# Nelson-Aalen estimator for cumulative hazard

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 $$T_j. And let $$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 $$\{T_j\}$$?

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.