Solution. (a) Origin: randomisation (or the first heart attack); event: a second heart attack; censoring: end of the trial, loss to follow-up, death from other causes (a competing risk). (b) Origin: switching on; event: failure; the \(30\) survivors are censored at the time of the \(20\)th failure, which is Type II censoring. (c) Origin: graduation; event: starting a job; censoring: end of the survey, emigration, further study (if not counted as the event).

Solution. \((2.1, 1)\), \((4.7, 0)\), \((2.8, 1)\), \((1.4, 0)\), \((3.5, 0)\), \((2.2, 1)\), \((0.7, 1)\), \((2.0, 0)\). There are \(4\) deaths, and the total time at risk is \(2.1 + 4.7 + 2.8 + 1.4 + 3.5 + 2.2 + 0.7 + 2.0 = 19.4\) years, giving an estimated death rate of \(4/19.4 = 0.21\) a year if the rate is constant.

Solution. The censoring is informative: the patients who leave are at higher risk than those who stay. Methods that assume non-informative censoring treat them as if they had the same future as the remaining patients, so the estimated survival is too high. Where possible, such patients should be traced, or the reason for leaving analysed as an event in its own right.

Solution. Left censoring: a child already walks at the first visit, so the age at first walking is only known to be below the age at that visit. Interval censoring: a tumour is absent at one scan and present at the next, so its onset lies between the two dates. Left truncation: a study of the elderly enrols people at age \(65\), so anyone who died before \(65\) cannot appear, and each subject’s time at risk starts at their age at entry.

Solution. Being uncensored is itself related to the survival time: a subject is uncensored only if \(T \leq C\), which is more likely when \(T\) is short. Restricting to uncensored subjects therefore selects short survivors. The mean of all recorded times includes the censored subjects at their (too short) censoring times, which are on average longer than the deaths’ times, so it is less biased, though still badly.

Solution. Type I censoring at \(1000\) hours. The total time at risk is \(210 + 430 + 560 + 790 + 940 + 7 \times 1000 = 9930\) hours, so the rate is \(5/9930 = 0.000504\) per hour, or \(0.50\) per \(1000\) hours, and the estimated mean life is \(9930/5 = 1986\) hours.