Overview of Recent Developments for Interval-Censored Data - Interval-Censored-Time-to-Event Data: Methods and Applications - page 8

information for T. In other words, the probabilistic behavior of T remains
the same except that the original sample space T 0 is now reduced to
l = L < T R = r. If we use the other formulation, it simply means that
T is independent of U and V , which is clearly much easier to interpret and
to formulate than the first one. In practice, one question of interest is to
establish or derive the conditions under which the independent assumption
holds. It is clear that this is not easy or straightforward with respect to the
first format and for this, several authors described the so-called constant-sum
condition (Oller et al. (2004); Betensky (2000); Oller et al. (2007)). On the
other hand, it is relatively easier with respect to the second format. One can
find more discussion on this and some other formulations of the Interval-
censoring mechanism in Chapter 2.
As mentioned above, interval-censored data include right-censored data as
a special case. On the other hand, there exist several major differences be-
tween the two types of failure time data. One is that it is obvious that the
latter provides much more information than the former about the failure time
T. Also it is easy to see that the latter has a much simpler structure than
the former. For the asymptotic study of a statistical procedure developed for
right-censored data, an important tool is the counting process approach as
the data can be represented by the counting process formulation (Anderson
et al. (1995)). On the other hand, the counting process formulation and ap-
proach cannot be directly used for interval-censored data, which makes the
asymptotic study of the statistical procedures developed for interval-censored
data much harder or impossible. With respect to the censoring mechanism,
for the latter, there is only one variable, the right-censoring time variable, in-
volved although there exist different ways in which it is generated (Kalbfleisch
and Prentice (2002)). To describe the mechanism behind interval-censoring or
interval-censoring structure, in general, one needs two variables and as dis-
cussed above, there also exist different ways for their generation.