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Let $T_i$ is the survival time for individual $i$ $(i=1,2,\ldots, n)$ and $C_i$ be the time to censoring. Let $U_i=\min(T_i,C_i)$. And $\hat S(U_i)$ is the Kaplan-Meier estimator for the censoring distribution. Suppose $R_i$ and $Z_i$ are two indicator functions. Also, $p$ is a probability. Consider the following estimator of cumulative distribution function:

$$\hat F(t)= \sum_{i=1}^{n}\frac{I(T_i<C_i)(1-R_i+R_iZ_i/p)I(U_i\le t)}{\hat S(U_i)},$$ where $I(.)$ is an indicator function.

Now it is written that, with no censoring $\hat F(t)$ becomes

$$\hat F(t)= \sum_{i=1}^{n}(1-R_i+R_iZ_i/p)I(T_i\le t).$$

I understand that if there is no censoring, then $I(T_i<C_i)=1$, that is, we will always observe the survival time. Also, with no censoring $U_i=T_i$ and hence $I(U_i\le t)=I(T_i\le t)$.

But I do not understand why does Kaplan-Meier estimator for the censoring distribution, $\hat S(U_i)$, which appears in the above first equation vanish in the second equation with no censoring?

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  • $\begingroup$ Recall, KM and most survival methods are based on unique failure times, as the informativeness for survival comes from failures only -- not censored. Also, each object (patient, mouse, etc) contributes to a failure variable (0-censored, 1-failed) and a time variable [$t$=time to failure or time to censoring (end of study or loss to follow-up)]. If you look at the product limit variant of KM, there's not a censoring time and a failure time, but rather bin counts for failures as a function of time interval. $\endgroup$ – JoleT Dec 18 '16 at 1:57
  • $\begingroup$ Can you give a source to where you found this expression? I am not following the meaning of $R$ and $Z$. In Fine/Gray JASA 1999 Proportional Hazards Model for the Subdistribution of a Competing Risk, $Z$ is the modeled effect of covariates and $R$ is a cause-specific indicator matrix. You might want to start with a simpler case. $\endgroup$ – AdamO Jun 17 '18 at 13:05
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The Kaplan-Meier Curve does not disappear when there is complete data. The true survival function is S(t)=1-F(t). The Kaplan-Meier is also called the product limit estimator. If you look it up in wikipedia you will find the case of complete data and the form of the KM curve as a product.

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  • $\begingroup$ Does it mean the second equation of the post inaccurate? $\endgroup$ – user 31466 Dec 18 '16 at 2:29
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    $\begingroup$ There should only be one censoring time C in the study and that is the time you stop to finalize the result. In the complete data case all event times are less than C. So the indicator functions should all equal 1. $\endgroup$ – Michael R. Chernick Dec 18 '16 at 15:24
  • $\begingroup$ I now see how the substitutions for the complete case that you exhibit in equation 2 loses the survivor function in the denominator but I don't know what the indicator functions R_i and Z_i represent. Also what does the probability p refer to? $\endgroup$ – Michael R. Chernick Dec 18 '16 at 15:27
  • $\begingroup$ Were you able to find the formula for the KM estimator in wikipedia? $\endgroup$ – Michael R. Chernick Dec 18 '16 at 15:29
  • $\begingroup$ Go to en.wikipedia.org/wiki/Kaplan-Meier-estimator . $\endgroup$ – Michael R. Chernick Dec 18 '16 at 15:36

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