Lyn R. Whitaker

Estimating the Reliability of Systems Subject to Imperfect Repair

Abstract This study of statistical inference for repairable systems focuses on the development of estimation procedures for the life distribution F of a new system based on data on system lifetimes between consecutive repairs. The Brown—Proschan imperfect-repair model postulates that at failure the system is repaired to a condition as good as new with probability p, and is otherwise repaired to the condition just prior to failure. In treating issues of statistical inference for this model, the article first points out the lack of identifiability of the pair (p, F) as an index of the distribution of interfailure times T 1, T 2, …. It is then shown that data pairs (Ti, Zi ) (i = 1, 2, …) render the parameter pair (p, F) identifiable, where Zi is a Bernoulli variable that records the mode of repair (perfect or imperfect) following the ith failure. Under the assumption that data of the form {(Ti, Zi )} are drawn via inverse sampling until the occurrence of the mth perfect repair, the problem of estimating the...

Sizing the US destroyer fleet

Abstract For the US Navy to be successful, it must make good investments in combatant ships. Historically a vital component in these decisions is expert opinion. This paper illustrates that the use of quantitative methods in conjunction with expert opinion can add considerable insight. We use the analytic hierarchy process (AHP) to gather expert opinions. Then, distributions are derived based on these expert opinions, and integrated into a mixed integer programming model to derive a distribution for the “effectiveness” of a fleet with a particular mix of ships. These ideas are applied to the planning scenario for the 2015 conflict on the Korean Peninsula, one of the two key scenarios that the Department of Defense uses for planning.

Min and max scorings for two-sample ordinal data

Abstract : Ordinal response variables often occur in practice. For example, in clinical trials a subjects response to a drug regime might be categorized as negative, none, fair, or good. There are several common approaches to analyzing two-sample ordinal response data. These procedures applied to the same data can lead to contradictory conclusions. In an attempt to reconcile contradictory results and provide guidance to the practitioner, Kimledorf, Sampson and Whitaker (1992) propose an alternative approach. They find the scores which when assigned to the levels of the ordinal response variable maximize a two-sample test statistic and the scores that minimize that same statistic. Since many of the two-sample statistics are related by monotonic transformations, these extreme scores are in fact extreme scores for several test statistics. Both minimized and maximized test statistics falling into the rejection region clearly indicate a difference between the two populations or treatments. On the other hand if neither of the two extreme statistics fall in the rejection region then no matter what scores are used there will be no significant difference in the two populations. In this paper we review the KsW procedure and its implementation in SAS software.

Estimation of Multivariate Distributions under Stochastic Ordering

Abstract Let F and G be the cdfs of two p-dimensional multivariate distributions, such that F is stochastically larger than G. A straightforward derivation is given of the generalized maximum likelihood estimators of F and G, based on random samples from each population. An algorithmic approach to computing these estimators is described and motivating numerical examples are discussed. The special case when F and G correspond to multivariate ordinal contingency tables is also presented. The relationship of these results to those of Robertson and Wright (1974) is considered.

Estimating Distributions with Increasing Failure Rate in an Imperfect Repair Model

A failed system is repaired minimally if after failure, it is restored to the working condition of an identical system of the same age. We extend the nonparametric maximum likelihood estimator(MLE) of a systems lifetime distribution function to test units that are known to have an increasing failure rate. Such items comprise a significant portion of working components in industry. The order-restricted MLE is shown to be consistent. Similar results hold for the Brown-Proschan imperfect repair model, which dictates that a failed component is repaired perfectly with some unknown probability, and is otherwise repaired minimally. The estimators derived are motivated and illustrated by failure data in the nuclear industry. Failure times for groups of emergency diesel generators and motor-driven pumps are analyzed using the order-restricted methods. The order-restricted estimators are consistent and show distinct differences from the ordinary MLEs. Simulation results suggest significant improvement in reliability estimation is available in many cases when component failure data exhibit the IFR property.

Case study in modeling and simulation validation methodology

The military develops simulations to analyze nearly every aspect of defense. How accurate are these simulations and to what extent do they produce dependable results? Most guidance available to DoD analysts provides broad recommendations geared towards management and coordination of the validation processes. Here, we focus on practical validation from the analysts perspective in the form of a case study. The platform used is the theater missile defense (TMD) aspects of Extended Air Defense Simulation (EADSIM) and a new simulation called Wargame 2000. The focus is not to validate Wargame 2000 but to develop real, usable tools for analysis. Measures of effectiveness include defense battery search, engagement and intercept times against threat missiles. Insight is provided into developmental and data production issues making the validation process more effective and meaningful.

Suicide rates and methods in active duty military personnel, 2005 to 2011: a cohort study

Suicide rates have increased by 60% worldwide during the past 47 years and is a leading cause of death among 15- to 44-year-olds (1). In 2010, suicide was the 10th leading cause of death in the United States (2). The overall suicide rate in the U.S. military has also increased, almost doubling from 2001 to 2011 (3). Potential factors that have changed over time, such as deployments and mental health conditions, have helped clarify the reasons for the increased suicide rate in the U.S. military. A recent study of suicide risk among veterans found that deployment did not increase the risk for suicide (4), whereas other studies explored risk for suicide after psychiatric hospitalization (5) as well as psychosocial risk (6). Research and debate are ongoing regarding the various motivations for choosing a particular method of suicide (7, 8). Previous studies showed that men are more likely to use violent methods of suicide (for example, firearm related), whereas women are more likely to use nonviolent means (for example, poisoning) (8). Within the military, research suggests that the suicide risk is significantly greater among personnel whose occupations provide easy access to firearms than among those in other occupations (9). Some researchers have suggested that both psychological and biological differences exist between people who choose violent methods and those who use nonviolent ones (10, 11). Empirical evidence suggests that among military conscripts, previous problems in school may predict violent suicide attempts, which also may be a strong indicator of subsequent suicide (12). Indeed, violent suicide attempts have been linked very strongly to subsequent suicide completion (12). In fact, a person who attempts suicide by firearm has a risk for subsequent completed suicide about 5 times higher than that of people who attempt suicide by nonviolent means (hazard ratio [HR], 5.18 [95% CI, 1.27 to 21.24]) (12). A common limitation in previous analyses evaluating suicide trends in the U.S. military was a lack of consolidated service data. Before 2008, risk factor analyses for U.S. Department of Defense (DOD) suicides were performed in relatively small populations, primarily at the military service level, by using service-unique databases (6). Moreover, the bulk of research has focused on the army (4, 5, 1318), whereas studies including all services have been limited to survey-type data or have had limited follow-up (1618). The objective of the present study is to evaluate suicide rates among active duty military personnel across years and to identify differences among branches. Further, with regard to completed suicides, we aim to identify the groups of active duty military personnel who are at greatest risk for firearm-specific suicide. Methods A joint endeavor by the Defense Suicide Prevention Office and the U.S. Department of Veterans Affairs resulted in development of the Suicide Data Repository (SDR), which has mitigated the problem of insufficient consolidated data. The SDR combines data from the Centers for Disease Control and Prevention (CDC) via the National Death Index (NDI), as well as the Military Mortality Database, to provide a collection of demographic and military-specific information on all service members and veterans who committed suicide and had served in the armed forces since 1974. The SDR was fully established in 2013 and to our knowledge is the most comprehensive source of demographic and military-specific data on suicides in the U.S. military. The data for this study were provided by the Defense Manpower Data Center (DMDC). Study Population We used 2 data sets: The first, extracted from the SDR, contains demographic and military-specific data for each suicide; the second contains the monthly end strength, or personnel count on the last day of the month, for each demographic subpopulation. All subpopulation strata were combinations of the following: year, sex, age, race, marital status, education, age at enlistment, rank, and Armed Forces Qualification Test (AFQT) category (higher AFQT categories represent lower cognitive ability; for example, category IIIB or higher is equal to a percentile score <50). Data on active duty personnel were not fully available for suicides occurring before 2005 or for those occurring outside the United States. Therefore, our study population comprises all enlisted personnel (that is, nonofficers) of the U.S. military regular component (including the army, air force, navy, and marines) who committed suicide while on active duty stateside between 2005 and 2011. For the analysis of firearm-specific suicides, we attempted to exclude suicides among enlistees who did not have military service exposure and perhaps had an unrecognized predisposition to suicidality before entering the military; therefore, we included only those who had already completed training. The DOD Human Research Protection Program and Naval Postgraduate Schools Institutional Review Board approved the collection of the data for this study (NPS.2014.0073). Statistical Analysis Temporal Trends of Active Duty Military Suicide Using the combined data set containing both suicides and personnel counts, we determined the branch-specific suicide rates. Service branches have different missions and recruit and attract different types of personnel; because we could not control for these differences with our available data, we analyzed data for each branch separately. Predictors of Violent Methods Among Active Duty Military Suicides Using the cohort of completed suicides in the SDR data set, we identified predictors of firearm-specific suicide. Predictor variables included factors previously identified in the literature and a priori hypothesized to affect both the service and the outcome. To evaluate the total effect of each military branch on firearm-specific suicide, we considered several covariates, including age at death, rank, sex, education, race, marital status, religion, length of service at the time of death, AFQT score category, and primary military occupation (that is, infantry/special operations). We identified covariates to adjust for in multivariable models using a directed acyclic graph approach (representing the relationships among service, suicide, and other variables) to determine minimally sufficient adjustment sets (19) (Appendix Figure). The minimally sufficient adjustment set identified included infantry/special operations job classification, age, sex, AFQT score category, and education. We restricted the multivariable models identifying predictors of firearm-specific suicide to men, because more than 95% of all suicides were committed by men (only 1 female marine and 9 female navy personnel committed suicide). Among the navy and air force suicides, we did not consider infantry/special operations job classification in multivariable models because only 2 suicides were identified and this job classification is found more commonly in the army and marines. Appendix Figure. Directed acyclic graph evaluating the relationship between branch of service and firearm-specific suicide and potential confounders. AFQT = Armed Forces Qualification Test; ops = operations. Of 1416 suicides, 366 (25.8%) had missing data for firearm-specific suicide or covariates (Appendix Table 1). We used multiple imputation to address missing data, and we assumed data were missing at random. The variables included in the imputation models included method of suicide, AFQT score category, education, infantry/special operations job classification, sex, and age; imputations were run separately by branch. We specified conditional models and performed the imputations based on these conditional models. We generated multiple sets (m= 10) of imputed values, allowing us to account for the uncertainty inherent in using the imputed values in our models (20, 21). The imputed data sets were combined by applying Rubin rules (22, 23), which are used to appropriately adjust estimated SEs, and thus CIs and P values, to account for the additional uncertainty associated with data missingness. Because the mechanism of these missing data is unknown and may not be consistent with the missing-at-random assumption, these results should be interpreted cautiously. In a secondary analysis, we compared multiple imputation results with a complete case analysis approach, excluding observations with missing data. Adjusted odds ratios (aORs) are reported with 95% CI. Data were analyzed using R version 3.1.2 (R Foundation for Statistical Computing) (24). The base package was used for the logistic models, and the mice (Multiple Imputation by Chained Equations) package (25) was used for the multiple imputation analyses. Appendix Table 1. Selected Variables and Percentage With a Missing Cause of Suicide Role of the Funding Source This study was unfunded. Results A total of 1455 suicides occurred during 125 million person-months among the active duty, regular component enlisted personnel in the U.S. Army, Air Force, Marine Corps, and Navy from 2005 to 2011, with an average end strength for that period of 451000, 268000, 173000, and 283000, respectively. The highest suicide rates (per 100000) from 2005 to 2011 were in the army in 2009 and 2010 (29.44 and 29.15 suicides, respectively) (Figure and Table 1), whereas the lowest suicide rates were in the air force and navy in 2005 (9.95 and 9.79 suicides, respectively). From 2006 to 2011, the rates were higher among army personnel (19.13 to 29.44 cases per 100000) than among members of any other branch. Figure. Suicide rates per 100000 persons (2005 to 2011), by branch of service. Table 1. Suicide Rates per 100000 Persons for Active Duty, Regular Component Enlisted Personnel* Characteristics Of these suicides, 1416 occurred among nontrainees, comprising our suicide cohort (Table 2). Most (52.5%) were among army personnel, and approximately 95% occurred in men. The median age was 25 years (interquartile range

Order restricted estimators: some bias results

Stimulated by practical concerns in the use of order restricted estimators, we study some bias issues of isotonic estimators of ordered parameters. We obtain explicit expressions for biases of order restricted estimators of ordered means of two normal populations with common variance. Based on intuitive considerations, we also propose a new estimator of the common variance of two normal populations having ordered normal means and show that it has smaller bias than the maximum likelihood estimator of the variance and has smaller mean squared error than the usual unbiased estimator of the common variance.

Cost rate heuristics for semi-Markov decision processes

In response to the computational complexity of the dynamic programming/ backwards induction approach to the development of optimal policies for semiMarkov decision processes, we propose a class of heuristics resulting from an inductive process which proceeds forwards in time. These heuristics always choose actions in such a way as to minimize some measure of the current cost rate. We describe a procedure for calculating such cost rate heuristics. The quality of the performance of such policies is related to the speed of evolution (in a cost sense) of the process. A simple model of preventive maintenance is described in detail. Cost rate heuristics for this problem are calculated and assessed computationally. DYNAMIC PROGRAMMING; REPLACEMENT POLICY AMS 1991 SUBJECT CLASSIFICATION: PRIMARY 90C39

A sequential scheme to estimate the optimal age replacement policy

Consider a system which is subject to failure and must be replaced when this occurs. If it costs less to replace the system in advance before failure, it may be advantageous to use an age replacement policy. However, the optimal age to replace the system is unknown if the underlying failure distribution is unknown. This paper develops a scheme to update the current estimate of the optimal age replacement policy in an on-line fashion and simultaneously controlling costs by reducing system failures