Left And Right Censored Survival Times Philosophy Essay

remaining And Right illegalize Survival Times Philosophy EssayProvide a clear explanation of what is meant by go forthover censored and right censored survival of the fittest of the fittest time, and illustrate your answer with or so utilisations of how each may arise in a social acquirement context.Suppose that you rich person continuous time unemployment magic spell data. The data were derived using a armoury try with follow-up (i.e. interviews some time after the timeworn sampling date). You also know the date of the interview, at which time information about characteristics were collected, and whether or not the spell in progress at the stock sampling date was still in progress and, if not, the date the spell ended. By deduction, you support propose the length of time in the midst of the stock sample date and the date at which each person was last ascertained to be unemployed (the interview date for those still unemployed or some date between the stock sample date and interview date for those who got a job). However, you dont know the date at which each persons spell began, and nor therefore the length of each persons unemployment spell in total from start until last find oneselfd. With reference to expressions for the sample log-likelihood position, show that it is possible to estimate the parameters of an exponential work reckon regression work in this typeface. Also discuss, giving reasons, whether you could estimate a Weibull model with the same data.adapted from Wooldridge (2002, Ex. 20.3) Assume that you claim a random sample from the inflow to the state, and all survival times are right-censored.(i) Write down the sample log-likelihood function for this situation.(ii) Derive the special deterrent example of likelihood function accustomed in (i) when survival times follow the Gompertz distribution. Recall that the Gompertz model has hazard function q(t, X) = lexp (gt), where l = exp (b0 + b1X1 + b2X2 + + bkXk) and shape parameter g 0.(iii) Consider the Gompertz model in which the covariate vector X only contains a unceasing. Show that the Gompertz log likelihood cannot be maximized for real numbers b0 and g.(iv) From (iii), what do you conclude about estimating duration models from inflow sample data when all survival times are right censored?Table of ContentsLeft censored and Right censoredWhen we deal with observations the observation period is the difference between the time when experiment begins (time is zero) and when it terminates (let, time is T0 in general anatomy 01). But in many cases the entities under consideration (human/device) dont come to an end and in those cases we say that it has been suspended, abbreviated or censored. In many areas of social science and life testing, the line of business(s) may leave or enter after they have been put on test. The subject may leave our study in front completion (due to failure or death) or may enter late. To analyse much(prenominal) beha viour of human existence we are interested in left censored and right censored. Censoring occurs because sometimes our study of interest is lost to follow-up.Censored data means that the observations are known partially and it reflects the side of the dimension. Stephen P. Jenkins in his Survival Analysis wrote,A survival time is censored if all that is known is that it began or ended at bottom some particular interval of time, and thence the total spell length (from entry time until transition) is not known exactly.(Jenkins 2005, p. 4)Its a major problem in social science that some observations are censored but its very usual that our study of interest may not survive until the end period.Left CensoredLeft outlaw refers to the upshot that occurs at a time before a left bound. In this case we dont know the time when it started. (L Samartzis 2005-06)It is such a situation that we know the datum is below a certain value but we dont know how much it is.Say, for example, a patholog ical report is revealed which ensures that the patient of is suffering from cancer but we have no idea when the patient has been infected.Figure 01 illustrates the censoring situations where X refers the expresss in time when we actually start or finish monitoring the censored entities, except the beginning (of entity life, at time zero) and the end of the experimental observation period (time T0). Here Line C completes its spell and all other entities are interrupted.Here, a shows an entity that has already been operating for some unknown period of time, before we start monitoring it. This case is called left-censoring. (Dr. J Luis Romeu, n. d.)Figure 01 Left and Right CensoringIn a word left censoring means censoring occurs on the left side. If we ignore this type of censoring then there arise selectivity bias because left censoring volition overestimate the mean duration as longer spells tend to be observed more frequently than shorter spells. (Amemiya 1999)Right CensoredRigh t censoring refers to the event that occurs at a time after a right bound. In this case we dont know the time when it ended. (L Samartzis 2005-06)In duration models and survival analysis right censoring occurs very often because in many cases observations are known to be larger than some given value. In this case the only information we have is the right bound.Say, for example, we start with 500 light bulbs and this ordain be change after an assigned period of time. In this experiment censoring will occur on the right side because we exactly know the starting point of our experiment.In Figure 01, Line b shows an entity that has been monitored since the beginning of its life (i.e. at the start of the experiment) but which we have ceased to observe before the experiment ends (time T0) or it fails. That is, we observe the entity for some time, after which we are not able to monitor it any more. This other type of truncation is known as right censoring. (Dr. J Luis Romeu, n. d) simili tude between left and right censoring with the help of an exampleSuppose, a social scientist is interested in analysing the adverse affect of taking illegal drugs in a particular area (may be Colchester). The researcher is willing to determine the distribution of the time until first Marijuana use among high school boys in that area. The question to be answered by the school boys isWhen did you first use Marijuana?Let us consider two hypothetical repliesRespondent 01I have used it but cannot remember just when the first time was.Respondent 02I never used it.In case of the 1st respondent the event had occurred but exact date at which he started using Marijuana is totally unknown. This is an example of left censored.On the other hand, in the 2nd case the event not yet occurred but there may be the possibility of taking Marijuana in some future dates. Unlike the left censored the censoring occurs on the right side and thus this is an example of right censored. (Klein and Moeschberger 2 003, p. 70-71)(b) Stock Sample with follow-upThe important things to be considered in this example areThis is a continuous time unemployment spell data.The data were derived using a stock sample with follow-up which is a different name of left truncation (delayed entry) and their applications are similar to deal. This type of data is most commonly used by economists. (Jenkins 2005, p. 5)The stock sample dates are still in progresses which doom that there are some observations that are right censored.Let us define,Ti = Total spell lengthf (Ti) = Probability density function (slope of Failure function) at time TiS (Ti) = Survival function at time Ti (Ti) = Hazard function at time TiS (ti) = The date at which the stock sample was drawnCi = Censoring indicatorXi = Vector of observed covariatesb = Parameter to be estimatedN = Sample size of itThere are two types of contributors,Those who leave the state of interest.Those who stay in our state of interest.So the likelihood function wil l be,N N = f(Ti) / S (ti) Ci S (Ti) / S (ti) 1- Cii = 1 i= 1 promptly by definition of hazard function, we haveN = (Ti) Ci S (Ti) / S (ti) i = 1NOr, log = Ci log (Ti) + log S (Ti) log S (ti) Equation no 01 i = 1Equation no 01 clearly states the log-likelihood function of the example. Now its not difficult to consider the Exponential and Weibull model to estimate the parameters.For Exponential poser caseWe know that the Exponential model has the following hazard function (Ti) = where l = exp(bX)Now, by definition the survival function can be obtained from the hazard function by the equation belowtS(t) = exp ( (u)du ) Equation no 02 0So the survival function of the Exponential model is S(t) = exp (-t ). Now plugging the value of the hazard and survival function of the Exponential model in the log-likelihood function (Equation no 01) we get the Exponential hazard regression model which is as followsNlog = Ci log + log exp (-T ) log exp (-t ) i = 1NOr, log = Ci (bX) T t i = 1Once we get the value of the variables we can easily calculate the log-likelihood function of the Exponential hazard regression model.For Weibull Model caseExponential model is a special case of Weibull model which has the following hazard function (Ti) = t-1 where l = exp(bX)When = 1 the model describes the Exponential model thus it is nothing but a special case of Weibull model. From equation no 02 the survival function of Weibull model is,S(t) = exp (-t )Plugging the value in the log-likelihood function (Equation no 01) we get the Weibull model,Nlog = Ci log t-1 + log exp (- T) log exp (- t ) i = 1NOr, log = Ci (bX) + Ci log + Ci ( 1) log t T t i = 1Like the exponential model we can easily calculate the Weibull model when we have the data of the model. The estimation can be obtained from the above log-likelihood function for the given data.But its a matter of judgment that which model will be the best-fitted? The result depends on the val ue of and its critical value of the t-statistic (the p-value). The critical t-statistic value of will decide which model is appropriate for the given data. If the value of is greater than 1 and significant then it is wise to consider the Weibull model rather than the exponential model.(c) Adapted from Wooldridge (2002, Ex. 20.3)The problem of estimating the censoring and time varying covariates is not possible to handle by the Ordinary Least Square (OLS) method rather it is addressed by the estimation based on Maximum Likelihood (ML) method. But before going to estimate we should identify the type of process that generates the data i.e. the type of sampling scheme.The random sample from the inflow to the state is one of the five sampling schemes analyzed in social science. (Jenkins 2005, p. 61)Given the random sample, letXi = Vector of observed covariates = Vector of unknown parametersN = Random sample sizeti = Length of timeCi= Censoring indicatorCi = 1 if uncensoredCi = 0 if ce nsoredThe conditional likelihood observations can be written asf( ti Xi, ) Ci 1 F (ti Xi, ) 1- Ciwhere uncensored and censored subjects are in product form. (Cox and Oakes 1992, p. 33)(i)If all observations are right censored, Ci = 0 and hence the log-likelihood function isN log 1 F (ti Xi, ) Equation no 03i=1(ii)Gompertz model has hazard function q(t, X) = lexp (gt)where l = exp(b0 + b1X1 + b2X2 + + bkXk) and shape parameter g 0By definition, survival function S(t) istS(t) = exp ( (u)du ) recall Equation no 02 0Now the survival function in Gompertz model isS(t) = exp / g exp ( gt ) + ( / g ) S(t) = exp ( / g) 1 exp (gt) And consequently the failure function isF(t) = 1- exp ( / g) 1 exp (gt)So the log-likelihood function for Gompertz distribution (from Equation no 03) isN log 1 1 + exp ( / g) 1 exp (gt) i=1N= log exp ( / g) 1 exp (gt) i=1N= ( / g) 1 exp (gt) Equation no 04 i=1(iii)In Gompertz distribution when the covariate vector Xi only contains a constant implies that l = exp (b0) where without this condition l = exp (b0 + b1X1 + b2X2 + + bkXk). In this conditional case the observed covariates Xi is defined only by the constant term b0.Hence the log-likelihood function (from Equation no 04) isN= ( / g) 1 exp (gt) where l = exp( b0 )i=1N= (exp( b0 ) / g) 1 exp (gt) Equation no 05 i=1Given positive value of t and g the value of 1 exp (gt) will ceaselessly be negative and consequently the value of equation no 05 will be negative. So we can maximise the likelihood function only by maximising b.But when the value of b the exp (b0) . So for any positive value of g (nevertheless to mention that t is also positive) the log-likelihood function (containing only constant of covariate vector Xi ) will lead to b getting more positive values without any bound.We can also rule out the minimization of log-likelihood function by minimising exp (b0) across b. For the value of b the exp (b0) 0. The values of b are getting m ore and more negative and it will go beyond calculation.Hence, the Gompertz log-likelihood cannot be maximized only for the real numbers b0 and g.(iv)From (iii) we observed that Gompertz log-likelihood cannot be maximised for only real numbers b0 and g. So it is not possible to estimate the Gompertz models from any given flow data when all survival times are right censored. Actually this might be a special case when all data under consideration are right censored and covariate vector Xi contains only a constant.(d) ReferencesAmemiya T. (1999), A note on left censoring, Analysis of Panels and limit Dependent Variables Models, Edited by Hsiao, C., Lahiri, K., Lee, Lung-Fei, and Pesaran, M. H., Cambridge Cambridge University Press.Cox, D. R. and Oakes, D. (1992), Analysis of Survival Data, 1st edition (Reprinted by University Press, Cambridge), London Chapman Hall.Jenkins, Stephen P. (2005), Survival Analysis (unpublished), , July 2005, Accessed on 07 April 2010.Klein, J. P. and Moes chberger, M. L. (2003), Survival Analysis Techniques for Censored and Truncated Data, 2nd Edition, New York Springer-Verlag.Romeu, Jorge L., (n. d.), Reliability and Advanced Information Technology Research with Alion Science and Technology, Online at , Accessed on 08 April 2010.Samartzis, Lefteris (n. d), Survival and Censored Data, Semester Project, Winter 2005-2006, Online at , Accessed on 08 April 2010. The End