# Estimator of failure by cause j at time t

I am reading Dirk Moore's Applied Survival Analysis Using R page 124.

Let $$S(t)$$ be cumulative survival curve of population facing cause of death due to a set of cause $$\{1,2,\dots,n\}$$ where $$1,2,\dots, n$$ are abstraction of causes by labeling.

Let $$h_i(t)$$ be hazard rate of cause $$i$$ at time $$t$$.

Then cumulative probability of a person dying due to cause $$i$$ is $$F_i(t)=P(T\leq t,cause=i)=\int_0^tdu h_i(u)S(u)$$.

Given D distinct failure times, $$t_1,\dots, t_n$$ and cause of death for each failure, one can estimate $$\hat{h}_i(t_j)$$ and $$\hat{S}(t_i)$$.

"The probability of failure from cause $$k$$ at time $$t_i$$ is the product of $$\hat{S}(t_{i-1})$$ and $$h_k(t_{i})$$ risk of dying from cause $$k$$. The probability of death due to cause $$k$$ to failure at time $$t$$(i.e. $$F_i(t)$$'s estimator) is given by $$\sum_{t_i\leq t}\hat{S}(t_{i-1})\hat{h}(t_i)$$."

Why the $$du$$ part is missing here? Technically, I would agree with $$\sum_{t_i\leq t}\hat{S}(t_{i-1})\hat{h}(t_i)(t_i-t_{i-1})$$ as an estimator of $$F_i(t)$$. The other issue of that sum estimator is that it does not have correct scaling dimension. If I scale time by a factor, that expression will scale inversely to that factor whereas $$F_i(t)$$'s integral does not care about that scaling.

My approach is to consider

$$P(T\leq t, cause=i)$$

$$\approx \sum_{t_j\leq t} P(t_{j-1}

$$\approx \sum_{t_j\leq t}h_i(t_{j})S(t_{j-1})(t_j-t_{j-1})$$

where last line uses $$P(t_{j-1}t_{j-1})P(T>t_{j-1})$$ and $$h_i(t_{j})\approx \frac{P(t_j.

It seems that the other source also indicates that estimator without $$(t_j-t_{j-1})$$ is correct as in https://www.publichealth.columbia.edu/research/population-health-methods/competing-risk-analysis

What is wrong with my thought process?

In your terminology, you thus can always specify $$(t_j - t_{j-1})=1$$ during the analysis, and your favored formula becomes that in the book. When you display the results you just go back to the original time scale.
• I think the biggest issue is that we may not have $t_j-t_{j-1}=1$ in real data set. Somehow the integral formula given does not remember the scale of time here whereas the given estimator formula does remember time scale. I do agree that time scale does not matter for KM curve construction other than some indictor function depending upon time multiplied. The difference of time always carries a unit of time. I think there is something hiding behind the scene here which makes that formula unbiased. Any reference on this construction? Aug 5, 2021 at 19:30
• @user45765 you also have to remember that, with data analyzed via non-parametric or semi-parametric models, the hazard estimate $\hat h_i(t_j)$ is identically 0 except at event times, and estimated survival $\hat S(t)$ is constant between events. So the duration between events doesn't matter, just the situation immediately before and at an event time.