Prof Carla Moreira - The Statistical Analysis of Doubly Truncated Data

Because of the interval sampling, the age at diagnosis is doubly truncated by the pair , where the right‐truncation variable is the time in years from birth (date of onset, ) to 31 December 2003, and . The triplets , , with the values observed for were reported in Moreira and de Uña‐Álvarez (2010), while de Uña‐Álvarez (2020) included the cancer group in the statistical analysis. Ordinary descriptive statistics can be applied to the information gathered along this 5 year long window to compute, for instance, the average age at cancer diagnosis. However, if the goal is to describe the population of children eventually developing cancer, the double truncation issue should be acknowledged and properly corrected, so potential biases are avoided.

Interestingly, the observed values for range between and 14.5 (years); equivalently, the observed values for range between 0.5 and 19.5. This means that the lower and upper endpoints of and satisfy and . Thus, in this case, the target variable is observable on its whole support , and there are no identification issues for , the cdf of . Information on is summarized in Table 1.1.

Table 1.1Descriptive statistics for Childhood Cancer Data: sample size and mean (and standard deviation, SD) for the age at diagnosis (years).

Group			Mean (SD)
All		406	6.47 (4.50)
By gender	Female	178	6.43 (4.51)
	Male	228	6.51 (4.51)
By ICCC Group	Leukemia	107	6.30 (4.15)
	Lymphoma	57	8.66 (4.39)
	N. System Tumour	94	6.38 (4.29)
	Neuroblastoma	38	3.16 (3.47)
	Other	105	6.87 (4.70)
	Missing	5	3.92 (5.18)

This dataset is used in Chapters 2, 3and 5and is accessible in the DTDApackage in ChildCancer.

Kalbfleish and Lawless (1989) reported 494 cases of transfusion‐related AIDS, corresponding to individuals diagnosed prior to 1 July 1986 ( ). The variable of ultimate interest is the induction or incubation time, which is the time elapsed from HIV infection to AIDS. Importantly, HIV was unknown before 1982 ( ); this implies that cases developing AIDS prior to this date were not reported. Let denote the time from HIV infection to 1 July 1986 (in months), and introduce ; then, due to the interval sampling, only triplets satisfying were observed (Bilker and Wang, 1996). We restrict our analysis to the cases with consistent data, for which the infection could be attributed to a single transfusion or a short series of transfusions. This dataset is fully reported in Kalbfleish and Lawless (1989), p. 361.

The observed values of range from 0.5 to 89 (months), while ranges from to 45.5. This suggests that the lower limit of the support of is about , while the upper limit of the support of is about 99.5. As discussed in Chapter 2, in such a case the distribution of the incubation time is identifiable on the interval (months). The AIDS Blood Transfusion Data also includes information on the age of the individual at infection; see Table 1.2.