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This is a question to help me understand how the Nelson-Aalen is computed in practice. There is probably something simple that I don't see...

Let $\{T_j\}_{j=1}^n$ be survival times, and $\{T_j^*\}_{j=1}^n$ censoring times, so that the death of individual $j$ is observed if and only if $T_j<T_j^*$. And let $Z(t)=\sum\limits_{j=1}^n\mathbb 1_{t\le \tilde T_j}$ be the function counting the number of individuals at risk (uncensored and not dead).

The Nelson-Aalen estimator (assuming $\mathbb P[T_i=T_j, i\ne j]=0$, i.e., no simultaneous deaths) is defined as $$\hat A(t)=\sum\limits_{T_j\le t} \frac{1}{Z(T_j)}$$

where the sum is over all deaths that happen up to time $t$.

Suppose we have censored data of the type $\{(\tilde T_j, D_j)\}_{j=1}^n$, where $\tilde T_j=\min(T_j, T_j^*)$ and $D_j=\mathbb 1_{T_j\le T_j^*}$ ($D_j=1$ if and only if individual $j$ dies before being censored).

How can we compute the Nelson-Aalen estimator from this data if we don't have access to $\{T_j\}$?

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The answer is the same as for constructing Kaplan-Meier curves for survival versus time with censoring. Cases that are censored contribute to the calculation (in this case, the denominators in the Nelson-Aalen formula) up to the censoring time. In your terminology, if case $k$ is censored, then that case is included in $Z(t)$ through its censoring time $T_k^*$. After that case $k$ is ignored, so you don't require information about $T_k$. Thus you combine information from the cases for which you know $T_j$ values with as much as you can get from the cases for which you only know $T_j^*$

For those who come later to this question, note that the formula in the question is for a single cohort. An extension of this Nelson-Aalen formula, with $Z(t)$ representing a risk-weighted sum, is used for example to estimate the baseline cumulative hazard in a Cox regression model.

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