Background: In medical research and survival analysis, it is common for an individual or item's failure to be attributable to multiple causes, also known as competing risks. This article focuses on examining the competing risks model as the data increasingly becomes type II censored and randomly removed. The model assumes that the causes of failure are independent and that the lifetimes of individuals are described by the Cox model. At each failure time, the number of items or people removed follows a Binomial distribution. The article derives estimators for the indefinite parameters in the model. The study presents a set of detailed data and includes a simulation study that also illustrates the results. Methods: Different reasons, frequently known as competing risks, are frequently embroiled in an individual's or an item's failure in medical research survival analysis. The competing risks shown under sort II dynamic censoring with random removals are the subject of this research. We get the maximum likelihood and inexact most extreme probability estimators of the obscure parameters. The asymptotic distribution of the maximum probability estimators is utilized to decide the CIs. Then, Monte Carlo simulations were applied to demonstrate the approach. The analyses were performed utilizing R 4.0.4 software. Results: For stroke, systolic blood pressure (SBP) and hypertension status are the only significant variables. In contrast, gender, body mass index (BMI), smoking status, the logarithm of urinary albumin and creatinine ratio, and diabetes status are significant variables for coronary heart disease (CHD) and other cardiovascular diseases (CVDs). The results suggest that significant risk factors differ for different types of CVD events. Conclusion: The outcomes of the simulation study indicate that progressively right-censored type II sampling designs outperformed the usual censored type II sampling designs. Therefore, the estimated parameters on the defined pattern setting are recommended. They can be used in many practical situations when competing risks occur, and progressive censoring could be considered.

In order to facilitate the interpretation of raw scores, they are usually converted to scale scores. In some cases, these conversions are a series of nonlinear transformations that can affect the conditional standard error of measurement throughout the scale of score. Therefore, the purpose of this study was to introduce methods for calculating the conditional standard error of measurement based on the strong true score theory. Furthermore, comparison of normalized and equipercentile nonlinear transformations on the raw scores of the academic achievements of the graduates of mathematical sciences in 2014 and their effect on conditional standard error of measurement was also conducted. So, in order to achieve these purposes, we used a sample of 3943 high school graduates of Mathematics and Physics in 2014 who had participated in national university entrance examination in 2015 randomly selected by National Organization of Educational Testing. The conditional standard error of measurement under these transformations was estimated based on the Binomial procedure of Brennan and Lee (1999) and Chang (2006) method based on the beta-Binomial distribution. The results of this study indicated that the conditional standard error of measurement of the Chang was smoother than Binomial procedure, but in both methods the estimated errors are larger for middle points and smaller for extreme points. Additionally, the conditional standard errors of measurement of equipercentile were always less than normalized tranformation, so the equipercentile method found to be better than normalized transformation.

In this paper, a new class of a weighted quadrature rule is represented as-------------------------------------------- where is a weight function, are interpolation nodes, are the corresponding weight coefficients and denotes the error term. The general form of interpolation nodes are considered as that and we obtain the explicit expressions of the coefficients using the q-Binomial theorem. We give an error analysis for the introduced formula and finally we illustrate its application with some numerical examples.

Background & Aim: Consider a sequence of independent Bernoulli trials with p denoting the probability of success at each trial. With this definition, the probability that the n th success proceed by r failures follows the negative Binomial distribution (NB). NB model has been derived from two different forms. At first, the NB can be thought as a Poisson-gamma mixture. The second form of the NB can be derived as a full member of a single parameter exponential family distribution, and therefore considered as a GLM (generalized linear models).Methods & Materials: We have described a new generalized NB (GNB) distribution with three parameters a, b and k obtained as a compound form of the generalized Poisson and gamma distributions. This distribution gives a very close fit for a large number of data and provides an appropriate model for numerous studies. The most important feature of this model is, its time dependent probabilities, and also it can be used for a variety of researches especially in the survival analysis.Results: This model has been illustrated with two datasets that are indirect measures of illness, along comparing the results of the fitting with NB. Results indicate too much satisfaction. Expected frequencies have been calculated for these data sets to show that the distribution provides a very satisfactory fit in different situations.Conclusion: Using GNB models allows analyzing very complex data. This distribution gives a very close fit for a large number of data and provides an appropriate model for numerous studies. With k=0 the model becomes the ordinary NB and with a=1, it becomes a new model which we call it the generalized geometric distribution with two parameters. The most important feature of this model is its time-dependent probabilities.

Sir,Psoriasis aff ects 2-3% of world’s population.Tuberculosis is the 7th cause of death all over the world, one-third of population is affected by the disease. Romania, in 2010, occupied the 6th position, after Kazakhstan, Moldavia, Georgia, Kirgizstan, and Tajikistan.[3] In 2011, a large retrospective study was carried out on the incidence of present and past tuberculosis within 1236 psoriasis patients, not previously treated with biologics, over 2004-2011.

so many natural phenomena of determining relationship and the effect of input variables on response variable in statistical studies may be different from the suggested model that the researcher selects for his study due to the occupant exists in the structure of data. It may be so influential on different distributions considered for response variables. The optimal properties of estimators evaluated and studied for two statistical variables considered for response variable and input variables in the suggested model. It has been simulated for study and real data has been also investigated. The results confirmed the superiority of a model which is close to the structure of the data.