Statistics and Its Interface Volume 14 (2021) 131–149 Lead time distribution for individuals with a screening history Ruiqi Liu, Dongfeng Wu ∗† , and Shesh N. Rai We derived the distribution of lead time for periodic screening in the future when an individual has a screening history with negative results. It is a mixture of a point mass at zero and a positive sub-PDF. The motivation comes from the reality that for people in older age, they may already have some screening exams for targeted cancer before and still look healthy and are asymptomatic at their current age. How to evaluate their future screening result is a challenge. We explored how the screening history would affect the lead time if one would be diagnosed with cancer in the future. Simulations were carried out on combinations of different initial screening age, current age, sensitivity, mean sojourn time, and screening schedule in the past and in the future. The method developed can be applied to periodic exams for any kind of chronic disease, such as cancer. We applied our new method of evaluating the lead time distribution for male and female heavy smokers using low-dose computed tomography in the National Lung Screening Trial. 1. INTRODUCTION Cancer is a group of diseases involving abnormal cell growth with potential to invade or spread to other parts of the body. Most types of cancer can be described using staging; stage I means that the cancer is small and has not grown deeply into nearby tissues, while stage IV means the cancer has spread to other parts of the body. Cancer stage at diagnosis helps determine which treatment is available and corresponding survival time. In general, patients with early-stage cancer have better prognosis and higher survival rate than those with late-stage cancer. Specifically, the 5- year survival rate for patients with early-stage lung cancer is approximately 50%, while it is only about 5% for stage IV lung cancer patients [1]. As the primary technique for early detection, the goal of screening is to detect the disease earlier before any symp- toms appear; so patients may receive earlier intervention and better treatment. Periodic screening is recommended for al- most all kinds of cancers, such as breast, lung, colon, cervical cancer, etc. [2] Several major randomized controlled cancer screening studies have been carried out since the 1960s: the ∗ Corresponding author. † ORCID: 0000-0002-6764-0083. Health Insurance Plan of Greater New York Project [3]; the Mayo Lung Project [4]; the Johns Hopkins Lung Project [5]; the Minnesota Colon Cancer Control Study [6]; the Prostate, Lung, Colorectal and Ovarian (PLCO) cancer screening trial [7] and the National Lung Screening Trial (NLST) [8, 9]. Early detection may mean more treatment choices and longer survivals for patients. However, since survival time is measured from the time of diagnosis, it could appear longer for screen-detected cases and screening may not truly con- tribute to overall survival. Lead time is the time interval between the time of early diagnosis using a screening exam and the time a clinical diagnosis would have been made with- out a screening. Therefore, to correctly estimate the survival time of screen-detected cases, it is critical to estimate the lead time first, and then subtract it from the overall survival. Hence, the lead time is an important factor when evaluating the effectiveness of a screening program [10]. A number of statistical methods were provided to esti- mate the mean and variance of lead time [11, 12, 13, 14]. Prorok [15] estimated the local lead time by focusing on the i-th screen-detected cases whose lead time is positive. However, he ignored the interval-incident cases, whose lead time is zero. Wu et al. [10] estimated the lead time for both screen-detected and interval-incident cases, where a person’s lifetime is treated as a fixed value. This model was applied to the Mayo Lung Project data to estimate the lead time when human lifetime was assumed to be 80 years. Later, Wu et al. [16] extended the model to make it more practical by treat- ing the lifetime as random, deriving its distribution from the actuarial life table of the US Social Security Adminis- tration [17]. The lead time for lung cancer screening using chest X-ray was estimated previously, when the lifetime was either fixed [18] or random [19]. Considering the advantage of low-dose helical computed tomography (LDCT) over tra- ditional chest X-ray, Liu et al. [20] estimated the lead time distribution for both genders using LDCT when the lifetime is random in the National Lung Screening Trials. All of the above methods were developed based on the assumption that an asymptomatic individual has not taken any screening exams at his/her current age, that is, there is no screening history. While in reality, participants aged 55 and older may already have had at least one (previous) screening exam in the past and look healthy right now. In this paper, we will introduce a lead time distribution model which can incorporate one’s screening history and derive the