Median Residual Lifetime Research Articles

Surveillance of the recurrence time of the effectiveness of national and region-level quarantines of COVID-19 pandemic in Chile.

The World Health Organization has recommended a range of social and health measures to mitigate the spread of coronavirus disease 2019 (COVID-19), including strategies such as quarantines, border closures, social distancing, and mask usage, among others. Specifically, the Chilean authorities implemented the "step-by-step" plan, built on the concept of dynamic quarantine. Numerous studies have examined the effectiveness of these quarantines in Chile during the pandemic, utilizing data published by the Chilean Ministry of Health. This study's primary aim was to enhance our understanding of quarantine effectiveness in Chile during the pandemic. We accomplished this by analyzing the distributional behavior of the time until the COVID-19 pandemic was deemed under control or not. In our study, we defined an event with two potential outcomes related to the instantaneous reproductive number (Rt), which signifies the time until a change in the event outcome occurs. Importantly, we did not predefine a specific temporal observation unit. These findings allowed us to complement the concept of effective quarantine by considering the dynamics generated by the protocols, such as phase 1 of the quarantine, in achieving natural herd immunity in response to the number of COVID-19 cases and Rt. We assessed the behavior of the mean and median residual lifetime until the initiation of controlled/uncontrolled episodes of the COVID-19 pandemic based on Rt in all regions of Chile. Despite variations in the distribution of residual times for controlled/uncontrolled episodes in different regions, there was a similar observation during the period considered (between March 2020 and March 2021): the mitigation measures did not produce a clear positive effect for controlling the epidemic. The residual times of episodes with Rr,t > 1 were not different from those episodes with Rr,t ≤ 1.

Modeling Reliability Engineering Data Using Scale-Invariant Quasi-Inverse Lindley Model

An important property that any lifetime model should satisfy is scale invariance. In this paper, a new scale-invariant quasi-inverse Lindley (QIL) model is presented and studied. Its basic properties, including moments, quantiles, skewness, kurtosis, and Lorenz curve, have been investigated. In addition, the well-known dynamic reliability measures, such as failure rate (FR), reversed failure rate (RFR), mean residual life (MRL), mean inactivity time (MIT), quantile residual life (QRL), and quantile inactivity time (QIT) are discussed. The FR function considers the decreasing or upside-down bathtub-shaped, and the MRL and median residual lifetime may have a bathtub-shaped form. The parameters of the model are estimated by applying the maximum likelihood method and the expectation-maximization (EM) algorithm. The EM algorithm is an iterative method suitable for models with a latent variable, for example, when we have mixture or competing risk models. A simulation study is then conducted to examine the consistency and efficiency of the estimators and compare them. The simulation study shows that the EM approach provides a better estimation of the parameters. Finally, the proposed model is fitted to a reliability engineering data set along with some alternatives. The Akaike information criterion (AIC), Kolmogorov-Smirnov (K-S), Cramer-von Mises (CVM), and Anderson Darling (AD) statistics are used to compare the considered models.

Comparison of potential long-term costs for preventive dentistry treatment of post-orthodontic labial versus lingual enamel cavitations and esthetically relevant white-spot lesions: a simulation study with different scenarios

BackgroundPost-orthodontic white-spot lesions (WSL) in esthetically relevant incisor and canine areas impair dentofacial esthetics, and preventive dentistry treatment is definitely required in case of enamel cavitations. The incidence of lingual post-orthodontic WSL and cavitation following lingual MB treatment has been reported to be distinctively decreased compared to labial MB treatment. Moreover, lingual WSL do not impair dentofacial esthetics. It was the objective of this study to calculate consequential costs of preventive dental care necessary to recover labial or lingual post-orthodontic cavitations as well as esthetically relevant WSL following either labial or lingual MB interventions.MethodsMB treatments (labial / lingual) were simulated in 1,000,000 patients between the ages of 12-18Y, with a median residual life time expectancy of 58Y based on local mortality tables. Range of MB Tx duration was 9–45 mo. Frequencies of post-orthodontic (labial / lingual) enamel damages were derived from large-scale WSL incidence studies. Anterior composite survival rates were based on a systematic review on the subject. Within the context of the German dental fee system (GOZ 2.3 and 3.5 fee increments), simulation of costs for enamel damage treatment and re-treatment (maximum: 5x) were based on single-surface composite restorations for lingual or labial cavitations and labial WSL treatment; and lingual WSL fluoridation.ResultsOverall mean total costs for Tx and re-Tx of both WSLs and cavitations may sum up to 1718.91 Eur in the high-cost (GOZ 3.5) scenario for conventional MB cases, versus 19.94 Eur for lingually treated cases, given that renewal of simulated single-surface restorations takes place at 15-year intervals. When focussing on patients diagnosed with least of one WSL, and/or cavitation, these mean costs increase up to 2332.35 Eur for conventionally treated MB patients, or 65.03 Eur for lingual MB patients.ConclusionCosts for repeated treatment of post-orthodontic enamel damages produced by conventional vestibular fixed appliances may easily exceed the initially higher costs associated with lingual orthodontic treatment. Judged economically in the long term, lingual MB Tx may be considered as a more cost-effective solution for a correction of malocclusion.

Effect of tuberculosis on immune restoration among HIV-infected patients receiving antiretroviral therapy

ABSTRACTIn this article, time to immune recovery during antiretroviral therapy was estimated and compared between HIV-infected children with and without tuberculosis (TB). CD4 T-cell restoration was used as a criterion for determining immune recovery. The median residual lifetime function, which is more intuitive and robust compared to the frequently used measures of lifetime data, was used to estimate time to CD4 T-cell restoration. The median residual lifetime is not influenced by extreme observations and heavy-tailed distributions which are commonly encountered in clinical studies. Permutation-based methods were used to compare the CD4 T-cell restoration times between the two groups of patients. Our results indicate that children with TB had uniformly higher median residual lifetimes to immune recovery compared to those without TB. Although TB was associated with slower CD4 T-cell restoration, the differences between the restoration times of the two groups were not statistically significant.

Estimation of the simultaneous confidence regions for the ratios of quantile residual lifetimes

ABSTRACTIn many clinical studies, especially when the associated diseases can be incurable, it is critical to evaluate the effect of treatments on mean residual lifetime or quantile residual lifetime. In this article, for right-censored survival data, we concern the construction of simultaneous confidence regions for the ratios of quantile residual lifetimes between treatment and control groups. We propose two approaches: the first is based on the score-type test of Jeong et al. [Nonparametric inference on median residual life function. Biometrics. 2008;64:157–163], while the second is based on the empirical likelihood ratio test of Zhou and Jeong [Empirical likelihood ratio test for median and mean residual lifetime. Statist. Med. 2011;30:152–159]. The performance of the associated coverage probability and the expected coverage area are investigated via a simulation study. The proposed method is illustrated using a real data set.

Comparison of Several Confidence Intervals for Median Residual Lifetime with Left-truncated and Right-censored Data

For left-truncated and right-censored data, the technique proposed by Brookmeyer and Crowley (1982) is extended to construct a point-wise confidence interval for median residual lifetime. This procedure is computationally simpler than the score type confidence interval in Jeong et al. (2008) and empirical likelihood ratio confidence interval in Zhou and Jeong (2011). Further, transformations of the estimator are applied to improve the approximation to the asymptotic distribution for small sample sizes. A simulation study is conducted to investigate the accuracy of these confidence intervals and the implementation of these confidence intervals to two real datasets is illustrated.

Confidence intervals for the ratio of two median residual lifetimes with left-truncated and right-censored data.

The confidence intervals for the ratio of two median residual lifetimes are developed for left-truncated and right-censored data. The approach of Su and Wei (1993) is first extended by replacing the Kaplan-Meier survival estimator with the estimator of the conditional survival function (Lynden-Bell, 1971). This procedure does not involve a nonparametric estimation of the probability density function of the failure time. However, the Su and Wei type confidence intervals are very conservative even for larger sample size. Therefore, this article proposes an alternative confidence interval for the ratio of two median residual lifetimes, which is not only without nonparametric estimation of the density function of failure times but is also computationally simpler than the Su and Wei type confidence interval. A simulation study is conducted to examine the accuracy of these confidence intervals and the implementation of these confidence intervals to two real data sets is illustrated.

Inference for the median residual life function in sequential multiple assignment randomized trials

In survival analysis, median residual lifetime is often used as a summary measure to assess treatment effectiveness; it is not clear, however, how such a quantity could be estimated for a given dynamic treatment regimen using data from sequential randomized clinical trials. We propose a method to estimate a dynamic treatment regimen-specific median residual life (MERL) function from sequential multiple assignment randomized trials. We present the MERL estimator, which is based on inverse probability weighting, as well as, two variance estimates for the MERL estimator. One variance estimate follows from Lunceford, Davidian and Tsiatis' 2002 survival function-based variance estimate and the other uses the sandwich estimator. The MERL estimator is evaluated, and its two variance estimates are compared through simulation studies, showing that the estimator and both variance estimates produce approximately unbiased results in large samples. To demonstrate our methods, the estimator has been applied to data from a sequentially randomized leukemia clinical trial.

Censored quantile regression for residual lifetimes

We propose a regression method that studies covariate effects on the conditional quantiles of residual lifetimes at a certain followup time point. This can be particularly useful in cancer studies, where more patients survive cancers initially and a patient's residual life expectancy is used to compare the efficacy of secondary or adjuvant therapies. The new method provides a consistent estimator that often exhibits smaller standard error in real and simulated examples, compared to the existing method of Jung et al. (2009). It also provides a simple empirical likelihood inference method that does not require estimating the covariance matrix of the estimator or resampling. We apply the new method to a breast cancer study (NSABP Protocol B-04, Fisher et al. (2002)) and estimate median residual lifetimes at various followup time points, adjusting for important prognostic factors.

Longevity of soccer players: an investigation of all German internationals from 1908 to 2006

Leisure-time physical activity is associated with better health and a reduced risk of all-cause mortality. It is unclear if this association is also present with a high level of physical activity as it is found in professional athletes. In a population-based retrospective cohort study, we compared the survival experience of all soccer players participating for Germany in international matches between 1908 and 2006 to that of the general population. To summarize survival experience, we calculated cumulative relative survival ratios (RSRs) from a life table. We included data of 812 international players, of which 428 (=52.7%) died during follow-up. In all 13 intervals, cumulative observed survival was smaller than cumulative expected survival, resulting in cumulative RSRs being <1. The cumulative RSRs are statistically significantly different from 1 in all but the last interval. This impaired survival experience of the internationals translates into a loss of median residual lifetime of 1.9 years [95% confidence interval: 0.6, 3.2] years at the entry time into the cohort. This loss is mainly driven by the mortality of internationals from the earlier half of the observation period. Reasons for this might be poorer medical care in former times, internationals being killed in action during World War II, and a changing distribution of causes of death during the 20th century.

Empirical likelihood ratio test for median and mean residual lifetime

Prompted by several recent papers on the inference on median and mean residual life time, we note that the testing involving the mean or median residual life function in censored survival data can be obtained by an easy application of the general empirical likelihood ratio test. This approach has several advantages: (1) There is no need to estimate the variance/covariance at all, which may become prohibitively complicated for other procedures that require the estimation of such. (2) When inverting the tests to obtain confidence regions/intervals, this procedure inherits all the good properties of a likelihood ratio test. (3) Free software implementation of the test is readily available.

Life expectancy of beagles

Life expectancy of beagles was calculated from 303 beagles (162 female and 141 male) with known birth and death dates. Most of the dogs were either spayed or castrated during the first year of their life. Both natural death and euthanasia due to terminal illness were considered as a death event. These beagles received regular veterinary medical care during their life.Gender had no effect on median survival of beagles (male = 12.82 years; female = 12.87 years; p = 0.610). Therefore, combined data of both male and female beagles were used for the calculation of the life expectancy. The age specific median residual lifetime (age in years/median residual lifetime in years) was 0.5/12.8, 1.5/12.1, 2.5/11.1, 3.5/10.1, 4.5/9.1, 5.5/8.2, 6.5/7.2, 7.5/6.3, 8.5/5.4, 9.5/4.5, 10.5/3.7, 11.5/2.8, 12.5/2.1, 13.5/1.6, 14.5/1.2, and 15.5/0.6. Mortality increased dramatically in beagles after 9 years of age (95% CI: 8.7, 10.0 years). The results of this study suggest that earlier diagnostic or therapeutic intervention is warranted in beagles beginning at 9 years of age. This study was supported by Hill's Pet Nutrition, Inc.

Life expectancy of American Domestic Shorthair cats

Life expectancy of cats was calculated from 196 cats (115 female and 81 male) with known birth and death dates. Most of the cats were neutered during the first year of their life. Both natural death and euthanasia due to terminal illness were considered as a death event. All cats received regular veterinary medical care during their life.Gender had no effect on median survival of cats (male = 12.6 years; female = 13.1 years; p = 0.490). Therefore, combined data of male and female cats were used for the calculation of the life expectancy. The maximum survival time was 17.95 years in this group of cats. The age specific median residual lifetime (age in years/median residual lifetime in years) was 0.5/12.9, 1.5/12.3, 2.5/11.4, 3.5/10.5, 4.5/9.5, 5.5/8.5, 6.5/7.6, 7.5/6.7, 8.5/5.8, 9.5/5.0, 10.5/4.3, 11.5/3.6, 12.5/3.0, 13.5/2.3, 14.5/1.6, 15.5/0.9, 16.5/0.8, and 17.5/0.56. Mortality increased dramatically in cats after 9.7 years of age (95% CI: 8.8, 10.6 years). The results of this study suggest that earlier diagnostic or therapeutic intervention is warranted in cats beginning at 10 years of age. This study was supported by Hill's Pet Nutrition, Inc.

Regression on Quantile Residual Life

A time-specific log-linear regression method on quantile residual lifetime is proposed. Under the proposed regression model, any quantile of a time-to-event distribution among survivors beyond a certain time point is associated with selected covariates under right censoring. Consistency and asymptotic normality of the regression estimator are established. An asymptotic test statistic is proposed to evaluate the covariate effects on the quantile residual lifetimes at a specific time point. Evaluation of the test statistic does not require estimation of the variance-covariance matrix of the regression estimators, which involves the probability density function of the survival distribution with censoring. Simulation studies are performed to assess finite sample properties of the regression parameter estimator and test statistic. The new regression method is applied to a breast cancer data set with long-term follow-up to estimate the patients' median residual lifetimes, adjusting for important prognostic factors.

Cumulative damage model based on two-mode fatigue damage bounds

Based on earlier work by the author, a macro-model of damage accumulation is developed that circumvents crack tip micro-mechanics through introduction of two-mode damage curves. In such model, the actual microcrack configuration is replaced by an ‘equivalent’ crack while the actual heterogeneous medium is replaced by an effective homogeneous and isotropic medium in which the crack is embedded. By the simultaneous presence of slip and opening-mode damage during each cycle, the two-mode damage curves reflect conditions at a crack tip in the effective medium. Such curves are obtained from conditions imposed at four discrete microstructure barriers. With all barriers established in terms of the fatigue limit. The fatigue process is thus divided into three stages, the analogues of the three creep stages under constant stress. Two length scales are defined from order of magnitude considerations: a macroscale defining a representative volume element (RVE) and a microscale defining a much smaller volume of grain size order; both volumes surrounding the ‘equivalent’ crack tip. The presence of a sufficiently large number of grains of random size and orientation makes the RVE representative of the medium as a whole; in such a medium, the location and orientation of the ‘equivalent’ crack becomes immaterial. While the RVE is dominated by monotonic slip, the microvolume represents a region of cyclic slip; both volumes surround the crack tip, in line with fracture mechanics concepts. This model is then applied to cumulative damage and bounds on mean residual lifetime are obtained for two-stage high–low (H–L) and low–high (L–H) cycling. The presence of microstructure barriers proves essential to damage irreversibility in L–H cycling, leading to a division of the entire region behind the S-N curve into three regions dominated by part or by all three fatigue stages. A statistical analysis based on the two-mode bounds leads to a median residual lifetime and to bounds on the variance. It is argued that the variance can only be bracketed by the present model but not uniquely defined, unless ‘instantaneous’ cyclic hardening is introduced as an additional random variable. Bounds on crack growth rate under constant amplitude cycling are also derived that effectively coincide in the macrocrack regime, thus explaining the small crack rate scatter observed. An experimental method is proposed whereby long fatigue lifetimes, including the fatigue limit, can be indirectly determined from a crack growth rate drop that occurs at the transition from a predominantly slip to a predominantly opening-mode interval of crack growth.

Nonparametric Inference on Median Residual Life Function

A simple approach to the estimation of the median residual lifetime is proposed for a single group by inverting a function of the Kaplan-Meier estimators. A test statistic is proposed to compare two median residual lifetimes at any fixed time point. The test statistic does not involve estimation of the underlying probability density function of failure times under censoring. Extensive simulation studies are performed to validate the proposed test statistic in terms of type I error probabilities and powers at various time points. One of the oldest data sets from the National Surgical Adjuvant Breast and Bowel Project (NSABP), which has more than a quarter century of follow-up, is used to illustrate the method. The analysis results indicate that, without systematic post-operative therapy, a significant difference in median residual lifetimes between node-negative and node-positive breast cancer patients persists for about 10 years after surgery. The new estimates of the median residual lifetime could serve as a baseline for physicians to explain any incremental effects of post-operative treatments in terms of delaying breast cancer recurrence or prolonging remaining lifetimes of breast cancer patients.

On the median residual lifetime and its aging properties: A characterization theorem and applications

AbstractThis paper is devoted to study several aspects of the median residual life function (MERLF). In reliability studies, it is well known that, although the MERLF have several advantages over the mean residual life function (MRLF), the MRLF has the good property of uniquely determine a life distribution whereas either the median residual life function (MERLF) or an α‐percentile residual life do not have such good property. We shall give a characterization result where knowledge of both the MERLF and the survival function on an interval does uniquely determine the distribution. Moreover, in order to apply this characterization in practical situations, we propose a method to estimate the necessary information of the survival function. Relationships between analytical properties of the survival function and its associated MERLF are also obtained. Bryson and Siddiqui [J Am Statist Assoc 64 (1969), 1472–1483] proved relationships among seven criteria for aging, out of which two contained the MRLF (decreasing MRLF and net decreasing MRLF). In this paper, we prove that the same pattern of relationships holds if the MRLF is replaced by the MERLF. We also examine the aging criteria corresponding to an increasing MERLF and show that there is no relation between the behavior (increasing or decreasing) of the MERLF and of the MRLF. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005

Bootstrapped confidence bands for percentile lifetime

We propose bootstrapped confidence bands for the percentile lifetime function. Our method is based on a joint approximation of the empirical and quantile processes and their bootstrapped counterparts. Modest simulations support the results. Confidence bands are applied to quantile and median residual lifetimes of tractor rear brakes.

Technical Note—When Does the βth Percentile Residual Life Function Determine the Distribution?

Knowledge of the βth percentile residual life function does not uniquely determine the underlying distribution. Additional information about the tail behavior of the distribution is needed. Knowledge of both the β1 and β2 percentile residual life functions does uniquely determine the distribution provided that β1 and β2 are noncommensurate. This corrects and extends the results of Schmittlein and Morrison (Schmittlein, D. C., D. G. Morrison. 1981. The median residual lifetime; a characterization theorem and an application. Opns. Res. 29 392–399.).

The Median Residual Lifetime: A Characterization Theorem and an Application

The concept of the median residual lifetime for a probability distribution is outlined and shown to have several advantages over its more common counterpart, the mean residual lifetime. A characterization theorem shows that a Pareto distribution of the second kind (or equivalently, a gamma mixture of exponentials) is the unique distribution with a linearly increasing median residual lifetime. An empirical example from the literature on strike durations is presented.