Stein's method and approximating the quantum harmonic oscillator.

Hall et al. (2014) recently proposed that quantum theory can be understood as the continuum limit of a deterministic theory in which there is a large, but finite, number of classical "worlds." A resulting Gaussian limit theorem for particle positions in the ground state, agreeing with quantum theory, was conjectured in Hall et al. (2014) and proven by McKeague and Levin (2016) using Stein's method. In this article we show how quantum position probability densities for higher energy levels beyond the ground state may arise as distributional fixed points in a new generalization of Stein's method These are then used to obtain a rate of distributional convergence for conjectured particle positions in the first energy level above the ground state to the (two-sided) Maxwell distribution; new techniques must be developed for this setting where the usual "density approach" Stein solution (see Chatterjee and Shao (2011)) has a singularity.

A probability metric for identifying high-performing facilities: an application for pay-for-performance programs.

BACKGROUND: Two approaches are commonly used for identifying high-performing facilities on a performance measure: one, that the facility is in a top quantile (eg, quintile or quartile); and two, that a confidence interval is below (or above) the average of the measure for all facilities. This type of yes/no designation often does not do well in distinguishing high-performing from average-performing facilities. OBJECTIVE: To illustrate an alternative continuous-valued metric for profiling facilities--the probability a facility is in a top quantile--and show the implications of using this metric for profiling and pay-for-performance. METHODS: We created a composite measure of quality from fiscal year 2007 data based on 28 quality indicators from 112 Veterans Health Administration nursing homes. A Bayesian hierarchical multivariate normal-binomial model was used to estimate shrunken rates of the 28 quality indicators, which were combined into a composite measure using opportunity-based weights. Rates were estimated using Markov Chain Monte Carlo methods as implemented in WinBUGS. The probability metric was calculated from the simulation replications. RESULTS: Our probability metric allowed better discrimination of high performers than the point or interval estimate of the composite score. In a pay-for-performance program, a smaller top quantile (eg, a quintile) resulted in more resources being allocated to the highest performers, whereas a larger top quantile (eg, being above the median) distinguished less among high performers and allocated more resources to average performers. CONCLUSION: The probability metric has potential but needs to be evaluated by stakeholders in different types of delivery systems.

Person-centered care practices and quality in Department of Veterans Affairs nursing homes: is there a relationship?

OBJECTIVE: To examine variation in culture change to a person-centered care (PCC) model, and the association between culture change and a composite measure of quality in 107 Department of Veterans Affairs nursing homes. METHODS: We examined the relationship between a composite quality measure calculated from 24 quality indicators (QIs) from the Minimum Data Set (that measure unfavorable events), and PCC summary scores calculated from the 6 domains of the Artifact of Culture Change Tool, using 3 different methods of calculating the summary scores. We also use a Bayesian hierarchical model to analyze the relationship between a latent construct measuring extent of culture change and the composite quality measure. RESULTS: Using the original Artifacts scores, the highest performing facility has a 2.9 times higher score than the lowest. There is a statistically significant relationship between the composite quality measure and each of the 3 summary Artifacts scores. Depending on whether original scores, standardized scores, or optimal scores are used, a facility at the 10th percentile in terms of culture change compared with one at the 90th percentile has 8.0%, 8.9%, or 10.3% more QI events. When PCC implementation is considered as a latent construct, 18 low performance PCC facilities have, on an average, 16.3% more QI events than 13 high performance facilities. CONCLUSIONS: Our results indicate that culture change to a PCC model is associated with higher Minimum Data Set-based quality. Longitudinal data are needed to better assess whether there is a causal relationship between the extent of culture change and quality.

Bringing responsibility for small area variations in hospitalization rates back to the hospital: the propensity to hospitalize index and a test of the Roemer's Law.

OBJECTIVE: To assign responsibility for variations in small area hospitalization rates to specific hospitals and to evaluate the Roemer's Law in a way that does not artificially induce correlation between bed supply and utilization. DATA SOURCES/STUDY SETTING: We used data on hospitalizations and outpatient treatment for 15 medical conditions of nonmanaged care Part B eligible Medicare enrollees of 65 years and older in Massachusetts in 2000. STUDY DESIGN: We used a Bayesian model to estimate each hospital's pool of potential patients and the fraction of the pool hospitalized (its propensity to hospitalize, PTH). To evaluate the Roemer's Law, we calculated the correlation between hospitals' PTH and beds per potential patient. Patient severity was measured using All Patient Refined Diagnosis Related Groups. RESULTS: We show that our approach does not artificially induce a correlation between beds and utilization whereas the traditional approach does. Nevertheless, our approach indicates a strong relationship between PTH and beds (r=0.56). Eighteen (of 66) hospitals had a high PTH that differed significantly from 16 hospitals with a low PTH. Average patient severity in the high PTH hospitals was lower than in the low PTH hospitals. Although the difference was not statistically significant (P=0.12), there was a medium effect size (0.58). DISCUSSION: Variation across hospitals in the PTH index, the strong relationship between beds and the PTH, and the lack of relationship between severity and the PTH suggest the importance of policies that limit bed growth of high PTH hospitals and create incentives for high PTH hospitals to reduce hospitalizations.

Accounting for variation in technical quality and patient satisfaction: the contribution of patient, provider, team, and medical center.

BACKGROUND: The delivery of healthcare depends on individual providers, coordination within teams, and the structure of the work setting. We analyzed the amount of variation in technical quality and patient satisfaction accounted for at the patient, provider, team, and medical center level. METHODS: Data abstracted from Veterans Health Administration patient medical records for 2007 were used to calculate measures of technical quality based on adherence to best practice guidelines in 5 domains. Outpatient satisfaction was obtained from a 2007 standardized national mail survey. Hierarchical linear models that accounted for the clustering of patients within providers, providers within teams, and teams within medical centers were used to partition the variation in technical quality and satisfaction across patients and components of the system (ie, providers, teams, and medical centers). RESULTS: Providers accounted for the largest percent of system-level variance for all technical quality domains, ranging from 46.5% to 71.9%. For the single-item measure of patient satisfaction, medical centers, teams, and providers accounted for about the same percent of system-level variance (31%-34%). For the doctor/patient interaction scale providers explained 59.9% of system-level variance, more than double that of teams and medical centers. For all the measures, the residual variance (composed of patient-level and random error) explained the largest proportion of the total variance. CONCLUSIONS: Providers explained the greatest amount of system-level variation in technical quality and patient satisfaction. However, in both of these domains, differences between patients were the predominant source of nonrandom variance.

Approximate models for aggregate data when individual-level data sets are very large or unavailable.

In this article, we study a Bayesian hierarchical model for profiling health-care facilities using approximately sufficient statistics for aggregate facility-level data when the patient-level data sets are very large or unavailable. Starting with a desired patient-level model, we give several approximate models and the corresponding summary statistics necessary to implement the approximations. The key idea is to use sufficient statistics from an approximate model fitted by matching up derivatives of the models' log-likelihood functions. This derivative matching approach leads to an approximation that performs better than the commonly used approximation given in the literature. The performance of several approximation approaches is compared using data on 5 quality indicators from 32 Veterans Administration nursing homes.

Estimating a composite measure of hospital quality from the Hospital Compare database: differences when using a Bayesian hierarchical latent variable model versus denominator-based weights.

BACKGROUND: A single composite measure calculated from individual quality indicators (QIs) is a useful measure of hospital performance and can be justified conceptually even when the indicators are not highly correlated with one another. OBJECTIVE: To compare 2 basic approaches for calculating a composite measure: an extension of the most widely-used approach, which weights individual indicators based on the number of people eligible for the indicator (referred to as denominator-based weights, DBWs), and a Bayesian hierarchical latent variable model (BLVM). METHODS: Using data for 15 QIs from 3275 hospitals in the Hospital Compare database, we calculated hospital ranks using several versions of DBWs and 2 BLVMs. Estimates in 1 BLVM were driven by differences in variances of the QIs (BLVM1) and estimates in the other by differences in the signal-to-noise ratios of the QIs (BLVM2). RESULTS: There was a high correlation in ranks among all of the DBW approaches and between those approaches and BLVM1. However, a high correlation does not necessarily mean that the same hospitals were ranked in the top or bottom quality deciles. In general, large hospitals were ranked in higher quality deciles by all of the approaches, though the effect was most apparent using BLVM2. CONCLUSIONS: Both conceptually and practically, hospital-specific DBWs are a reasonable approach for calculating a composite measure. However, this approach fails to take into account differences in the reliability of estimates from hospitals of different sizes, a big advantage of the Bayesian models.

Shrinkage estimators for a composite measure of quality conceptualized as a formative construct.

OBJECTIVE: To demonstrate the value of shrinkage estimators when calculating a composite quality measure as the weighted average of a set of individual quality indicators. DATA SOURCES: Rates of 28 quality indicators (QIs) calculated from the minimum dataset from residents of 112 Veterans Health Administration nursing homes in fiscal years 2005-2008. STUDY DESIGN: We compared composite scores calculated from the 28 QIs using both observed rates and shrunken rates derived from a Bayesian multivariate normal-binomial model. PRINCIPAL FINDINGS: Shrunken-rate composite scores, because they take into account unreliability of estimates from small samples and the correlation among QIs, have more intuitive appeal than observed-rate composite scores. Facilities can be profiled based on more policy-relevant measures than point estimates of composite scores, and interval estimates can be calculated without assuming the QIs are independent. Usually, shrunken-rate composite scores in 1 year are better able to predict the observed total number of QI events or the observed-rate composite scores in the following year than the initial year observed-rate composite scores. CONCLUSION: Shrinkage estimators can be useful when a composite measure is conceptualized as a formative construct.

Do variations in disease prevalence limit the usefulness of population-based hospitalization rates for studying variations in hospital admissions?

BACKGROUND: Studies of geographic variation in hospitalizations commonly examine age- and gender-adjusted population-based hospitalization rates (ie, the numbers of persons hospitalized relative to what is expected given the age/gender distributions in the area population). OBJECTIVE: To determine whether areas identified as extreme using population-based hospitalization rates remain extreme when ranked by disease-based hospitalization rates (the numbers of persons hospitalized relative to what is expected given the amount of disease in the area). DESIGN: The authors examined 1997 Medicare data on both inpatient admissions and outpatient visits of patients 65 years and older in each of 71 small areas in Massachusetts for 15 medical conditions. For each area, the number of people having each condition was calculated as the sum of those hospitalized plus those treated as outpatients only. The authors used hierarchical Bayesian modeling to estimate area-specific population-based hospitalization rates, disease-based hospitalization rates (DHRs), and disease prevalence. MAIN OUTCOME MEASURE: The extent to which the same areas were identified as extreme based on population-based hospitalization rates versus DHRs. RESULTS: Area-specific population-based hospitalization rates, DHRs, and disease prevalence varied substantially. Areas identified as extreme using population-based hospitalization rates often were not extreme when ranked by DHRs. For 11 of the 15 conditions, 5 or more of the 14 areas ranked in top and bottom deciles by population-based hospitalization rates were more likely than not (ie, with probability > or = 0.50) to be at least 2 deciles less extreme when ranked by DHRs. CONCLUSION: Differences in disease prevalence can limit the usefulness of population-based hospitalization rates for studying variations in hospital admissions.

Comparing the importance of disease rate versus practice style variations in explaining differences in small area hospitalization rates for two respiratory conditions.

Many studies have reported large variations in age- and sex-adjusted rates of hospitalizations across small geographic areas. These variations have often been attributed to differences in medical practice style which are not reflected in differences in health care outcomes. There is, however, another potentially important source of variation that has not been examined much in the literature: geographic differences in the age-sex adjusted size of the pool of patients who present with the disease and are candidates for hospitalization. Previous studies of small area variations in hospitalization rates have only used data on hospitalizations. Thus, it has not been possible to distinguish the extent to which differences in hospitalization rates are due to (i). differences in the chance that patients diagnosed with a disease are admitted to a hospital, which we refer to as the 'practice style effect,' versus (ii). geographic differences in the total amount of diagnosed disease, which we refer to as the 'disease effect.' Elementary methods for estimating the relative strength of the two effects directly from the data can be misleading, since equal amounts of variability in each effect result in unequal impacts on hospitalization rates. In this paper we describe a model-based approach for estimating the relative importance of the practice style effect and the disease effect in explaining variations in hospitalization rates. The key to our approach is the use of data on both inpatient and outpatient visits. We use 1997 Medicare data for two respiratory medical conditions across 71 small areas in Massachusetts: chronic bronchitis and emphysema, and bacterial pneumonia. Based on a Poisson model for the process generating hospitalizations and outpatient visits, we use a Bayesian framework and Gibbs sampling to compute and compare the correlation between the number of people hospitalized and each of these two sources of variation. Our results show that for the two conditions, disease rate variation explains at least as much of the variation in hospitalization rates as does practice style variation.