# A short note on Nelson-Aalen

Hi All

Last time I made a post deriving the variance of the Kaplan Meier survival function. I had intended to also give the Nelson Aalen estimate as well buts its variance is much harder to get. This post just highlights the difficulties that I had.

The Nelson Aalen estimate is $\hat{S}(t) = \prod_{j=1}^{k} \exp(-d_j/r_j)$. It feels a bit random to me, however David Collett points out that $\exp(-x) = 1 - x + x^2/2 - x^3/6 ...$ so for $x << 1 \exp(-x) \approx 1-x$ $\hat{S}(t) = \prod_{j=1}^{k} \exp(-d_j/r_j) \approx \prod_{j=1}^{k} )(1-d_j/r_j) \approx \prod_{j=1}^{k} )(r_j-d_j)/r_j$ which is the Kaplan Meier estimate.

Collett give the variance of the Nelson Aalen estimate as $Var[\hat{S}(t)] = \hat{S}(t)^2 \sum_{j=1}^k (d_j/r_j^2)$. I have been unable to verify the formula. If the same logic as my perfidious post is applied the variance of $var(\log[\hat{S}(t)])$ can be found to be $var(\log[\hat{S}(t)]) = \sum_{j=1}^{k} var[-d_j/r_j]$ and using everyone’s favorite identity ($var[g(x)] \approx [(d/dx) g(x)]^2 var(x)$) $var(\log[\hat{S}(t)])$ can also be found to be $var(\log[\hat{S}(t)]) =\hat{S}(t)^{-2} var[\hat{S}(t)]$. Putting to two sides together $\hat{S}(t)^{-2} var[\hat{S}(t)] = \sum_{j=1}^{k} var[-d_j/r_j]$ which is solved for $var[\hat{S}(t)]$ to give $var[\hat{S}(t)] = \hat{S}(t)^{2} \sum_{j=1}^{k} var[-d_j/r_j]$. So for $Var[\hat{S}(t)] = \hat{S}(t)^2 \sum_{j=1}^k (d_j/r_j^2)$ to be true $\sum_{j=1}^{k} var[-d_j/r_j] = \sum_{j=1}^k (d_j/r_j^2)$. I don’t see any clear reason why this is the case.

That is it for now. Tune in next time for hopefully some answers instead of just expositions on my inadequacies.