Parameter Estimation and Hypothesis Testing in Linear Models

Covariate-adaptive designs are widely used to balance covariates and maintain randomization in clinical trials. Adaptive designs for discrete covariates and their asymptotic properties have been well studied in the literature. However, important continuous covariates are often involved in clinical studies. Simply discretizing or categorizing continuous covariates can result in loss of information. The current understanding of adaptive designs with continuous covariates lacks a theoretical foundation as the existing works are entirely based on simulations. Consequently, conventional hypothesis testing in clinical trials using continuous covariates is still not well understood. In this paper, we establish a theoretical framework for hypothesis testing on adaptive designs with continuous covariates based on linear models. For testing treatment effects and significance of covariates, we obtain the asymptotic distributions of the test statistic under null and alternative hypotheses. Simulation studies are conducted under a class of covariate-adaptive designs, including the p-value-based method, the Su’s percentile method, the empirical cumulative-distribution method, the Kullback–Leibler divergence method, and the kernel-density method. Key findings about adaptive designs with independent covariates based on linear models are (1) hypothesis testing that compares treatment effects are conservative in terms of smaller type I error, (2) hypothesis testing using adaptive designs outperforms complete randomization method in terms of power, and (3) testing on significance of covariates is still valid.

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Efficient estimation of bounded gradient-drift diffusion models for affect on CPU and GPU

Behavior Research Methods ◽

10.3758/s13428-021-01674-7 ◽

2021 ◽

Author(s):

Tim Loossens ◽

Kristof Meers ◽

Niels Vanhasbroeck ◽

Nil Anarat ◽

Stijn Verdonck ◽

...

Keyword(s):

Parameter Estimation ◽

Continuous Time ◽

Graphics Processing Units ◽

Linear Models ◽

Likelihood Function ◽

Efficient Estimation ◽

Closed Form Expression ◽

Central Processing ◽

Research Fields ◽

Non Linear

AbstractComputational modeling plays an important role in a gamut of research fields. In affect research, continuous-time stochastic models are becoming increasingly popular. Recently, a non-linear, continuous-time, stochastic model has been introduced for affect dynamics, called the Affective Ising Model (AIM). The drawback of non-linear models like the AIM is that they generally come with serious computational challenges for parameter estimation and related statistical analyses. The likelihood function of the AIM does not have a closed form expression. Consequently, simulation based or numerical methods have to be considered in order to evaluate the likelihood function. Additionally, the likelihood function can have multiple local minima. Consequently, a global optimization heuristic is required and such heuristics generally require a large number of likelihood function evaluations. In this paper, a Julia software package is introduced that is dedicated to fitting the AIM. The package includes an implementation of a numeric algorithm for fast computations of the likelihood function, which can be run both on graphics processing units (GPU) and central processing units (CPU). The numerical method introduced in this paper is compared to the more traditional Euler-Maruyama method for solving stochastic differential equations. Furthermore, the estimation software is tested by means of a recovery study and estimation times are reported for benchmarks that were run on several computing devices (two different GPUs and three different CPUs). According to these results, a single parameter estimation can be obtained in less than thirty seconds using a mainstream NVIDIA GPU.

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The Application of Statistic Inference in the SMS Package Model

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.380-384.1129 ◽

2013 ◽

Vol 380-384 ◽

pp. 1129-1132

Author(s):

Miao Chao Chen ◽

Ting Zhou

Keyword(s):

Mathematical Model ◽

Parameter Estimation ◽

Hypothesis Testing ◽

Poisson Distribution ◽

Distribution Parameter ◽

Research Results ◽

Distribution Parameter Estimation

Hypothesis testing is one of the most important aspects in statistic inference. In this paper, we consider the SMS package problem of hypothesis testing. Firstly, we establish a mathematical model for SMS package problem. Secondly, we use the knowledge of Poisson distribution, parameter estimation and hypothesis testing to analyze this model, and the research results have proved the validity of the method.

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