Question

Twenty units were placed on stress. Readouts were done at the end of each week at 168,332 , 504,672,840, and 1008 hours. The failure counts observed at the readouts were 0,4,5,3,2, and 2 , respectively. At the end of weeks 2,3,4, and 5, one randomly chosen surviving unit was removed for failure analysis. Using the Kaplan-Meier PLE for interval data, estimate the CDF at the observed failure times. Estimate the standard error of the CDF estimate using Greenwood's formula. Provide 90% pointwise confidence intervals at each readout using the normal approximation. Redo the confidence interval estimates by applying the log and logistic transformations

Answer 1

The student's question involves calculating the Cumulative Distribution Function using the Kaplan-Meier Product-Limit Estimator, estimating the standard error with Greenwood's formula, and constructing 90% confidence intervals using a normal approximation with log and logistic transformations.

Step-by-step explanation:

The student is asking about the Kaplan-Meier Product-Limit Estimator (PLE) which is used for estimating the Cumulative Distribution Function (CDF) of time-to-event data, in this case, failure times of units under stress. The process involves calculating survival probabilities at each observed time point, adjusting for censored data, and then obtaining the estimate of the CDF at those points. Greenwood's formula provides an estimate of the variance of the Kaplan-Meier estimator, which can be used to construct confidence intervals for the CDF.

To perform the analysis, we work through the failure times and compute the survival probability at each time point. For each interval, we use the number of units at risk at the beginning of the interval and the number of events (failures) that occurred. Greenwood's formula uses these values to estimate the standard error of the Kaplan-Meier estimator at each time point. Pointwise confidence intervals can then be constructed using the normal approximation, and adjusted using log and logistic transformations to ensure they stay within appropriate bounds (e.g., 0 to 1 for probabilities).

For 90% pointwise confidence intervals, a Z-score corresponding to the 90% confidence level is used along with the estimated standard error to find the upper and lower bounds of the interval. When performing the log and logistic transformations, the interval is transformed, the confidence interval is calculated, and then the interval is back-transformed to the original scale to give bounds for the estimate.