Bo Henry Lindqvist

On the Statistical Modeling and Analysis of Repairable Systems

We review basic modeling approaches for failure and mainte- nance data from repairable systems. In particular we consider imperfect re- pair models, defined in terms of virtual age processes, and the trend-renewal process which extends the nonhomogeneous Poisson process and the renewal process. In the case where several systems of the same kind are observed, we show how observed covariates and unobserved heterogeneity can be included in the models. We also consider various approaches to trend testing. Modern reliability data bases usually contain information on the type of failure, the type of maintenance and so forth in addition to the failure times themselves. Basing our work on recent literature we present a framework where the ob- served events are modeled as marked point processes, with marks labeling the types of events. Throughout the paper the emphasis is more on modeling than on statistical inference.

Breast cancer tumor growth estimated through mammography screening data

IntroductionKnowledge of tumor growth is important in the planning and evaluation of screening programs, clinical trials, and epidemiological studies. Studies of tumor growth rates in humans are usually based on small and selected samples. In the present study based on the Norwegian Breast Cancer Screening Program, tumor growth was estimated from a large population using a new estimating procedure/model.MethodsA likelihood-based estimating procedure was used, where both tumor growth and the screen test sensitivity were modeled as continuously increasing functions of tumor size. The method was applied to cancer incidence and tumor measurement data from 395,188 women aged 50 to 69 years.ResultsTumor growth varied considerably between subjects, with 5% of tumors taking less than 1.2 months to grow from 10 mm to 20 mm in diameter, and another 5% taking more than 6.3 years. The mean time a tumor needed to grow from 10 mm to 20 mm in diameter was estimated as 1.7 years, increasing with age. The screen test sensitivity was estimated to increase sharply with tumor size, rising from 26% at 5 mm to 91% at 10 mm. Compared with previously used Markov models for tumor progression, the applied model gave considerably higher model fit (85% increased predictive power) and provided estimates directly linked to tumor size.ConclusionScreening data with tumor measurements can provide population-based estimates of tumor growth and screen test sensitivity directly linked to tumor size. There is a large variation in breast cancer tumor growth, with faster growth among younger women.

The Trend-Renewal Process for Statistical Analysis of Repairable Systems

The most commonly used models for the failure process of a repairable system are nonhomogeneous Poisson processes, corresponding to minimal repairs, and renewal processes, corresponding to perfect repairs. This article introduces and studies a more general model for recurrent events, the trend-renewal process (TRP). The TRP is a time-transformed renewal process having both the ordinary renewal process and the nonhomogeneous Poisson process as special cases. Parametric inference in the TRP model is studied, with emphasis on the case in which several systems are observed in the presence of a possible unobserved heterogeneity between systems.

Mathematical and statistical methods in reliability

Reliability Theory in the Past and Present Centuries General Aspects of Reliability Modelling Reliability of Networks and Systems Stochastic Modelling and Optimization in Reliability Modelling in Survival and Reliability Analysis Statistical Methods for Degradation Data Statistical Methods for Maintained Systems Statistical Inference in Survival Analysis Software Reliability Methods.

Association of probability measures on partially ordered spaces

A notion of association of probability measures on partially ordered (Polish) spaces is introduced and its basic properties are investigated. This generalizes the classical notion of association among random variables, due to Esary, Proschan, and Walkup. The relation between association and monotone stochastic kernels is investigated and a theorem of Jogdeo is generalized. The general theory is applied to stochastic processes with both discrete and continuous time parameter and partially ordered state spaces. Also, an application to mixtures of statistical experiments is included.

TTT-based tests for trend in repairable systems data

Abstract A major aspect of analysis of failure data for repairable systems is the testing for a possible trend in interfailure times. This paper reviews some important and popular graphical methods and tests for the nonhomogeneous Poisson process model. In particular, the total time on test (TTT) plot is considered, and trend tests based on the TTT-statistic are motivated and derived. In particular, a test based on the Anderson-Darling statistic is suggested. The tests are evaluated and compared in a simulation study, both with respect to the achievement of correct significance level and rejection power. The considered alternatives to ‘no trend’ are the log-linear, power law and a class of bathtub-shaped intensity functions. The simulation study involves single systems, as well as the case where several independent systems of the same kind are observed.

Variability in methotrexate serum and cerebrospinal fluid pharmacokinetics in children with acute lymphocytic leukemia: relation to assay methodology and physiological variables

Methotrexate (MTX) steady state concentrations were evaluated in 42 children who had received high-dose infusions (6-8 g/m2) for acute lymphocytic leukemia. Concentrations in serum and cerebrospinal fluid (CSF) measured by immunoassay were found to be highly variable. Reanalysis by a reference high-pressure liquid chromatography method ruled out analytical factors as a source of this variability. The correlation coefficient between the analytical methods was 0.77 for the serum data and 0.88 for the CSF data. The variability of serum and CSF concentrations was higher in younger patients (serum; P = 0.05 and CSF; P = 0.18), and the CSF concentration decreased with decreasing age and in later courses. Body surface area, body mass index, weight, and gender were not significantly related to MTX variability. We conclude that the pronounced pharmacokinetic variability seen during MTX infusions remains largely unexplained.

Estimating mean sojourn time and screening test sensitivity in breast cancer mammography screening: new results.

Objective: To assess if new screening techniques, increased use of hormone replacement therapy, or the transition from breast cancer screening trials to large scale screening programmes may influence the average time in preclinical screening detectable phase (mean sojourn time [MST]) or screening test sensitivity (STS). Setting: Screening and interval data for 395,188 women participating in the Norwegian Breast Cancer Screening Programme (NBCSP). Methods: Weighted non-linear least-square regression estimates using a tree step Markov chain model, and a sensitivity analysis of the possible impact by opportunistic screening between ordinary breast cancer screening rounds. Results: MST was estimated to 6.1 (95% confidence interval [CI] 5.1–7.0) years for women aged 50–59 years, and 7.9 (95% CI 6.0–7.9) years for those aged 60–69 years. Correspondingly, STS was estimated to 58% (95% CI 52–64 %) and 73 % (67–78 %), respectively. Simulations revealed that opportunistic screening may give a moderate estimation bias towards higher MST and lower STS. Assuming a probable 21% higher background incidence, due to increased hormone replacement therapy use, MST estimates decreased to 3.9 and 5.0 years for the two age groups, and STS increased to 75 and 85%. Conclusions: The new estimates indicate that screening detectable phase is longer than that found in previous mammography trials/programmes, but also that the sensitivity of the screening test is lower. Overall, the NBCSP detects more cancer cases than most previous trials/programmes.

Growth rates in epidemic models: Application to a model for HIV/AIDS progression

The most common quantity used to describe the growth of an epidemic when modelling infectious diseases is the basic reproduction number R0. While R0 is most appropriate for epidemics with short-lasting infections, long-lasting infections such as HIV/AIDS may call for the use of growth rates with other properties. For a group of multi-state compartment models we define both R0, the actual reproduction number Ra(t), and the intrinsic growth rate r. We study the relationship between these different reproduction numbers and growth rates and take a brief look at how they could be estimated from actual observed data. The work is illustrated by a model for HIV/AIDS progression among homosexual men in England and Wales. We conclude that other measures of growth, in addition to R0, give important supplementary information.

Modeling of dependent competing risks by first passage times of Wiener processes

Consider the competing risks situation for a component which may be subject to either a failure or a preventive maintenance action, where the latter will prevent the failure. It is then reasonable to expect a dependence between the time to failure and the time to preventive maintenance. This paper briefly reviews some modeling approaches and introduces a new approach based on modeling of the degradation of a component by means of Wiener processes, with failure corresponding to the first crossing of a certain level, and potential time for maintenance corresponding to the crossing of a certain lower degradation level.