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Data issue from a study concerned by the association between diabetes and retinopathy ([DRSRG76]). For each of the 197 diabetic patients, one eye was subject to a laser treatment, the other one remaining untreated, and time to retinopathy was recorded for both eyes, so that two events were studied: retinopathy for the treated eye, and retinopathy for the untreated eye. The patient type of diabetes was furthermore known as juvenile if detected before age of 20 and adult otherwise. Censoring rates according type of diabetes for each event are described in Table (3).

We wished to compare types of diabetes. Let Ti and T2 be the time to blindness for the treated eye and the untreated eye, with respective marginal survival function F ij and F2j in group j (j =juvenile diabetes or adult diabetes). The null tested hypothesis is:

H : I Fi,juvenile = Fi,adult [ F2,juvenile = F2,adult

Results of Ho testing against an unspecified alternative are produced in Table

Table 3. Description of the sample of 197 diabetic patients - sizes and censoring proportions according type of diabetes and eye treatment.

n

Censoring in %

Treated eye

Adult diabetes 83

78.3

Juvenile diabetes 114

68.4

All 197

72.6

Untreated eye

Adult diabetes 83

39.8

Juvenile diabetes 114

55.3

All 197

48.7

Table 4. Test of comparison of marginal survival functions for treated and untreated eyes according type of diabetes with nominal level 5%. p-values are expressed in %; LR: sum of the marginal logrank test statistics; WL: Wei and Lachin's test statistic; WLW: Wei, Lin and Weissfeld' Wald test statistic; K: the proposed test statistic (9).

Test statistic value 5.953 8.146 8.089 7.141 p-value 0.051 0.017 0.018 0.028

We observe that the sum of the marginal logrank statistics p-value is close to 5%, that could question about not rejecting Ho; Wei and Lachin's and Wei, Lin and Weissfeld's test statistics values are equal, and greater than the value of the test statistic (9), as noticed with the simulations study.

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