How do left-censored data and right-censored data impact Cox Regression? My application is not a traditional survival analysis scenario. However, I believe survival analysis methods, e.g., Cox regression, can be a possible solution. In particular, my dataset contains two kinds of data: 


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*Left-censored: For example, I know a patient has been dead at time t8. But the exact time point that the patient died is unknown. Also, in the nature of the problem, it is impossible to know.

*Right-censored: Similar in the traditional case. For example, I know a patient was alive from t0-t6. But it is unknown when the patient died after t6.


In both case, the exact "death point" is unknown. My data only contains these two cases.
I plan to use widely used survival analysis packages (e.g. lifelines) to solve this problem:


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*Left-censored: For the example above, I will label it as "duration: t8, event:1". (note that t8 is NOT the exact time that the patient dies)

*Right-censored: For the example above, I will label it as "duration: t6, event 0".


My questions:


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*Does using Cox regression make any sense?

*How does these two types of data impact the final model? Say, under-estimate/over-estimate the baseline hazard?

*Are these any other models can better handle this case, instead of Cox regression?

 A: EDIT: see AdamO's answer as well. 
unfortunately that solution won't work. One can't encode left-censorship and right censorship together like that. Lifelines currently (as of v0.14.1) doesn't have an interval censorship model, so you'll have to use something like the solution here: https://stats.stackexchange.com/a/198748/11867
A: For left censored data, they are excluded from follow-up at the point of censoring and coded as a non-event. That is in fact the definition of censoring. Also, you need to know what "time 0" is. So if you don't understand that, you can't include right censored data. You will get different results if the time under observation goes from 0 to 6 as compared to 12 to 18. The last consideration is that you can't just sample observations on the basis of being failures. It will inflate the incidence and bias the hazard ratios. Cox regression works, but I think you need a clearer understanding of how censoring is encountered. A practical description of the data is warranted.
