Log-logistic Research Articles

Parametric regression model for response time in clinical trials – a bayesian approach

In this paper an attempt has been made to model the censored survival data using Bayesian regressions with Markov Chain Monte Carlo (MCMC) methods. Bayesian LogNormal (LN) regression model are found to be providing better fit than the other Bayesian regression models namely Exponential (E), Generalized Exponential (GE), Webull (W), LogLogistic (LL) and Gamma (G).

Log-logistic distribution for survival data analysis using MCMC.

This paper focuses on the application of Markov Chain Monte Carlo (MCMC) technique for estimating the parameters of log-logistic (LL) distribution which is dependent on a complete sample. To find Bayesian estimates for the parameters of the LL model OpenBUGS—established software for Bayesian analysis based on MCMC technique, is employed. It is presumed that samples for independent non informative set of priors for estimating LL parameters are drawn from posterior density function. A proposed module was developed and incorporated in OpenBUGS to estimate the Bayes estimators of the LL distribution. It is shown that statistically consistent parameter estimates and their respective credible intervals can be constructed through the use of OpenBUGS. Finally comparison of maximum likelihood estimate and Bayes estimates is carried out using three plots. Additively through this research it is established that computationally MCMC technique can be effortlessly put into practice. Elaborate procedure for applying MCMC, to estimate parameters of LL model, is demonstrated by making use of real survival data relating to bladder cancer patients.

Probability Density Function of EMG Signals based on Hand Movements in Time and Frequency Domain

This paper attempts to estimate the probability density function of hand movements by using EMG signals. Several hand grasps generated from different hand movements, we have analyzed Tip and Lateral. Four well known pdf functions for good fitness of test are Log Logistics (3P), Johnson, Dagum (4P) and Burr (4P), that have been tested. The probability density function has been carried out in time domain and FFT domain as well as in DWT domain. It was observed that there are different distributions for different hand movements, which describe the samples most accurately with the movements of hand with respect to two channels; channel 1 and channel 2. In this scenario, channels 1 is placed on upper limb and channel 2 placed on lower limbs as a reference channel. Although, Burr distribution and Log logistic distribution along with that Dagum has been a good fit for most of the data, it is shown in this paper that Non Negative distribution (Dagum (4P) and Burr (4P) distribution) is a better choice for estimating the Tip and Lateral hand movements. General Terms This method is used for classification purpose. This is the another method of classification of data set in probability density function. This is used as a clinic/ engineering field as a designing of prosthetic arm and hand.

Some Useful Properties of Log-Logistic Random Variables for Health Care Simulations

A log-logistic (LL) random variable is one whose logarithm has a logistic distribution. Since the logistic distribution is similar to the normal distribution, log-logistic random variables are similar to log-normal (LN) random variables. However, many of the important properties of LN random variables can only be described using integrals, while the corresponding properties of LL random variables can be described using simple algebra. LL random variables may therefore be a useful alternative to LN random variable for computer simulation of operating room processes or other health care applications, especially when they fit the data more closely. We review the properties of LL random variables, and derive some relationships of the mean residual time to the median residual time. We describe methods of fitting LL distributions to observed data, and discuss potential advantages of using them for simulation of operating room utilization.

Eddy Current Testing and Evaluation of Far-Side Corrosion Around Rivet in Jet-Engine Intake of Aging Supersonic Aircraft

This study proposes an automated inspection system to inspect far-side corrosion around rivets in the F-5 supersonic aircraft. The system includes a linear integrated Hall sensor array, a C-type exciter, and an automated scanning system. The inspection system was tested on a modified elliptical intake having artificial defects with 10-mm diameter and 0.12–1.11-mm depth. An algorithm was produced to detect the defects. The probability of detection (POD) curve was obtained with hits/misses data using a log–logistics model. Defect with depths of 0.67 and 0.98 mm were detected with 90 % POD and lower 95 % confidence bound.

Modeling heterogeneity for bivariate survival data by shared gamma frailty regression model

In the analysis of survival data with parametric models, it is well known that the Weibull model is not suitable for modeling survival data where the hazard rate is non-monotonic. For such cases, where hazard rates are bathtub-shaped or unimodal (or hump-shaped), log-logistic, lognormal, Birnbaun-Saunders, and inverse Gaussian models are used for the computational simplicity and popularity among users. When models are inadequate and inappropriate, compound Rayleigh, arctangent, generalized Weibull, and Weibull-Pareto composite models are also used. Out of these models log-logistic (LL) model is frequently used. The log-logistic distribution (LLD) has the advantage of having simple algebraic expressions for its survivor and hazard functions and a closed form for its distribution function. In this paper, we consider gamma distribution as frailty distribution and LLD as baseline distribution for bivariate survival times. The problem of analyzing and estimating parameters of bivariate LLD with shared gamma frailty is of interest and the focus of this paper. We introduce Bayesian estimation procedure using Markov Chain Monte Carlo (MCMC) technique to estimate the parameters involved in the proposed model. We present a simulation study and two real data examples to compute Bayesian estimates of the parameters and their standard errors and then compare the true values of the parameters with the estimated values for different sample sizes. A search of the literature suggests there is currently no work has been done for bivariate log-logistic regression model with shared gamma frailty using Bayesian approach.

Characterization of Life Distributions Using Log-odds Rate in Discrete Aging

For studying and modeling the time to failure of a system or component, many reliability practitioners used the hazard rate and its monotone behaviors. However, nowadays, there are two problems. First, the modern components have high reliability and, second, their distributions are usually have non monotone hazard rate, such as, the truncated normal, Burr XII, and inverse Gaussian distributions. So, modeling these data based on the hazard rate models seems to be too stringent. Zimmer et al. (1998) and Wang et al. (2003, 2008) introduced and studied a new time to failure model in continuous distributions based on log-odds rate (LOR) which is comparable to the model based on the hazard rate. There are many components and devices in industry, that have discrete distributions with non monotone hazard rate, so, in this article, we introduce the discrete log-odds rate which is different from its analog in continuous case. Also, an alternative discrete reversed hazard rate which we called it the second reversed rate of failure in discrete times is also defined here. It is shown that the failure time distributions can be characterized by the discrete LOR. Moreover, we show that the discrete logistic and log logistics distributions have property of a constant discrete LOR with respect to t and ln t, respectively. Furthermore, properties of some distributions with monotone discrete LOR, such as the discrete Burr XII, discrete Weibull, and discrete truncated normal are obtained.

Comparison of two fitting methods for the log‐logistic distribution

We investigate generalized probability weighted moments (GPWM) and maximum likelihood (ML) fitting methods in the two‐parameter log‐logistic (LL) model. Parameter and quantiles estimators are computed along with their asymptotic variances and covariances. A comparison of these methods is done by simulation. It is concluded that for estimating β, GPWM can provide better results than the ML method. However, for estimating quantiles, GPWM provides better results only for very small sample sizes, especially when the distribution is quite asymmetrical. Although presently, LL is not one of the distributions frequently used in hydrology, we agree with some authors that it merits wider use in hydrological practice. For a clearer idea on the merits of LL, we compare it with three other distributions for fitting flood data from 114 hydrometric stations in Canada. The results support our view regarding the good fitting potential of the LL distribution to extreme hydrologic data.

A note on the applicability of log-Gumbel and log-logistic probability distributions in hydrological analyses: II. Assumed pdf

Applicability of log-Gumbel (LG) and log-logistic (LL) probability distributions in hydrological studies is critically examined under real conditions, where the assumed distribution differs from the true one. The set of alternative distributions consists of five two-parameter distributions with zero lower bound, including LG and LL as well as lognormal (LN), linear diffusion analogy (LD) and gamma (Ga) distributions. The log-Gumbel distribution is considered as both a false and a true distribution. The model error of upper quantiles and of the first two moments is analytically derived for three estimation methods: the method of moments (MOM), the linear moments method (LMM) and the maximum likelihood method (MLM). These estimation methods are used as methods of approximation of one distribution by another distribution. As recommended in the first of this two-part series of papers, MLM turns out to be the worst method, if the assumed LG or LL distribution is not the true one. It produces a huge bias of upper quantiles, which is at least one order higher than that of the other two methods. However, the reverse case, i.e. acceptance of LN, LD or Ga as a hypothetical distribution, while the LG or LL distribution is the true one, gives the MLM bias of reasonable magnitude in upper quantiles. Therefore, one should avoid choosing the LG and LL distributions in flood frequency analysis, especially if MLM is to be applied.

A note on the applicability of log-Gumbel and log-logistic probability distributions in hydrological analyses: I. Known pdf

Two probability density functions (pdf), popular in hydrological analyses, namely the log-Gumbel (LG) and log-logistic (LL), are discussed with respect to (a) their applicability to hydrological data and (b) the drawbacks resulting from their mathematical properties. This paper—the first in a two-part series—examines a classical problem in which the considered pdf is assumed to be the true distribution. The most significant drawback is the existence of the statistical moments of LG and LL for a very limited range of parameters. For these parameters, a very rapid increase of the skewness coefficient, as a function of the coefficient of variation, is observed (especially for the log-Gumbel distribution), which is seldom observed in the hydrological data. These probability distributions can be applied with confidence only to extreme situations. For other cases, there is an important disagreement between empirical data and theoretical distributions in their tails, which is very important for the characterization of the distribution asymmetry. The limited range of shape parameters in both distributions makes the analyses (such as the method of moments), that make use of the interpretation of moments, inconvenient. It is also shown that the often-used L-moments are not sufficient for the characterization of the location, scale and shape parameters of pdfs, particularly in the case where attention is paid to the tail part of probability distributions. The maximum likelihood method guarantees an asymptotic convergence of the estimators beyond the domain of the existence of the first two moments (or L-moments), but it is not sensitive enough to the upper tails shape.

Median time from seroconversion to AIDS in Italian HIV-positive haemophiliacs: different parametric estimates.

The purpose of this study was to estimate the median incubation time between human immunodeficiency virus (HIV) infection and onset of acquired immunodeficiency syndrome (AIDS), using three parametric models and six estimates of seroconversion time. Study subjects were 732 HIV-positive haemophiliacs enrolled in the Italian Registry of patients with congenital coagulation disorders. Seroconversion time was estimated for each subject according to six different criteria, based on three distributions of seroconversion (uniform, uniform on three sub-intervals and truncated Weibull) and two indices synthesizing each distribution (median and median of three random values). The estimated seroconversion times were subsequently used as starting points in the analysis of incubation. This was performed applying Kaplan-Meier non-parametric survival analysis, and fitting to incubation data three probability density functions, representing three different situations with respect to the hazard of developing AIDS following seroconversion (namely Weibull (WE), generalized exponential (GE) and log-logistic (LL)). The cumulative incidence over an 8-year period ranged from 14.9 to 17.8 per cent when applying the Kaplan-Meier method, from 14.1 to 17.2 per cent when using the WE function, from 14.5 to 17.3 per cent when using the GE function and from 14.4 to 17.3 per cent when using the LL function, depending on the estimate of seroconversion time used. Similarly, the median incubation times ranged from 12.6 to 15.0 years with the WE function, from 14.0 to 16.5 years with the GE function, and from 13.4 to 16.1 years with the LL function. The presence of a bound on the increase of the hazard function seems to affect the incubation more strongly than the eventual decrease following the attainment of the maximum risk. This may be due to the decrease in the hazard beginning when most of the seropositive subjects have already developed AIDS.