Introduction to censoring and truncation

Q1: Censoring? Truncation？

1.Censoring 删失: 由于某些随机原因导致观察没有完成，而这些导致删失的原因必须和我们感兴趣的事件所独立【比如关心癌症病人生存时间，某个志愿者因为车祸等其他随机原因死亡，他的死亡发生和我们关心的肺癌无关，所以属于删失】；

删失类型:
(1). 右删失(right censoring):对象参与研究直到某个时间,且关心的事件仍未发生，但是没有继续参与观察。
根据没有继续参与观察的原因右删失又分为Ⅰ/Ⅱ/Ⅲ三种类型:
Ⅰ型右删失:由于观察时间和费用原因，某些个体在观测期结束还未发生关心事件,这种称为Ⅰ型(定时)右删失；
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Ⅱ型右删失:约定观察到特定数目r个个体发生关心事件时终止研究，比如设备测试,约定直到n个设备中有r个设备失效时终止测试；
Ⅲ型右删失:试验个体因为其他原因而撤离试验，这种被称作Ⅲ型右删失，又称作 随机删失.
(2). 左删失:研究对象在时刻 $t$ 时开始接受观察，但是我们关心的事件在此之前就已经发生了,这种称为左删失，比如调查第一次吸食大麻时间，被调查者在被调查时说吸食过.但是记不清具体时间了，这种就是典型的左删失。
(3). 区间删失:仅知道存活时间存在于某种时间内，但是不知道精确时间，这种称为区间删失。比如我们只知道发生心绞痛的年龄是在近两年内，这种就是区间删失。

2.截尾(Truncation): 截尾是删失的一种变体，它与删失本质的不同是导致数据不完全的原因是系统性的而非随机性的，比如人为淘汰一些个体。

截尾类型： 左截尾、右截尾 $\to$
(1).左截尾: 比如要进入退休中心必须 $>$ 65岁，这个65岁就是系统性的截尾时间，此时观察者是一定不可能获取没有达到65岁人群的特定情况的，这种就是典型左截尾情形；
• Left truncation occurs when, due to the structure of the study design, we can only observe those individuals whose event time is greater than
some truncation threshold.
(2).右截尾: 比如对于已经感染艾滋病毒但未发病的个体研究者是无法把他们纳入到观察对象中的，注意到这些人的发病是一定会发生的，但是由于这个病的系统性特性，无法获取他们的信息。
• Right truncation is when only individuals with event times less than some threshold are included in the study.

3.左截尾与左删失的区别： 在左截尾的研究中，根本就没有考虑那些不在范围之内的样本个体，比如我们根本就不考虑年龄小于65岁的人，这是系统性选择的；
左删失是会关注这些个体的信息的，这些个体无法被观测是因为随机原因造成的，但我们仍会去考虑这些个体。

Q2.Examples of censored data?

1.Patient A takes part in the study until her death at age $T_A$ . Her survival time is uncensored.------uncensored

2.Patient B takes part in the study until age $T_B$ . He then leaves the study. His survival time is censored: we know it is at least $T_B$ but we don’t know it precisely.-------Ⅲ型右删失

3.Patient C takes part in the study until age $T_C$ . She then is hit by a car and dies. Her survival time with regard to the event of interest, namely death through lung cancer, is also censored: we know it is at least $T_C$ but the unfortuitous accident has censored it.------Ⅲ型右删失

4.Leukæmia patients are given a drug or a placebo. Survival time is the
duration from remission to relapse. The study ends at 52 weeks with
some patients yet to relapse.------Ⅰ型右删失

5.The age at which children are able to count from 1–10 at school. Some children are already able to count before joining school.------
Left censoring—times to the left of the data collection period are not
known precisely

6.Death times at a retirement community------Left truncation—people who died before going to the community do not enter the study at all.

7.Early in the AIDS outbreak, patients with AIDS were recruited to
study the time from infection with HIV to development of AIDS. At
the time of the study, many people were infected