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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

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

$\approx \sum_{t_j\leq t} P(t_{j-1}<T\leq t_j,cause=i)$

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

where last line uses $P(t_{j-1}<T\leq t_{j},cause=i)=P(t_{j-1}<T\leq t_j,cause=i|T>t_{j-1})P(T>t_{j-1})$ and $h_i(t_{j})\approx \frac{P(t_j<t<t_j+(t_{j+1}-t_j),cause=i)}{(t_{j+1}-t_j)}$.

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?

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I think that the confusion arises when you move from the continuous-time representation of the cumulative probability to the necessarily discrete-time data analysis needed to estimate that probability.

For the Kaplan-Meier type of competing-risk analysis being illustrated in that section of the book, all that matters is the ordering of event times. Unlike for parametric survival models, the actual times don't fundamentally matter for these non-parametric analyses or for related Cox semi-parametric regressions. Whether there is 1 day or one year between, say, the 29th and the 30th event in order doesn't enter into the calculation.

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.

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  • $\begingroup$ 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? $\endgroup$
    – user45765
    Aug 5, 2021 at 19:30
  • $\begingroup$ @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. $\endgroup$
    – EdM
    Aug 5, 2021 at 19:57

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