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The exact distribution of overlapping counting patterns associated with a sequence of homogeneous Markov-dependent multi-state trials: An application to SECON. (English) Zbl 0901.60059
The distribution of the number of non-overlapping consecutive \(k\) successes \((N_{n,k})\) in a sequence of \(n\) Bernoulli trials has been widely studied and applied to many practical situations. For instance, the probability of \(N_{n,k} =0\) is the reliability of a consecutive-\(k\)-out-of-\(n:F\) system, and the tail probability of \(N_{n,k} <m\) is the reliability of an \(m\)-consecutive-\(k\)-out-of-\(n:F\) system. When the counting method is overlapping permissible and the sequence restrictions relax to the case of homogeneous Markov-dependent multi-state trials, then the resulting distribution problem of \(N_{n,k}\) can be used to model a variety of real-world processes, but such cases have not been investigated in the literature. In this article, we present some results based on the finite Markov chain imbedding technique for the exact distribution of \(N_{n,k}\) with \(k=2\), and focus on its application to an important measurement of continuity in the health-care sector called SECON. Here continuity means that a patient sees the same health-care provider (physicians or nurse) on two consecutive visits. The objective of this article is to present new mathematical formulations for and to link the medical care system to consecutive-\(k\)-out-of-\(n\) systems.
MSC:
60K10 Applications of renewal theory (reliability, demand theory, etc.)
90B25 Reliability, availability, maintenance, inspection in operations research
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[1] Bollinger, R.C., Discrete computation for consecutive-k-out-of-n: F system, IEEE transactions on reliability, R31, 444-446, (1982) · Zbl 0502.90040
[2] Chao, M.T.; Fu, J.C., The reliability of large series systems under Markov structure, Advances in applied probability, 23, 894-908, (1991) · Zbl 0795.60082
[3] Chao, M.T.; Fu, J.C.; Koutras, M.V., A survey of reliability studies of consecutive-k-out-of-n: F systems and related systems, IEEE transactions on reliability, R44, 120-127, (1995)
[4] Fu, J.C.; Lou, W.Y., On reliabilities of certain large linearly connected engineering systems, Statistics & probability letters, 12, 291-296, (1991) · Zbl 0736.60083
[5] Papastavridis, S.G., A limit theorem for the reliability of a consecutive-k-out-of-n: F system, Advances in applied probability, 19, 746-748, (1987) · Zbl 0626.60086
[6] Papastavridis, S.G., m-consecutive-k-out-of-n: F systems, IEEE transactions on reliability, R39, 386-388, (1990) · Zbl 0721.60101
[7] Fu, J.C.; Lou, W.Y.W., On the exact distribution of SECON and its application, () · Zbl 1053.62511
[8] Fu, J.C.; Koutras, M.V., Distribution theory of runs: a Markov chain approach, Journal of the American statistical association, 89, 1050-1058, (1994) · Zbl 0806.60011
[9] Rogers, J.; Curtis, P., The concept and measurement of continuity in primary care, American journal of public health, 70, 122-127, (1980)
[10] Blendon, R.J.; Knox, R.A.; Brodie, M.; Benson, J.M.; Chervinsky, G., Americans compare managed care, medicare, and fee for service, Journal of American health policy, 4, 42-47, (1994)
[11] Jellinek, M.S.; Nurcombe, B., Two wrongs don’t make a right: managed care, mental health, and the marketplace, Jama, 270, 1737-1739, (1993)
[12] Breslau, N.; Reeb, K., Continuity of care in a university-based practice, Journal of medical education, 50, 965-969, (1975)
[13] Ejlertsson, G.; Berg, S., Continuity-of-care measures: an analytic and empirical comparison, Med-ical care, 22, 231-239, (1984)
[14] Eriksson, E.A., Continuity-of-care measures: random assignment of patients to providers and the impact of utilization level, Medical care, 28, 180-190, (1990)
[15] Eriksson, E.A.; Mattsson, L.G., Quantitative measurement of continuity of care, Medical care, 21, 858-875, (1983)
[16] Steinwachs, D.M., Measuring provider continuity in ambulatory care, Medical care, 17, 551-565, (1979)
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