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Background  In epidemiological studies, subjects are often followed for a period during which study outcomes are measured at selected time points, such as by diagnostic testing performed on biological samples collected at each visit. Although test results may indicate the presence or absence of a disease or condition, they cannot provide information on when exactly it occurred. Such study designs generate arbitrarily censored time-to-event data, which can include left, interval and right censoring. Adding to this complexity, the data may be clustered such that observations within the same cluster are not independent, such as time to recovery of an infectious disease of family or community members. This data structure is observed when evaluating circumcision's effect on clearance of penile high risk human papillomavirus (HR-HPV) infections using data collected from the male circumcision(MC) trial conducted in Rakai, Uganda, where the multiple infections within individual and HPV testings performed at trial follow-up visits gave rise to the clustered data with arbitrary censoring.    Methods  We describe the use of parametric proportional hazards frailty models and accelerated failure time frailty models to examine the relationship between explanatory variables and the survival outcomes that are subject to arbitrary censoring, while accounting for the correlation within clusters. Standard software such as SAS can be used for parameter estimation.    Results  Circumcision's effect on HPV infection was a secondary end point in the Rakai MC trial, and HPV genotyping was conducted for penile samples of a subset of trial participants collected at enrollment, 6, 12 and 24-month follow up visits. At enrollment, 36.7% intervention arm men (immediate circumcision) and 36.6% control arm men (delayed circumcision at 2 years) were infected with HR-HPV, with the number of infections per man being 1-5. The proposed models were used to examine whether MC facilitated clearance of the prevalent infections. Results show that clearance of multiple infections within each man is highly correlated, and clearance was 60% faster if a man was circumcised.    Conclusions  Parametric frailty models provide viable ways to study the relationship between exposure variables and clustered survival outcome that is subject to arbitrary censoring, as is often observed in HPV epidemiology studies.

Parametric frailty models for clustered data with arbitrary censoring: application to effect of male circumcision on HPV clearance