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Most literature on survival models assumes that the data is either a collection of individual survival times or right-censored individual survival times (so you know when some subjects failed but for others only that they have survived to time t).

However - I have some data that looks like this:

T .   Survival (%)

0 .   100
1 .   95
2 .   93
3 .   87

Essentially, just an empirical survival function.

Edit: The total number of subjects $n$ is also known.

How would I go about expressing the likelihood for say, and exponential model ($S(t) = e^{-\lambda t}, f(t) = \lambda e^{-\lambda t}$) for this data?

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You can't derive a likelihood from that. The percent surviving is just that: a percent. You need to know if it's 10 or 100... or 1,000 who are at risk at baseline.

However, if the survival distribution above has no censored observations, you can calculate the area under the survival curve as the mean survival, which immediately tells you what the estimate of $\lambda$ is for those data. Not knowing the weights simply means you can't construct a CI around that value, nor can you actually write down the likelihood.

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  • $\begingroup$ Actually, the total number of subjects is known. Editing into OP for clarity. Knowing that number of initial subjects, can the likelihood be calculated? My goal is to apply a Bayesian inference to this data with a prior on $\lambda$, so I'm mostly wondering if the structure of the data supports that. $\endgroup$
    – LoLa
    Commented Dec 10, 2019 at 17:58

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