Bayesian Adaptive Designs in Drug Development
Emmanuel Lesaffre, Gianluca Baio, Bruno Boulanger in Bayesian Methods in Pharmaceutical Research, 2020
The Exponential distribution is particularly easy to work with when analyzing failure-time data. Exponentially distributed data are positive and the posterior distribution is available directly if one uses a Gamma prior distribution for the rate parameter, since this distribution is conjugate for an Exponential sampling distribution or likelihood. In most cases, however, the Exponential distribution may not characterize the patients’ failure times well and one will use a different probability model for the observations, such as another parametric model (e.g. the Weibull distribution) or a semi-parametric proportional hazards regression model. Even if one uses a different model when one analyzes the study’s data, the Gamma-Exponential model may be adequate for adapting the randomization probabilities. As part of developing the study design and writing the protocol, one should explore the robustness of the Gamma-Exponential model via simulation studies that may use alternative sampling distributions to characterize likely study data.
Survival Modeling II: Time-to-Event Regression Models
Gary L. Rosner, Purushottam W. Laud, Wesley O. Johnson in Bayesian Thinking in Biostatistics, 2021
Piecewise Exponential Model. We now describe a fairly straightforward statistical model for the baseline hazard function. The exponential distribution is a simple one-parameter model that sometimes is useful when modeling times until an event. The usefulness of this sampling model derives from its relationship to a Poisson process. That is, if one considers that the number of events that occur in an interval follows a Poisson distribution with parameter λ, then the time in between the events will follow an exponential distribution with parameter λ. We have seen that the gamma distribution is conjugate for the exponential distribution, so a gamma prior on λ will lead to a gamma posterior.
Preliminaries: Welcome to the Statistical Inference Club. Some Basic Concepts in Experimental Decision Making
Albert Vexler, Alan D. Hutson, Xiwei Chen in Statistical Testing Strategies in the Health Sciences, 2017
Common parametric distribution functions employed in clinical experiments include, but are not limited to, normal or Gaussian, lognormal, t, χ2, gamma, F, binomial, uniform, Wishart, and Poisson distributions. In reality, there have been thousands of models that have been published, studied, and available for use via software packages. As noted before, parametric distribution functions can be defined up to finite set of parameters (Lindsey 1996). For example, in the one-dimensional case, the normal (Gaussian) distribution has the notation N(μ, σ2), which corresponds to density function defined as ,where the parameters μ and σ2 represent the mean and variance of the population and the value of x is over the entire real line. The shape of the density has been described famously as the bell-shaped curve. The values of the parameters μ and σ2 may be assumed to be unknown. If the random variable X has a normal distribution, then Y = exp(X) has a lognormal distribution. Other examples include the gamma distribution, denoted Gamma(α, β), with shape parameter α and rate parameter β. The density function for the gamma distribution is given as (βα/Γ(α))xα−1 exp(−βx), x > 0. The exponential distribution, denoted exp(λ), has rate parameter λ with the corresponding density function given as λexp(−λx), x > 0. Note that the exponential distribution is a special case of the gamma distribution with α = 1. It is often the case that simpler, well-known models are nested within more complex models with additional parameters. While the more complex models may fit the data better, there is a trade-off in terms of efficiency when a simpler model also fits the data well.
Design of variables sampling plans based on lifetime-performance index in presence of hybrid censoring scheme
Published in Journal of Applied Statistics, 2019
Ritwik Bhattacharya, Muhammad Aslam
Let us consider X representing the lifetime random variable. In practice, it is not always clock-time or chronological. For example, a number of pages output for a computer printer machine can be considered as its lifetime. However, lifetime clearly represents a larger-the-better type quality characteristic, that is, longer lifetime represents a better quality product. Therefore, a quantity called lower specification limit, say L, is generally associated with the lifetime. Lifetime exceeds L is highly expected. A dimensionless process capability index 18]) X following as exponential distribution with cumulative distribution function is given by 3]), and secondly, the maximum likelihood estimator of the model parameter exists with its explicit form. Moreover, the exact sampling distribution of the estimator is derived in the literature.
Optimal design of repetitive group sampling plans for Weibull and gamma distributions with applications and comparison to the Birnbaum–Saunders distribution
Published in Journal of Applied Statistics, 2018
S. Balamurali, P. Jeyadurga, M. Usha
Weibull distribution is one of the most widely used lifetime distributions and also it is a general case of exponential and Rayleigh distributions. That is, Weibulll distribution converges to exponential distribution and Rayleigh distribution when shape parameter is 1 and 2, respectively. Although the Weibull distribution is an appropriate model for the analysis of material strength, software reliability, lifetime prediction and the reliability evaluation of power systems, recently this distribution has also been used in some other fields such as geophysics and food science (see, for example, Perez et al. [42], Oliveira et al. [41]). Gamma distribution is frequently used as lifetime distribution and this distribution also converges to exponential distribution when the shape parameter is 1. The application of gamma distribution is found in many fields such as reliability, meteorology and financial services ([19,21,35]). Recently, both distributions have been used in designing the sampling plans for ensuring the product lifetime. In view of this, we consider these two distributions to design the MDSRGS plan for assuring the mean life of the product.
Queueing theory techniques and its real applications to health care systems – Outpatient visits
Published in International Journal of Healthcare Management, 2021
Peter O. Peter, R. Sivasamy
In specifying a queueing model, we must make some necessary assumptions about the probabilistic nature of the arrival and service processes. The most common assumption to make about the arrivals is that they follow a Poisson process. This results from the fact that the number of arrivals at any given point has a poison distribution. So if N(t) is the number of arrivals during a time period of duration t and N(t) has a Poisson distribution, thenλ is called the rate and it is the expected number of arrivals per unit of time. Another way to characterize the Poisson is that the time between the consecutive arrivals known as the inter-arrival times (IA) has an exponential distribution.
Related Knowledge Centers
- Conditional Probability
- Failure Rate
- Gamma Distribution
- Power Law
- Memorylessness
- Infinite Divisibility
- Standard Deviation
- Order Statistic
- Law of Total Expectation
- Convolution of Probability Distributions