American Journal of Mathematics and Statistics 2014, 4(5): 205-213 DOI: 10.5923/j.ajms.20140405.01 Parametric Models and Future Event Prediction Base on Right Censored Data Joseph. O. Okello1,*, D. Abdou Ka2 1Pan Africa University Institute of Basic Sciences, Technology and Innovation (PAUISTI), Nairobi, Kenya 2Department of statistics, University of Gaston Berger, Senegal and a visiting lecturer to (PAUISTI), Nairobi, Kenya Abstract In this paper, we developed a parametric displacement models base on the time that the Internally Displaced Persons (IDPs) took to return from IDPs Camps to their ancestral homes in Northern Uganda. The objective is to analyze the displaced proportion of the IDPs using suitable time-to-event parametric models. The accelerated failure time (AFT) models (Weibull, Exponential and log-logistic) were considered. A retrospective data of seven years study of 590 subjects is considered. Maximum likelihood method together with the Davidon-Fletcher- Powell optimization algorithm in MATLAB is used in the estimation of the parameters of the models. The estimated displaced proportions of these AFT models are used in predicting the displaced proportion of the IDPs at a time t. Weibull and exponential models provided better estimates of the displaced proportion of the IDPs due to their good convergence power to four decimal points and predicted the 2027 and 2044 respectively as the year when the displaced proportion can be approximated to be zero. Keywords Survival Techniques, Parametric Models (Weibull, Exponential and Log-logistic), Displaced Proportion, Retrospective Data, Lord Resistance Army (LRA) in which Weibull regression model showed a superior fit. 1. Introduction There is already a very wide literature on parametric distribution (Weibull, Exponential and Log-logistic) in The underlying foundation of most inferential statistical analyzing the time-to-event data, for instance [11], [13], analysis is the concept of probability distribution. An [16], and [21]. The Weibull and Exponential regression understanding of probability distribution is critical in using model have been used in medical research in [19] to model quantitative methods such as hypothesis testing, regression survival data of CABG patients. On the other hand analysis, and time-series analysis. The mathematical researchers such as: [1], [6], [9], [10], [11] [12], [13] and [14] expression that describes the individual probabilities that a provide literature on parametric regression analysis of random variable will take on each of a set of specified values time-to-event data. The primary advantage of Weibull is known as its probability density function. In life data analysis has been stressed out by [1] as the ability to provide analysis, the practitioner attempts to make predictions about reasonably accurate failure analysis and failure forecasts the life of all products in the population by fitting a statistical with extremely small samples and providing a simple and distribution to life data from a representative sample of units. useful graphical plot of the failure data. Furthermore, [1] The parameterized distribution for the data set can then be maintain that AFT interpretation is usually presented in used to estimate important life characteristics of the product coefficients where Positive coefficient means increasing that such as reliability or probability of failure at a specific time, covariate extends the time until failure which is the opposite the mean life and the failure rate. of the proportional hazard covariate coefficient In this paper, we present the parametric distributions of the interpretation where positive coefficient increases the hazard, accelerated failure time models (Weibull, exponential and therefore decreasing the time until failure. Exponential Log-logistic) for the analysis of the time the internally model is a special case of the Weibull distribution model. In displaced persons took to return from IDPs camps to their spite of the wide literature provided on parametric models, ancestral homes. The paper takes the form of case study in we feel that there is some important attribute worth which 590 families displaced by the lord resistance army in discussing. First, although there are large literatures on Northern Uganda were studied. The data was previously application of parametric regression models in estimating modelled parametrically by [18]to test for the distribution fit the time to event, most of the events of interest are always negative occurrences such as death from a certain disease, * Corresponding author: [email protected] (Joseph. O. Okello) failure of a machine parts and above all too much leaning Published online at http://journal.sapub.org/ajms toward hard sciences. Furthermore many data sets have Copyright © 2014 Scientific & Academic Publishing. All Rights Reserved been consider in the study of such kind but there is no 206 Joseph. O. Okello et al.: Parametric Models and Future Event Prediction Base on Right Censored Data attempt involving the time-to-return of the internally origin is when the Ugandan Government declared the displaced persons to their ancestral homes after war which villages safe in 2006 after signing of the truce and formation is a very important social attribute. In this paper, we use the of satellite camps. parametric regression models (Weibull, exponential and According to [20], time-to-event analysis is frequently Log-logistic) to predict the time that all the internally used with retrospective data in which subjects are asked to displaced persons would have returned to their ancestral recall the dates when the events of interest happened to them. homes after being displaced by the Lord Resistance army This was the case employed in this study where subjects war in Northern Uganda using a sample of 590 displaced were asked to recall the year when they returned to their families from seven different villages in Otuke district. The ancestral homes and the censored subjects’ information were idea is to estimate the parameters of the distribution of extracted from the record kept by the Local Council interest and used it in the formulation of the displaced Chairpersons of the seven villages. Our study therefore function which has the same properties as that of a survival considered a retrospective data of 590 subjects that were function in medical research or reliability function in previously studied by Okello, Odongo and AbdouKa in [18]. engineering research. The displaced function estimates the The study period was between the years 2007 to 2013. The displaced proportions of the IDPs at a given time. uncensored subjects were those whose return times were known and the censored subjects were those whose return time is unknown may be because they had not yet returned to 2. Analysis Techniques their ancestral homes by the end of 2013 or had died within the study time. This generated a right censored data set. According to [3], parametric, non-parametric and Several researches have been conducted using the semi-parametric techniques are the three well known technique of time-to-event analysis for many case studies. techniques used for analyzing the time-to-event data, each Although much of the work in this paper pays much attention with its own limitation but parametric approach is thought to to internally displaced persons return time and prediction of yield better results provided the assumption made in the the return event, the explored methods of parametric model analysis are correct. With Parametric models, the outcome is are much more general. They can be applied to any study of assumed to follow a certain known distribution. There are a time-to-event analysis. number of texts that discuss comprehensively parametric time-to-event-models such as; [5], [7], [9], [13], [15], [17] and [19]. For instance [15], suggests that exponential, 3. Methodology Weibull, lognormal and gamma distribution are the most commonly used parametric models in analyzing 3.1. Introduction time-to-event data. In this paper, the parameters of Weibull, Exponential and According to [4], Survival analysis is a phrase used to Log-logistic distribution are estimated based on censored describe the analysis of data in the form of time from a data of the IDPs return time to their ancestral homes. The well-defined time origin until the occurrence of the particular uncensored observation under this study were the subjects event of interest or the end point of the study. On the other who have resumed their ancestral homes within the hand, [2] defined survival analysis as a class of statistical predetermine study period and the censored subjects are techniques used for studying the occurrence and timing of those whose time of return are not known. The status of the events. They were originally designed for the event of death subjects was defined as: occurrence and hence name survival analysis. The techniques is extremely useful for studying many different 1 푓 푡ℎ푒 푡ℎ 푝푒푟푠표푛 푟푒푡푢푛푒푑 푡푚푒 푠 푘푛표푤푛 훿 = kinds of events in both the social and natural sciences 0 푓 푡ℎ푒 푡ℎ 푝푒푟푠표푛 푟푒푡푢푛푒푑 푡푚푒 푠 푢푛푘푛표푤푛 research, such as the onset of disease in Biostatistics, equipment failures in engineering, earthquakes, automobile The status contribution to the likelihood function for the accidents, stock market crashes, revolutions, job subject who have returned to their ancestral home would be terminations, births, marriages, divorces, promotions in job 푓 푡; 휃 and for those who have not returned to their places, retirements,