type implementations for fitting mixed effects models, which we show empirically can be hundreds of times faster than the common EM-type implementations. In our limited simulations, they also compare well with the routines in S-PLUS® and Stata® in terms of both speed and reliability. The central idea of the working parameter approach is to search for efficient data augmentation schemes for implementing the EM algorithm by minimizing the augmented information over the working parameter, and in the mixed effects setting this leads to a transfer of the mixed effects variances into the regression slope parameters. We also describe a variation for computing the restricted maximum likelihood estimate and an adaptive algorithm that takes advantage of both the standard and the alternative EM-type implementations.     

利用互联网大数据预测季度GDP增速的方法研究

大数据为季度GDP走势预测创新研究带来重要突破口。本文利用百度等网站的互联网大数据,基于代表性高维数据机器学习(和深度学习)模型,对我国2011-2018年季度GDP增速深入进行预测分析。研究发现,对模型中的随机干扰因素作出一定分布的统计假设,有助于降低预测误差,任由模型通过大量数据机械地学习和完善并不总是有利于模型预测能力的提升;采用对解释变量集添加惩罚约束的方法,可以有效地处理互联网大数据维度较高的棘手问题;预测季度GDP增速的最优大数据解释变量集的稳定性较高。     

Spatial sampling design for parameter estimation of the covariance function

We study the spatial optimal sampling design for covariance parameter estimation. The spatial process is modeled as a Gaussian random field and maximum likelihood (ML) is used to estimate the covariance parameters. We use the log determinant of the inverse Fisher information matrix as the design criterion and run simulations to investigate the relationship between the inverse Fisher information matrix and the covariance matrix of the ML estimates. A simulated annealing algorithm is developed to search for an optimal design among all possible designs on a fine grid. Since the design criterion depends on the unknown parameters, we define relative efficiency of a design and consider minimax and Bayesian criteria to find designs that are robust for a range of parameter values. Simulation results are presented for the Matérn class of covariance functions.     

Optimal collapsing of mixture distributions in robust recursive estimation

Several authors have discussed Kalman filtering procedures using a mixture of normals as a model for the distributions of the noise in the observation and/or the state space equations. Under this model, resulting posteriors involve a mixture of normal distributions, and a “collapsing method” must be found in order to keep the recursive procedure simple. We prove that the Kullback-Leibler distance between the mixture posterior and that of a single normal distribution is minimized when we choose the mean and variance of the single normal distribution to be the mean and variance of the mixture posterior. Hence, “collapsing by moments” is optimal in this sense. We then develop the resulting optimal algorithm for “Kalman filtering” for this situation, and illustrate its performance with an example.