 |
 |
dear allstatters, i asked a question last month about survival data which are interval censored at both ends. a few replies pointed out that these data are sometimes called doubly censored data - although I think that this can also mean something else to some authors (namely, data for which the origin is interval censored, but the survival time is observed or right-censored). Other replies are summarised below, and the original post is given at the bottom. Ben ------------------------------------------------------------------------ - Luuk Gras suggested two papers of interest: Gruttola and Lagakos: Analysis of doubly-censored survival data, with application to AIDS. Biometrics. 1989 Mar;45(1):1-11 Pan, W. A multiple imputation approach to regression analysis for doubly censored data with applications to AIDS studies. Biometrics 2001; 57: 1245-1250 ------------------------------------------------------------------------ - Ian White suggested a Markov transition model: I guess your observation times are fairly widely spaced, otherwise you wouldn't be worrying. But widely spaced times means that an interval could contain 2 changes - e.g. your example individual could have become false and then true again between t_0 and t_1. So you could try a Markov model for both transitions (true->false and false->true) with constant transition rates - I believe there is software around for this. But you probably don't want to assume constant transition rates. If you're prepared to assume double changes don't happen, then you could treat it as a missing data problem for the first change time. First fit the model for outcome 1 and use it to multiply impute the missing times of first change. Then fitting the model for outcome 3 becomes straightforward. Because estimating the baseline time for outcome 3 depends on the parameters of the model for outcome 1, I don't think you can deal with outcome 3 without bringing in an analysis of outcome 1. A full analysis would use the joint likelihood of outcomes 1 and 3, which involves intergration over both event times & would probably be hard to maximise. ------------------------------------------------------------------------ - : -----Original Message----- : From: Cowling, Ben : Sent: 06 January 2004 15:31 : To: [log in to unmask] : Subject: survival data which are interval censored at both ends : : : dear allstatters : : i have a problem with survival data that are interval : censored at both ends. i am working on a large : observational database, where for each patient i know the : value of a boolean variable X at various times T. The : starting value at baseline is 'True', so the data for one : individual might look like this: : : T X : t_0 True : t_1 True : t_2 False : t_3 False : t_4 True : : The observation times t_0, t_1, ... are irregularly spaced : through time, and are not the same for each individual. : : I am interested in three survival outcomes for these data, : but have so far only been able to look at the first two: : : Outcome 1. the time from t_0 until X becomes 'False' - this : is simple survival analysis with an interval censored : survival time, in this example the outcome occurs in the : interval (t_1, t_2), and i can use SAS or s-plus to fit a : variety of parametric survival models allowing for : left/right/interval censoring. The log-likelihood : contribution for the example individual above would be S(t_2) : - S(t_1), where S(.) represents the survivor function. : : Outcome 2. the time from t_0 until X becomes 'True' after : being 'False' (so this is conditional on X being 'false' at : some point after t_0). Again, this is interval censored : survival data, and SAS or s-plus can be used, although there : is an issue about truncation if individuals who are always : 'True' are excluded. : : Outcome 3. the time from X becoming 'False' until it becomes : 'True' again - the "rebound" time. In this case, I know that : the minimum possible rebound time is (t_3 - t_2), and the : maximum possible rebound time is (t_4 - t_1), but how can i : analyse these data? What about rescaling the time axis to : allow this individual to contribute the term [ S(t_4 - t_1) - : S(t_3 - t_2) ] to the log-likelihood? : : : thanks for any help : : i'll summarise any responses and post to the list. : : : Ben : : : -- : : Ben Cowling : Department of Infectious Disease Epidemiology : Division of Primary Care and Population Health Sciences : Faculty of Medicine : Imperial College London : Norfolk Place : London : W2 1PG : email: [log in to unmask] :