2.5.2.2 Algorithm: Non-Parametric Distribution Analysis (Right Censoring)

The Right/Exact Censor data are typically represented using two components: a time variable and a censoring indicator. The time variable records the observed duration up to either the event occurrence or the end of observation, while the censoring indicator specifies whether the event was actually observed (exact/uncensored) or whether the observation was censored (right-censored).

Kaplan-Meier method

Let \(n\) be the total number of units whose survival times, censored or not, are available. Relabel the \(n\) survival times in order of increasing magnitude such that \(t_{(1)} \le t_{(2)} \le ... \le t_{(n)}\).

Survival Probabilities

• Cumulative Survival Probabilities

  \[ \hat{S}(t) = \prod_{t_{(r)} \le t} \frac{n-r}{n-r+1} \]

where \(r\) runs through those positive integers for which \(t_{(r)} \le t\) and \(t_{(r)}\) is uncensored.

• Variance of Cumulative Survival Probabilities

  \[Var[\hat{S}(t)] = [\hat{S}(t)]^2\sum_r \frac{1}{(n-r)(n-r+1)} \]

Hazards

• Hazard Estimates

  \[\hat{h}(t)=\frac{1}{n- r +1}\]

where \(r\) corresponds to the largest index that \(t_{(r)} \le t\)

Mean time to failure

• Mean time to failure \(\mu\) equals the area under the estimated survival function. if the times to death are ordered as \(t^{(1)} \le t^{(2)} \le...\le t^{(m)}\) (if there are \(m\) uncensored observations) , then

  \[\hat{\mu} = t^{(1)} + \hat{S}(t^{(1)})(t^{(2)}-t^{(1)})+\hat{S}(t^{(2)})(t^{(3)}-t^{(2)})+...+\hat{S}(t^{(m-1)})(t^{(m)}-t^{(m-1)})\]

• Variance of \(\mu\)

  \[Var[\hat{\mu}]=\sum_r\frac{A^2_r}{(n-r)(n-r+1)}\]

where

• \(A_r\) is the area under curve \(\hat{S}(t)\) to the right of \(t^{(r)}\).

Actuarial estimation method

See Actuarial estimation method

Reference

1. Elisa T. Lee (1992). "Statistical Methods for Survival Data Analysis (3rd Ed.)". John Wiley & Sons, Inc, Chapter 4